\section{Introduction}
For $n\in \mathbb N$ and $N = 2^n$, let $V_N \subseteq \mathbb Z^2$ be a box of side
length $N$ whose lower left corner is located at the origin. Let $\mathbb{BD}_j$ denote the
collection of boxes of the form $([0, 2^{j}-1] \cap \mathbb{Z})^2 +
(i_1 2^j, i_2 2^j)$ where $i_1, i_2$ and $j$ are nonnegative integers. For $v\in \mathbb{Z}^2$,
let $\mathbb{BD}_j(v)$ be the (unique) box in $\mathbb{BD}_j$ that contains $v$. Let
$\{a_{B}\}_{k\geq 0, B\in \mathbb{BD}_k}$ be a collection of i.i.d.\ standard Gaussian variables.
We define a branching random walk  (indexed over $V_N$) with Gaussian increments by
\begin{equation}\label{eq-def-BRW}
     \mathcal{R}_z^N = \sum\limits_{k = 0}^{n-1} \sum_{B \in \mathbb{B}\mathbb{D}_k(z)} a_{B}.
\end{equation}
For any rectangle $V\subseteq \mathbb Z^2$, we denote by $\Pi_{LR}^V$ the collection of all paths
$\pi$ in $V$ (considered as a subgraph of $\mathbb{Z}^2$) that connects the left boundary of $V$
to its right boundary. We refer to such a path as a (left-right) \emph{crossing} of $V$. For a
fixed $\gamma > 0$ we define
$$D_{\gamma, LR}(V_N) = \min_{\pi \in \Pi_{LR}^{V_N}}\sum\limits_{z \in \pi} \mathrm{e}^{\gamma \mathcal{R}_z^N},$$
to be the first passage percolation distance between the two boundaries of $V_N$ where we assign
each vertex $v$ a weight of $e^{\gamma \mathcal R_v^N}$.
\begin{theorem}\label{thm-main}
For a fixed $\gamma_0$ that is small, and all $\gamma < \gamma_0$, we have that
$$\mathbb E D_{\gamma, LR}(V_N) = O( N^{1 - \gamma^2/10})\,.$$
\end{theorem}

\begin{remark}
Our proof in fact gives an upper bound on the expected weight for the geodesic crossing through a rectangle. As a result, one can show that the expected weight for the geodesic connecting two fixed vertices has exponent strictly less than 1, by constructing a sequence of ($\log N$ many) rectangles with geometrically growing size that connect these two vertices.
\end{remark}

\subsection{A brief discussion on motivation and background}

In a broad context, our work is motivated by studying first passage percolation on random media with heavy correlation. The main contribution of our work, is to demonstrate an instance that in a hierarchical random field the exponent of the FPP distance can be strictly less than 1 (despite the fact that the a straight line has weight with exponent strictly larger than 1, and that the majority of the vertices are of values $O(\sqrt{\log N})$). This is, of course, rather different from the classical FPP where the edge/vertex weights are independent and identically distributed. We refer to \cite{ADH15, GK12} and references therein for a review of classical first passage percolation.

In the specific instance of branching random walk, our work is closely related to the first passage percolation problem when the vertex weight is given by exponentiating a two-dimensional discrete Gaussian free
field (GFF). Inspired by the current work, a similar upper bound on the FPP distance in this case has been recently obtained in
\cite{DG16}. The FPP on two-dimensional discrete GFF, as we expect, is of fundamental importance in understanding the random metric associated with the Liouville quantum gravity (LQG) \cite{P81, DS11, RV14}. We remark that the random metric of LQG is a major open problem, even ``just'' to make rigorous sense of it. In a recent series of works of Miller and Sheffield, much understanding has been obtained (more on the continuum set up) in the special case of $\gamma = \sqrt{8/3}$;
see \cite{MS15, MS16} and references therein. Another recent work \cite{GHS16} has provided some bounds on the scaling exponent for a type of LQG metric, though we emphasize that their definition is of very different flavor from the FPP perspective considered in our paper (in particular, no mathematical connection can be drawn between these works at the moment). We refrain ourselves from an extensive discussion on background of LQG in this article. As a final remark to the connection of LQG metric, we note that there are other candidate discrete metric (associated with LQG or GFF) that have been proposed, and it is possible that these metrics with suitable normalization would give a more desirable
scaling limit than that of FPP. However, we feel that in the level of precision of the present article and that of \cite{DG16}, it is quite possible that the fundamental mathematical structures (and thus obstacles) are common for these related notions of metrics.

BRW is perhaps the simplest construction that approximates a log-correlated Gaussian field, of which
the GFF is a special instance. A number of other properties, especially those regarding to the extreme values of the field, have been proved to exhibit universal behavior among log-correlated Gaussian fields
(c.f., \cite{Madaule, DRZ15}). Indeed, it is typical that properties were first proved for BRW, and then later for GFF; and it is also typical that the proof in the case of GFF were substantially enlightened by the understanding on BRW.
However, caution is required when drawing heuristic conclusion on GFF based on BRW for FPP: as demonstrated in \cite{DZ15}, there exists a family of log-correlated Gaussian fields whose exponent for FPP can be made arbitrarily close to 1 (as $K$ grows to $\infty$) if one is allowed to perturb the covariance entry-wise by an additive amount that is at most $K$ (and thus the scaling exponent for FPP is non-universal among log-correlated fields). That being said, we remark that while the mathematical details in \cite{DG16} are substantially more involved and delicate, these two papers do share the same very basic multi-scale analysis proof framework.

\subsection{Main ideas of our approach}

Our approach is inspired by the rescaling argument employed in the proof of the connectivity of fractal percolation process \cite{CCD88, DM90, Chayes95}, the idea of which went back to \cite{ACCFR}. For the purpose of induction, we will in fact work with crossings through a rectangle rather than a square. In what follows, we should consider $\gamma$ as a small positive number (but fixed as $N \to \infty$).

We wish to carry out a inductive construction and proof for light path crossing the rectangle: we will take advantage of the hierarchical structure of BRW and will construct the BRW along the way as we construct light path in different scales. In order to properly
describe our procedure, we need a number of definitions. Let $\Gamma = \Gamma(\gamma) \in [\alpha/\gamma^2, \alpha/\gamma^2 + 2]$ be an odd, positive integer for
some $\alpha$ to be selected, and let $V_N^{\Gamma} = ([0, \Gamma N - 1] \times [0, N - 1])
\cap \mathbb{Z}^2$. The reason for choosing $\Gamma$ odd is mere technical convenience which will
become apparent later. We will also work under the assumption that $\gamma\ll 1/\alpha \ll 1$ (small or large enough for our bounds or inequalities to hold although we keep this
implicit in our discussions). We define a Gaussian process $\{\mathcal{R}_z^N: z \in V_{N}^{\Gamma}\}$ on $V_N^{\Gamma}$ as follows:
\begin{itemize}
\item $\{\mathcal{R}_z^N: z \in V_{N;j}^\Gamma\}$ is a BRW on $V_{N;j}^\Gamma = ([(j-1)N, jN - 1] \times [0, N-1]) \cap \mathbb{Z}^2$ where $j \in [\Gamma]$.
\item $\{\mathcal{R}_z^N: z \in V_{N;j}^\Gamma\}$'s are independent of each other for $j\in [\Gamma]$.
\end{itemize}
So $\{\mathcal{R}_z^N: z \in V_{N}^{\Gamma}\}$ is basically a concatenation of $\Gamma$
independent BRWs placed side by side. We can view $V_{N}^{\Gamma}$ as a $\Gamma N \times N$
rectangle divided into $\Gamma$ cells of dimension $N \times N$. Similarly, $V_{2N}^{\Gamma}$ can
be viewed as a $2\Gamma \times 2$ rectangle divided into 4 sub-rectangles of dimension $\Gamma N
\times N$ each of which is a copy of $V_{N}^{\Gamma}$ (see Figure~\ref{fig:geodesic_new} below for
an illustration).
\begin{figure}[!htb]
 \centering
\begin{tikzpicture}[semithick, scale = 1.1]
\draw [dashed] (0, -1.9) -- (0, 1.9);
\draw [dashed] (-5.9, 0) -- (5.9, 0);

\draw (0.1, 0.1) rectangle (1.9, 1.9);
\node [scale = 1] at (1, 1) {$a_{B_{n; 1, 2, 1}}$};
\draw (2.1, 0.1) rectangle (3.9, 1.9);
\node [scale = 1] at (3, 1) {$a_{B_{n; 1, 2, 2}}$};
\draw (4.1, 0.1) rectangle (5.9, 1.9);
\node [scale = 1] at (5, 1) {$a_{B_{n; 1, 2, 3}}$};

\draw [decorate,decoration={brace,amplitude=10pt, raise=4pt},yshift=0pt](0.1, 1.9) -- (5.9, 1.9);
\node [scale = 1, above] at (3, 2.5) {$V_{N; 1, 2}^\Gamma$};

\draw (-0.1, 0.1) rectangle (-1.9, 1.9);
\node [scale = 1] at (-1, 1) {$a_{B_{n; 1, 1, 3}}$};
\draw (-2.1, 0.1) rectangle (-3.9, 1.9);
\node [scale = 1] at (-3, 1) {$a_{B_{n; 1, 1, 2}}$};
\draw (-4.1, 0.1) rectangle (-5.9, 1.9);
\node [scale = 1] at (-5, 1) {$a_{B_{n; 1, 1, 1}}$};

\draw [decorate,decoration={brace,amplitude=10pt, raise=4pt},yshift=0pt](-5.9, 1.9) -- (-0.1, 1.9);
\node [scale = 1, above] at (-3, 2.5) {$V_{N; 1, 1}^\Gamma$};

\draw (0.1, -0.1) rectangle (1.9, -1.9);
\node [scale = 1] at (-1, -1) {$a_{B_{n; 2, 1, 3}}$};
\draw (2.1, -0.1) rectangle (3.9, -1.9);
\node [scale = 1] at (-3, -1) {$a_{B_{n; 2, 1, 2}}$};
\draw (4.1, -0.1) rectangle (5.9, -1.9);
\node [scale = 1] at (-5, -1) {$a_{B_{n; 2, 1, 1}}$};

\draw [decorate,decoration={brace,amplitude=10pt, mirror, raise=4pt},yshift=0pt](-5.9, -1.9) -- (-0.1, -1.9);
\node [scale = 1, below] at (-3, -2.5) {$V_{N; 2, 1}^\Gamma$};

\draw (-0.1, -0.1) rectangle (-1.9, -1.9);
\node [scale = 1] at (1, -1) {$a_{B_{n; 2, 2, 1}}$};
\draw (-2.1, -0.1) rectangle (-3.9, -1.9);
\node [scale = 1] at (3, -1) {$a_{B_{n; 2, 2, 2}}$};
\draw (-4.1, -0.1) rectangle (-5.9, -1.9);
\node [scale = 1] at (5, -1) {$a_{B_{n; 2, 2, 3}}$};

\draw [decorate,decoration={brace,amplitude=10pt,mirror, raise=4pt},yshift=0pt] (0.1, -1.9) -- (5.9, -1.9);
\node [scale = 1, below] at (3, -2.5) {$V_{N; 2, 2}^\Gamma$};
\end{tikzpicture}
 \caption{{\bf $(n+1)$-th stage of the recursive construction of BRW on $V_{2N}^\Gamma$}. Here $\Gamma = 3$.}
 \label{fig:geodesic_new}
\end{figure}

We wish to construct a light path in $V_{2N}^\Gamma$ based on constructions in $V_N^\Gamma$. As the path crosses through the rectangle of $V_{2N}^\Gamma$, we will make a choice of whether the path will stay in the upper or lower sub-rectangles based on the Gaussian values associated with the boxes of dimension $N\times N$, and thus we will switch back and forth between top and bottom layers. Our goal is to show that the expected weight of the crossing we constructed expands by a factor less than 2 when the size of the rectangles doubles. Apart from doubling of dimension at each level,there are a number of reasons that the weight will expand: (1) For standard Gaussian variable, we have $\mathbb E e^{\gamma X} = 1 + \tfrac{\gamma^2}{2} + O(\gamma^3)$; (2) We will need to make a construction to connect the paths on the left half and the right half of the rectangle; (3) Every time we switch between top and bottom layer, we will have to add a top-down crossing in the switching location. What we would possibly gain is from the ``variation'' of these Gaussian variables associated with $N\times N$ boxes, and thus choosing the favorable layer would reduce the weight of our path. There are a couple of subtle points in carrying out this plan, as we explain now.

In order to see how switching will reduce our cost, we first consider a toy problem where we simply compute $\mathbb E \min (X, Y)$ for independent standard Gaussian variables $X$ and $Y$. It turns out for this purpose it is more convenient to write $X = \frac{X+Y}{2} + \frac{X-Y}{2}$ and $Y = \frac{X+Y}{2} - \frac{X-Y}{2}$, and thus $\mathbb E \min(X, Y) = -\mathbb E |\frac{X-Y}{2}| = \tfrac{-1}{\sqrt{\pi}}$. If we carry out inductive procedure in this fashion, what we gain from the switching will not be able to beat the aforementioned expansions. This is because, in the language of the toy model, $\min(X, Y)$ is itself a random variable that can be written as $\frac{X+Y}{2} - \frac{|X-Y|}{2}$, where we gain $-\frac{1}{\sqrt{\pi}}$ by averaging over $(X-Y)$ but we get nothing by averaging over $X+Y$ (which is independent of $(X-Y)$). In fact, the better strategy would be to save the variation from $(X+Y)$ and use it in the optimizations in the next scale. But it is inconvenient to carry out this strategy, as it would require to do an induction based on quantities beyond the expected weight of the crossings in different scales. We resolve this issue by represent the BRW in a different way, which is tailored for our application here.

Another point is that, in the aforementioned optimization procedure, one has to be rather strategic in switchings as there is a cost for that. Eventually, this is reduced to a question of computing the following \emph{regularized total variation} for the Brownian motion, defined by
\begin{equation}\label{eq-regularized-total-variation}
\mathbb E \max_k \max_{0 = t_0 < t_1 < t_2 <\ldots < t_k < t_{k+1} = 1} \sum_{i=1}^{k+1} |B_{t_i} - B_{t_{i-1}}| - \lambda k\,.
\end{equation}
Here $B_{[0, 1]}$ is a standard Brownian motion, and $\lambda > 0$ is the term that measures the
switching cost. We will use a recent result of \cite{Dunlap} on the asymptotic value of
\eqref{eq-regularized-total-variation} as $\lambda \to 0$.\\

\noindent {\bf Notation convention.} In the rest of the paper we refer to the vertices of $\mathbb{Z}^2$
as \emph{points}. For any point $z \in \mathbb{Z}^2$, the horizontal and vertical coordinates of $z$ are
denoted by $z_x$ and $z_y$ respectively. If $z_y = 0$, we represent $z$ simply as $z_x$, which
should be clear from the context. A (finite) \emph{path} $\pi$ is a finite sequence $(v_0, v_1, \cdots, v_m)$ of points such that for each $i \geq 0$, $v_i$ and $v_{i + 1}$ are neighbors in $\mathbb{Z}^2$. In each of the pairs of non-negative integers $(n, N), (\ell,
L), (\ell', L')$ and $(\ell'', L'')$, the second element is always assumed to be 2 raised to the
power of first element. For functions $F(.)$ and $G(.)$, we write $F = O(G)$ if there exists an
absolute constant $C > 0$ such that $F \leq C G$ everywhere in the domain. We write $F =
\Omega(G)$ if $G = O(F)$.

\noindent {\bf Acknowledgement.} We warmly thank Steve Lalley for numerous helpful discussions and
constant encouragement on the project.

\section{Preliminaries}
\subsection{Coarsening of paths and $\mathcal L$-segments}
The $L$-\emph{coarsening} of a (finite) path $\pi = (v_0, v_1, \cdots, v_m)$ in $\mathbb{Z}^2$ is
defined as follows. Cover the lattice $\mathbb{Z}^2$ with $L \times L$ (non-overlapping) boxes such that one such box is $V_L$. Denote the center of a box $B$ as $c(B)$. For $v \in \mathbb{Z}^2$, let
$B_L(v)$ be the unique box containing $v$. Now define a sequence of integers $m_0, m_1,\cdots$ recursively as $m_0 = 0$ and,
$$m_j = \inf\{m \geq i \geq m_{j-1} : v_i \notin B_L(v_{m_{j-1}})\}\mbox{ for }j \geq 1\,.$$
Here we adopt the standard convention that infimum of an empty set is $\infty$. Let $p_{L, \pi}$
be the last integer $j$ such that $m_j$ is finite. The $L$-coarsening of $\pi$, denoted as
$\pi_{L, coarse}$, can now be defined as the sequence $\big(c(B_L(v_{m_0})), c(B_L(v_{m_1})),
\cdots, c(B_L(v_{m_{p_{L, \pi}}}))\big)$. We can also treat $\pi_{L, coarse}$ as a path in
$\mathbb{R}^2$ that connects successive points in this list by a line segment. See
Figure~\ref{fig:coarsening} for an illustration. Notice that the coarsening of a simple (i.e.
self-avoiding) path is not necessarily simple. However if the $L$-coarsening of a (left-right)
crossing $\pi^*$ of $V_{2L}^\Gamma$ is simple, then the $2L$-coarsening of $\pi^*$ is also simple.
We will use this fact when we carry out the induction step.\\
\begin{figure}[!htb]
	\centering
	\begin{tikzpicture}[semithick, scale = 2]
	\foreach \x  in {-3.5, -2.6, -1.7}
	{ \foreach \y  in {0, 0.9, 1.8}
		{ \draw (\x, \y) rectangle (\x + 0.8, \y + 0.8);

		}
	}

	\draw [thick] (-3.5, 1.3) -- (-3.2, 1.3) -- (-3.2, 1.7);
	\draw [thick] (-3.2, 1.8) -- (-3.2, 2.2) -- (-2.7, 2.2);
	\draw [thick] (-2.6, 2.2) -- (-2.2, 2.2) -- (-2.2, 1.8);
	\draw [thick] (-2.2, 1.7) -- (-2.2, 1.2) -- (-2.6, 1.2);
	\draw [thick] (-2.7, 1.2) -- (-3.1, 1.2) -- (-3.1, 0.9);
	\draw [thick] (-3.1, 0.8) -- (-3.1, 0.4) -- (-2.7, 0.4);
	\draw [thick] (-2.6, 0.4) -- (-1.8, 0.4);
	\draw [thick] (-1.7, 0.4) -- (-0.9, 0.4);

	\foreach \x  in {3.5, 2.6, 1.7}
	{ \foreach \y  in {0, 0.9, 1.8}
		{ \draw (\x, \y) rectangle (\x - 0.8, \y + 0.8);
			\fill (\x - 0.4, \y + 0.4) circle [radius = 0.04];

		}
	}

	\draw [thick, dashed] (1.3, 1.3) -- (1.3, 2.2) -- (2.2, 2.2) -- (2.2, 1.3) -- (1.3, 1.3) --(1.3, 0.4) -- (2.2, 0.4) -- (3.1, 0.4);

	\draw [->] (-0.8, 1.3) -- (0.8, 1.3);

	\end{tikzpicture}
	\caption{{\bf $L$-coarsening of a path}. The dashed path on the right represents the $L$-
		coarsening of the solid path on the left.}
	\label{fig:coarsening}
\end{figure}An \emph{$\mathcal L$-segment at level $\ell$} is a path $\pi$ whose $2^\ell$-coarsening is one of
two shapes shown in Figure~\ref{fig:L_segment}.
\begin{figure}[!htb]
	\centering
	\begin{tikzpicture}[semithick, scale = 2]
	\draw (-0.6, -0.6) rectangle (0.6, 0.6);
	\fill (0, 0) circle [radius = 0.04];
	\draw (0.7, -0.6) rectangle (1.9, 0.6);
	\fill (1.3, 0) circle [radius = 0.04];
	\draw (-0.6, 0.7) rectangle (0.6, 1.9);
	\fill (0, 1.3) circle [radius = 0.04];

	\draw [thick, dashed] (0, 1.3) -- (0, 0) -- (1.3, 0);

	\draw (2.5, 0.7) rectangle (3.7, 1.9);
	\fill (3.1, 1.3) circle [radius = 0.04];
	\draw (3.8, 0.7) rectangle (5, 1.9);
	\fill (4.4, 1.3) circle [radius = 0.04];
	\draw (2.5, -0.6) rectangle (3.7, 0.6);
	\fill (3.1, 0) circle [radius = 0.04];

	\draw [thick, dashed] (3.1, 0) -- (3.1, 1.3) -- (4.4, 1.3);
	\end{tikzpicture}
	\caption{{\bf Two possible $2^\ell$-coarsenings of an $\mathcal L$-segment at level $\ell$}.}
	\label{fig:L_segment}
\end{figure}
\subsection{Switched sign construction for branching random walk}
We first introduce a few more notations. As described in the introduction, we view
$V_{2N}^{\Gamma}$ as a $2\Gamma \times 2$ rectangle divided into 4 sub-rectangles of dimension
$\Gamma N \times N$ each of which is a copy of $V_{N}^{\Gamma}$ (see Figure~\ref{fig:geodesic_new}
for an illustration). Denote these 4 subrectangles (or the copies of $V_N^{\Gamma}$) as
$\{V_{N;i,k}^{\Gamma}\}_{i\in [2], k \in [2]}$ in the usual order. We can also shift the origin by
$u = (\Gamma i 2^{n+1}, j 2^{n+1})$ for $i, j \in \mathbb{N} \cup \{0\}$ and denote the corresponding
subrectangles (respectively rectangles) by $\{V_{N;i,k}^{\Gamma, u}\}_{i\in [2], k \in [2]}$
(respectively $V_{2N}^{\Gamma, u}$). Similarly we can define the fields $\{\mathcal{R}_{z; i,
k}^{N, u}: z \in V_{N;i,k}^{\Gamma, u}\}$ or $\{\mathcal{R}_{z}^{N, u}: z \in V_{N}^{\Gamma, u}\}$. The $j$-th constituent cell in $V_{N;i,k}^{\Gamma, u}$ (which is box of side length $N$) will be denoted as $B_{n;i,k,j}^u$ where $j \in [\Gamma]$. We observe that the Gaussian field $\{\mathcal{R}_{z}^{2N, u}: z \in V_{2N}^{\Gamma, u}\}$ is obtained by adding $a_{B_{n;i,k,j}^u}$ to the field $\{\mathcal{R}_{z; i, k}^{N, u}: z \in V_{N;i,k}^{\Gamma, u}\}$ in the box $B_{n;i,k,j}^u$ for $i \in [2], k \in [2]$ and $j \in
[\Gamma]$ (See Figure~\ref{fig:geodesic_new}). We will omit the additional superscript $u$ in all these notations when $u = 0$.

We will represent BRW by a construction in the fashion of switching-signs, which is tailored to
our inductive  construction for light crossings. To this end, we denote the collection of all
rectangles of the form $([0, 2^\ell - 1] \times [0, 2^{\ell+1} - 1]) \cap \mathbb{Z}^2 + (i2^\ell,
j2^\ell)$ by $\mathbb{B}\mathbb{D}_{k,2}$ where $i, j, \ell$ are nonnegative integers. Let
$A_{\ell,\Gamma}$ be the collection of all points of the form $(\Gamma i 2^\ell, j
2^\ell)$. Denote by $R_{\ell; k, j}^u$ the rectangle formed by the boxes $B_{\ell;1, k, j}^u$
and $B_{\ell;2, k, j}^u$ when $x \in A_{\ell+1, \Gamma}$. Note that $R_{\ell; k, j}^u\in
\mathbb{B}\mathbb{D}_{\ell, 2}$. Let $\{a_{B}\}_{B \in \mathbb{B}\mathbb{D}_{k,2}, k \geq 0}$ be a
collection of i.i.d.\ standard normal random variables, that is independent of $\{a_{B}\}_{k\geq
0, B\in \mathbb{BD}_k}$. We will now construct a family of Gaussian fields $\{\chi_z^{L, u}: z \in V_{2^\ell}^{\Gamma, u}\}$ for $u \in A_{\ell, \Gamma}$ and $\ell =0, 1, 2, \ldots, n$
recursively as follows:
\begin{itemize}
\item $\chi_z^{1, u} = 0$ for all $z \in V_1^{\Gamma, u}$.
\item Let $u \in A_{\ell+1, \Gamma}$ and $\{\chi_{z; i, k}^{L, u}: z \in
V_{L;i,k}^{\Gamma, u}\}$ be the field on $V_{L; i,k}^{\Gamma, u}$ constructed in level $\ell$. For
$z \in B_{\ell; 2, k, j}^u$, define
$$\chi_z^{2L, u} = \chi_{z; i, k}^{L, u} + (-1)^{(k - 1)\Gamma + j}b_\ell a_{\tilde{R}_{\ell; k,
j}^u} + (-1)^i c_\ell a_{B_{\ell; 2, k, j}^u}\,,$$
where $b_\ell = \sqrt{(1 - 4^{-\ell})/3}, c_\ell = \sqrt{2(1 - 4^{-\ell})/3}$ and
$\tilde{R}_{\ell; k, j}^u$ is the rectangle defined as follows:
$$\tilde{R}_{\ell; k, j}^u = \begin{cases}
R_{\ell; k, j}^u &\mbox{ if } (k-1)\Gamma + j\mbox{ is odd}\,, \\
R_{\ell; k, j-1}^u &\mbox{ if } (k-1)\Gamma + j\mbox{ is even and }j > 1\,,\\
R_{\ell; k-1, \Gamma}^u &\mbox{ if } (k-1)\Gamma + j\mbox{ is even and }j = 1\,.
\end{cases}$$
\end{itemize}
For future reference we will denote $Z_{\ell; i, k, j} = (-1)^{(k - 1)\Gamma + j}b_\ell
a_{\tilde{R}_{\ell; k, j}} + (-1)^i c_\ell a_{B_{\ell; 2, k, j}}$. Similarly as before, we will drop the superscript $u$ whenever $u = 0$. See Figure~\ref{fig:switch_BRW} for an
illustration of this construction at level $\ell + 1$. Notice that since $c_\ell^2 =
2b_\ell^2$, distribution of the field $\{\chi_z^N: z \in V_{N; j}^\Gamma\}$ is symmetric with respect to symmetries of the box $V_{N; j}^\Gamma$. This fact will be used later when we construct crossings through $V_{N;j}^\Gamma$'s in vertical direction. Finally, we define
$$\tilde{\chi}_z^{N, u}= b_na_{V_{N; j}^u} + \chi_z^{N, u} \mbox{ for }u \in A_{n, \Gamma} \mbox{ and } z \in V_{N; j}^{\Gamma, u} = u + V_{N; j}^\Gamma\,.$$
\begin{figure}[!htb]
 \centering
\begin{tikzpicture}[semithick, scale = 1.2]
\draw (-4.7, 0.1) -- (-1.7, 0.1);
\node [scale = 1] at (-3.2, 1.6) {$-b_\ell a_{R_{\ell; 1, 1}} - c_\ell a_{B_{\ell; 2, 1, 1}}$};
\draw (-1.5, 0.1) rectangle (1.5, 3.1);
\node [scale = 1] at (0, 1.6) {$b_\ell a_{R_{\ell; 1, 1}} - c_\ell a_{B_{\ell; 2, 1, 2}}$};
\draw (1.7, 0.1)  rectangle (4.7, 3.1);
\node [scale = 1] at (3.2, 1.6) {$-b_\ell a_{R_{\ell; 1, 3}} - c_\ell a_{B_{\ell; 2, 1, 3}}$};
\draw [decorate,decoration={brace, amplitude = 10pt, raise=4pt},yshift=0pt](-4.7, 3.1) -- (4.7, 3.1);
\node [scale = 1, above] at (0, 3.5) {$V^\Gamma_{L; 1, 1}$};

\draw (-4.7, -0.1) rectangle (-1.7, -0.1);
\node [scale = 1] at (-3.2, -1.6) {$-b_\ell a_{R_{\ell; 1, 1}} + c_\ell a_{B_{\ell; 2, 1, 1}}$};
\draw (-1.5, -0.1) rectangle  (1.5, -3.1);
\node [scale = 1] at (0, -1.6) {$b_\ell a_{R_{\ell; 1, 1}} + c_\ell a_{B_{\ell; 2, 1, 2}}$};
\draw (1.7, -0.1)  rectangle  (4.7, -3.1);
\node [scale = 1] at (3.2, -1.6) {$-b_\ell a_{R_{\ell; 1, 3}} + c_\ell a_{B_{\ell; 2, 1, 3}}$};
\draw [decorate,decoration={brace, amplitude = 10pt, mirror, raise=4pt},yshift=0pt](-4.7, -3.1) -- (4.7, -3.1);
\node [scale = 1, below] at (0, -3.5) {$V^\Gamma_{L; 2, 1}$};

\draw [red] (-4.7, -3.1) -- (-4.7, 3.1) -- (-1.7, 3.1) -- (-1.7, -3.1) -- (-4.7, -3.1);

\end{tikzpicture}
 \caption{{\bf Construction of the field $\chi_z^{2L}$ on $V_{L;i,1}^{\Gamma}$}. Here $\Gamma =
 3$. The red lines indicate the boundary of $R_{\ell; 1, 1}$.}
 \label{fig:switch_BRW}
\end{figure}
\begin{lemma}
\label{lem:switch_brw}
The Gaussian fields $\{\tilde{\chi}_z^{N}: z \in V_N^\Gamma\}$ and
$\{\mathcal R_z^{N}: z \in V_N^{\Gamma}\}$ are identically distributed.
\end{lemma}
\begin{proof}
It is not difficult to check that the fields $\{\tilde{\chi}_z^N: z \in V_{N; j}\}$'s are
independent and identically distributed for $j \in [\Gamma]$. Hence it suffices to verify that we
have the correct covariances between field values at all pairs of vertices inside $V_N$. Toward
this end take a pair $(u, v)$ of vertices in $V_N$ which were separated until level $\ell$,  i.e.,
$\mbox{Cov}(\mathcal R_{u}^N, \mathcal R_{v}^N) = n - \ell$. The covariance between $\chi_u^N$ and
$\chi_v^N$ is given by
$$\mbox{Cov}(\chi_u^N, \chi_v^N)  = -(1 - 4^{-\ell})/3 + \mbox{$\sum_{\ell + 1 \leq \ell \leq n}$}(1 -
4^{-\ell}) = n - \ell - (1 - 4^{-n})/3\,.$$
Consequently $\mbox{Cov}(\tilde{\chi}_u^N, \tilde{\chi}_v^N) = \mbox{Cov}(\chi_u^N, \chi_v^N) + (1 - 4^{-n})/3
=  n - \ell$. Hence the lemma follows.
\end{proof}
\begin{remark}
It is clear that it suffices to work with $\{\chi_z^N: z\in V_N^\Gamma\}$ rather than the BRW in
what follows. Using this construction, in every scale of our optimization, we crucially have effective variance $c_\ell^2 \approx \frac{2}{3}$ that is larger than $1/2$ (which would be the variance we effectively used if we work with the more canonical construction of BRW).
\end{remark}
\subsection{Regularized total variation of Brownian motion}
\label{sec:total_variation}
In this subsection we present a result from \cite{Dunlap} that provides a lower bound on the functional defined in \eqref{eq-regularized-total-variation}. This is only one part of the main result in \cite{Dunlap} which also provides an upper bound.

\begin{theorem}
\label{thm:brownian_switch}
Let $\{B_t\}_{0 \leq t \leq 1}$ be a Standard Brownian Motion and $1 > \lambda > 0$. For any partition $P = [t_0, t_1, \cdots, t_{k+1}]$ of $[0, 1]$ where $0 = t_0 < t_1 < \cdots < t_{k+1} = 1$, define the quantity $\Phi_{P, \lambda}$ as
$$\Phi_{P, \lambda} = \mbox{$\sum_{0 \leq i \leq k}$} |B_{t_{i + 1}} - B_{t_i}|  - \lambda k\,.$$
Then there exists a (random) partition $\mathcal P = [t_0^*, t_1^*, \cdots, t_{k^* + 1}^*]$ of $[0, 1]$ such that the following conditions hold:
\begin{enumerate}[(a)]
\item The random vector $(t_1^*, \cdots, t_{k^*}^*, k^*)$ is measurable relative to $\sigma(\{B_t\}_{0 \leq t \leq 1})$.
\item $\mathbb E \Phi_{P, \lambda} \geq \frac{1}{\lambda}$.
\item $B_{t_1^*} - B_{t_0^*}, B_{t_2^*} - B_{t_1^*}, \cdots, B_{t_{k^* + 1}^*} - B_{t_{k^*}^*}$ have alternating signs. Furthermore $\P(B_{t_1^*} - B_{t_0^*} > 0) = 0.5$.
\item $|B_{t_{j + 1}^*} - B_{t_j^*}| \geq \lambda$ for all $1 \leq j \leq k^* - 1$.
\end{enumerate}
\end{theorem}
\section{Inductive construction for light crossings}
\subsection{The induction hypotheses}
We will construct a light (left-right) crossing through $V_{L}^\Gamma$ (or equivalently
$V_{L}^{\Gamma, u}$) inductively for $L = 2^\ell$ and $\ell = 0, \ldots, n$. Throughout this
section whenever we refer to a crossing of $V_L^{\Gamma, u}$, it is implicitly assumed
that the underlying field is $\{\chi_z^{L, u}: z \in V_{L}^{\Gamma, u}\}$ unless stated
otherwise. For technical convenience, let us assume that for each $\ell \geq 0$ and $u \in
A_{\ell+1, \Gamma}$ we have an independent standard Brownian motion $\{W_t^{u, \ell}\}_{0 \leq t \leq 2\Gamma}$ that is also independent with $\{a_{B}\}_{B \in \mathbb{B}\mathbb{D}_{k,2}, k \geq 0}$. We sample $a_{B^u_{\ell; 2, k, j}}$'s as increments of this process at appropriate time points. We are now ready to state our induction hypotheses. For $\ell = 0$, we
have only one option, i.e., to take the straight line as our crossing. This crossing has weight exactly $\Gamma$. Suppose that for each $1 \leq \ell' \leq \ell$ and $u \in A_{\ell', \Gamma}$, we have a crossing $\pi^{*,u,\ell'}$ of $V_{2^{\ell'}}^{\Gamma, u}$, identically distributed for different $u$, that satisfies the following properties
\begin{enumerate}
\item \label{induct:hypo2} $\pi^{*,u,\ell'}_{L'/2, coarse}$ is simple and measurable relative to $\sigma\big(\{W_t^{u, \ell' - 1}\}_{0 \leq t \leq 2\Gamma}\big)$.
\item \label{induct:hypo3} Law of $\pi^{*,u,\ell'}_{L'/2, coarse}$ is invariant with respect
to reflection about the line $y = u_y + 2^{\ell' - 1} - 0.5$ i.e. the horizontal line passing through the center of $V_{2^{\ell'}}^{\Gamma, u}$.
\item \label{induct:hypo4}For $j \in [\Gamma]$, denote the line $x = u_x + jL' - 1$ as $\mathbb L_{jL'}$. Let $z_{j, \ell'}^u$ be the point in the set $\{u, u + (0, L'/2), $ $u + (\Gamma L'/2, 0), u + (\Gamma L'/2, L'/2)\}$ such that the point where $\pi^{*, u, \ell'}$ hits $\mathbb L_{jL'}$ for the last time lies within $V_{L'/2}^{\Gamma, z_{j, \ell'}^u}$. Also for $j \in \mathbb N^+$, let $v_{j, L', L''}^u$ be the last point along $\pi^{*,u, \ell'}_{L'', coarse}$
that lies to the left of $\mathbb L_{jL'}$. Then $v_{j, L', L'_{100}}^u = v_{2j, L'/2,
L'_{100}}^{z_{j, \ell'}^u}$, i.e., the crossings $\pi^{*, u, \ell'}$ and $\pi^{*, z_{j, \ell'}^u,
\ell' - 1}$ hit $\mathbb L_{jL'}$ for the last time in the same $L'_{100} \times L'_{100}$ box. Here $L'_{100} = \max(2^{-100} L', 1)$.
\item \label{induct:hypo5}$\mathbb E (\sum_{z \in \pi^{*, u, \ell'}}\mathrm{e}^{\gamma \chi_z^{\ell'}}) \leq 2\Gamma(2 - 0.2\gamma^2)^{\ell'}$.
\end{enumerate}
One of the consequences of our induction hypotheses, which is crucial for our analysis, is that
$v_{j, L, \delta L}^u$ is uniformly distributed for all $\ell' \leq \ell$ as given by the
following lemma.
\begin{lemma}
\label{lem:uniform}
Let $R_{\ell; i, k, j}^u$ be the unique $L_{100} \times L$ rectangle that shares its
right boundary with with $B_{\ell; i, k, j}^u$. Also let $\{B_{\ell; i, k, j, m}^u\}_{1
\leq m \leq L/L_{100}}$ be the collection of boxes in $\mathbb{BD}_{\log_2 L_{100}}$ that
comprise $R_{\ell; i, k, j}^u$. Now if the induction hypotheses \ref{induct:hypo2},
\ref{induct:hypo3} and \ref{induct:hypo4} hold for all $\ell' \leq \ell$, then $v_{j, L,
L_{100}}^u$ is distributed uniformly on the set $\{c_{\ell; i, k, j, m}^u\}_{1 \leq m
\leq L / L_{100}}$, where $c_{\ell; i, k, j,m}^u$ is the center of the box $B_{\ell; i, k,
j,m}^u$ (see Figure~\ref{fig:Uniform}).
\end{lemma}
\begin{proof}
We get, from hypothesis \ref{induct:hypo3}, that for any $2 \leq L' \leq L, j' \in [\Gamma]$ and
$x \in A_{\ell', \Gamma}$, $v_{j', L', L'/2}^u$ is equally likely to be above or below the line $y
= u_y + 2^{\ell' - 1} - 0.5$. We can recursively apply this argument for all the levels down to $\log_2 L_{100}$ in view of hypothesis \ref{induct:hypo2} which gives that the
$L'/2$-coarsenings of crossings at level $\ell'$ are independent for different $\ell'$. Combining these observations with hypothesis \ref{induct:hypo4} completes the proof of the lemma.
\end{proof}
\begin{figure}[!htb]
 \centering
\begin{tikzpicture}[semithick, scale = 2]
\draw (-3, -2) rectangle (2, 3);

\draw (1, -2) -- (1, -1) -- (2, -1);
\fill (1.5, -1.5) circle [radius = 0.04];
\node [scale = 1, below] at (1.5, -1.5) {$c_{\ell; i, k, j, 1}$};

\draw (2, -0.9) -- (1, -0.9) -- (1, 0.1) -- (2, 0.1);
\draw [thick, dotted] (1.5, 0.2) -- (1.5, 2.9);
\fill (1.5, -0.4) circle [radius = 0.04];
\node [scale = 1, below] at (1.5, -0.4) {$c_{\ell; i, k, j, 2}$};
\end{tikzpicture}
 \caption{{\bf The set of points $\{c_{\ell; i, k, j, m}\}_{1 \leq m \leq 2^{100}}$}.}
 \label{fig:Uniform}
\end{figure}
\subsection{The induction step}
Now we will carry out the induction step. It suffices to produce a crossing for $u = 0$. As
usual, we will drop $u$ from all the superscripts when $u$ is origin. Before going to more
detailed discussions let us give a broad overview of the strategy that we follow. \\
First we decide the $L$-coarsening of $\pi^{*, \ell + 1}$. It would consist of a sequence
$H_1, H_2, \cdots, H_{R + 1}$ of horizontal segments (possibly of length 0) from left to right such that every two successive segments are connected by a vertical segment. See Figure~\ref{fig:coarse_construct} for an illustration. Notice that this is the only possible option for $\pi^{*, \ell + 1}_{L, coarse}$, if it has to be simple as required by induction hypothesis~\ref{induct:hypo2}. So we can ``encode'' $\pi^{*, \ell + 1}_{L, coarse}$ as a sequence of $\{1, 2\}$ valued random variables $\{i_{j, k}\}_{j \in [\Gamma], k \in [2]}$, where $i_{k, j} = 1$ or $2$ accordingly as $\pi^{*,\ell + 1}_{L, coarse}$ enters the rectangle $R_{\ell; k, j}$ through $B_{\ell; 1, k, j}$ or $B_{\ell; 2, k, j}$ respectively. We refer to this as a \emph{switching strategy}. Having defined $L$-coarsening of the crossing, we now move on to actual construction. From the previous level we have four crossings $\{\pi^{*, \ell}_{i, k}\}_{i \in [2], k \in [2]}$ respectively in $\{V_{L;i,k}^\Gamma\}_{i \in [2], k \in [2]}$. We ``join'' the paths $\pi^{*, \ell}_{i, 1}$ and $\pi^{*, \ell}_{i, 2}$ into a crossing $\pi^{*, \ell+1}_i$ of $V_{2L}^\Gamma$ for $i \in [2]$. Since $L$-coarsening of crossings at level $\ell$ are also simple, it follows that the vertices of $\pi_i^{*, \ell + 1}$ that lie within some $B_{\ell; i, k, j}$ define a subpath of $\pi_i^{*, \ell + 1}$. Now we replace each $H_r$ with the subpath $h_r^{*, \ell + 1}$ of $\pi_i^{*, \ell + 1}$ that is contained within the $L \times L$ boxes that intersect $H_r$. Here $i$ is 1 or 2 accordingly as $H_r$ lies in the top or bottom row of $V_{2L}^\Gamma$ (See Figure~\ref{fig:coarse_construct} and the discussions preceding Figure~\ref{fig:geodesic_new}). Finally we connect $h_r^{*, \ell + 1}$ and $h_{r+1}^{*, \ell + 1}$
by an appropriate $\mathcal L$-segment $v_r^{*, \ell + 1}$ at level $\ell$. The paths $h_1^{*, \ell + 1}, v_1^{*, \ell + 1}, h_2^{*, \ell + 1}, \cdots, v_{R}^{*, \ell + 1}, h_{R + 1}^{*, \ell + 1}$ define the crossing $\pi^{*, \ell + 1}$. Note that $L$-coarsening of $\pi^{*, \ell + 1}$ is the very one that we started with.

\begin{figure}[!htb]
 \centering
\begin{tikzpicture}[semithick, scale = 1.1]
\draw (0.1, 0.1) rectangle (1.9, 1.9);

\draw (2.1, 0.1) rectangle (3.9, 1.9);

\draw (4.1, 0.1) rectangle (5.9, 1.9);

\draw (-0.1, 0.1) rectangle (-1.9, 1.9);

\draw (-2.1, 0.1) rectangle (-3.9, 1.9);

\draw (-4.1, 0.1) rectangle (-5.9, 1.9);

\draw (0.1, -0.1) rectangle (1.9, -1.9);

\draw (2.1, -0.1) rectangle (3.9, -1.9);

\draw (4.1, -0.1) rectangle (5.9, -1.9);

\draw (-0.1, -0.1) rectangle (-1.9, -1.9);

\draw (-2.1, -0.1) rectangle (-3.9, -1.9);

\draw (-4.1, -0.1) rectangle (-5.9, -1.9);
\draw [<->] (-5.9, -2) -- (-4.1, -2);
\node [scale = 1, below] at (-5, -2) {$L$};

\foreach \x in {-5, -3, -1, 1, 3, 5}{

\foreach \y in {-1, 1}{
         \fill (\x, \y) circle [radius = 0.05];
}

\draw [thick, dashed] (-5, -1) -- (-3, -1) -- (-3, 1) -- (-1, 1) -- (1, 1) -- (1, -1) -- (3, -1) -- (5, -1);
}

\end{tikzpicture}
 \caption{{\bf $L$-coarsening of $\pi^{*, \ell+1}$}. Here $\Gamma = 3$ and $R = 2$.}
 \label{fig:coarse_construct}
\end{figure}

We also need some notations in order to track the change in expected weight of crossings between
two levels. To this end define,
\begin{align*}
D_{\gamma, \ell} &= \sum_{v \in \pi^{*, \ell}} e^{\gamma \chi_v^L}\,, \mbox{ and } d_{\gamma, \ell}
= \mathbb E  D_{\gamma, \ell}\\
D_{\gamma, \ell, j} &=  \sum_{v\in \pi^{*,\ell} \cap V_{L, j} } e^{\gamma \chi_v^L}, \mbox{ and }
d_{\gamma, \ell, j}  = \mathbb E D_{\gamma, \ell, j} \mbox{ for } j\in [ \Gamma]\,.
\end{align*}
Let $\tilde{D}_{\gamma, \ell; k}$ the weight contributed to $\pi^{*, \ell + 1}$ by the paths
$\pi_{i, k}^{*, \ell}$'s. From what we described in the previous paragraph, we may bound
$\mathbb E\tilde{D}_{\gamma, \ell; k}$ as
\begin{equation}
\label{eq:weight_upper_bnd}
\mathbb E \tilde{D}_{\gamma, \ell; k} \leq \mathbb E \sum_{j = 1}^\Gamma d_{\gamma, \ell, j} \big(1 + \gamma
Z_{\ell; i_{j, k}, k, j} + \frac{a_{\ell + 1}^2\gamma^2}{2}\big) + \mathbb E \mathrm{Err}_{\gamma, \ell; k, 1} + \mathbb E \mathrm{Err}_{\gamma, \ell; k, 2}\,,
\end{equation}
where $a_{\ell+1}^2 = b_{\ell+1}^2 + c_{\ell+1}^2 = 1 - 4^{-(\ell + 1)}$, $\mathrm{Err}_{\gamma, \ell; k, 1} = \frac{\gamma^2}{2}\sum_{j = 1}^\Gamma d_{\gamma, \ell, j} (Z_{\ell; i_{j, k}, k,
j}^2 - a_{\ell + 1}^2)$ and
$$\mathrm{Err}_{\gamma, \ell; k, 2} = \sum_{j = 1}^\Gamma d_{\gamma, \ell, j} \big(\mathrm{e}^{\gamma Z_{\ell; i_{j, k}, k, j}} - (1 + \gamma Z_{\ell; i_{j, k}, k, j} + \frac{Z_{\ell; i_{j, k}, k, j}^2\gamma^2}{2})\big)\,.$$
It is easy to see that $\mathbb E |\mathrm{Err}_{\gamma ,\ell; k, 2}| = d_{\gamma, \ell}O(\gamma^3)$. On
the other hand, we get the following from expanding the terms in $\mathrm{Err}_{\gamma, \ell; k,
1}$:
\begin{align*}
\mathrm{Err}_{\gamma, \ell; k, 1} = \frac{b_\ell^2\gamma^2}{2}\sum_{j = 1}^\Gamma d_{\gamma, \ell, j} (a_{\tilde{R}_{\ell; k, j}}^2 - 1) + \frac{c_{\ell}^2\gamma^2}{2}\sum_{j = 1}^\Gamma d_{\gamma, \ell, j}(a_{B_{\ell; 2, k, j}}^2 - 1) \\
+ \gamma^2 b_\ell c_\ell (-1)^{(k-1)\Gamma}\sum_{j = 1}^\Gamma d_{\gamma, \ell, j}
a_{\tilde{R}_{\ell; k, j}} (-1)^{j + i_{j, k}}a_{B_{\ell; 2, k, j}}\,.
\end{align*}
Expectations of the first two terms in the expansion above are obviously 0. As to the third term,
recall that our switching mechanism is independent of the random variables $a_{\tilde{R}_{\ell; k,
j}}$'s by induction hypothesis~\ref{induct:hypo3}. Consequently $\mathbb E \mathrm{Err}_{\gamma, \ell; k,
1} = 0$. Thus we only need to reckon with the first term in the right hand side of
\eqref{eq:weight_upper_bnd} so far as the contributions from $\pi_{i, k}^{*, \ell}$'s are
concerned. This will, of course, depend on the particular switching strategy we adopt. Finally we
need to add the contributions from $v_r^{*, \ell + 1}$'s and the extra crossings that we make in
order to link $\pi_{i, 1}^{*, \ell}$ and $\pi_{i, 2}^{*, \ell}$ for $i \in [2]$.\par

Now we carry out the detailed computations. It is expected that $d_{\gamma, \ell, j}$ is
approximately uniform over $j\in [\Gamma]$. However, proving this requires some effort. It turns
out easier to treat the ``hypothetical'' case that $d_{\gamma, \ell}$ is dominated by a small
number of $d_{\gamma, \ell, j}$'s. We also include a case to deal with small values of $\ell$ when $c_\ell^2$ is not big enough for our ``main'' strategy to work.

\noindent {\bf Case 1.} $\sum_{j=1}^\Gamma d_{\gamma, \ell, j} \mathbf 1_{\{d_{\gamma, \ell, j}
\geq \Gamma^{-2/3} d_{\gamma, \ell}\}} \geq d_{\gamma, \ell} \Gamma^{-1/10}$. It is
easy to see that
\begin{equation}\label{eq-var-lower-bound}
\sum_{j=1}^\Gamma d_{\gamma, \ell, j}^2 \geq d_{\gamma, \ell}^2 \Gamma^{-13/15}\,.
\end{equation}
In this case our switching strategy is simple. For each $k = 1, 2$, we choose the rectangle that
minimizes $\sum_{j=1}^\Gamma d_{\gamma, \ell, j} Z_{\ell; i, k, j}$. Thus, we have for $k \in [2]$
\begin{align*}
\mathbb E \tilde{D}_{\gamma, \ell; k} \leq d_{\gamma, \ell}(1 + \gamma^2/ 2 + O(\gamma^3)) + \mathbb E
\min_{i=1, 2} \sum_{j=1}^\Gamma d_{\gamma, \ell, j} \gamma Z_{\ell; i, k, j} \\ \leq
d_{\gamma, \ell} (1 + O(\gamma^2) - \Omega(\gamma)\Gamma^{-13/30})\leq d_{\gamma, \ell} (1 -
\Omega(\gamma^{1.9}))\,,
\end{align*}
where the last inequality follows from our assumption that $\Gamma \geq \alpha / \gamma^2$. Let $u_k \in A_{\ell, \Gamma}$ be such that $V_L^{\Gamma, u_k}$ is
the minimizer. In order to connect $\pi^{*, u_1, \ell}$ and $\pi^{*, u_2, \ell}$ into a crossing of
$V_{2L}^{\Gamma}$, we need some additional paths. To this end we add two vertical crossings through the rectangles $V_{\mathrm{left}} = \big([(\Gamma-1) L , \Gamma L - 1] \times [0, 2L-1]\big) \cap \mathbb{Z}^2$ and $V_{\mathrm{right}} = \big([\Gamma L , (\Gamma + 1)L -
1] \times [0, 2L-1]\big) \cap \mathbb{Z}^2$. We also add two horizontal crossings through the rectangles $V_{\mathrm{up}} = \big([(\Gamma - 1)L, (\Gamma + 1) L - 1] \times [L, 2L-1]\big) \cap \mathbb{Z}^2$ and $V_{\mathrm{down}} = \big([(\Gamma - 1)L, (\Gamma + 1) L
- 1] \times [0, L-1]\big) \cap \mathbb{Z}^2$. All these
crossings have minimum weights. We can now form $\pi^{*, \ell+1}$ by concatenating $\pi^{*, u_1, \ell}, \pi^{*,
u_2, \ell}$ and these four crossings. See
Figure~\ref{fig:Case1} for an illustration. It is obvious from the construction that $\pi^{*, \ell + 1}_{L, coarse}$ is simple, symmetric in law with respect to the line $y = L - 0.5$ and entirely determined by the random variables $\{a_{B_{\ell; 2, k, j}}\}_{k \in [2],
j \in [\Gamma]}$. The only possible case where induction hypothesis \ref{induct:hypo4} could fail would be the
line $\mathbb L_{\Gamma L}$. But we do not need to
consider this line as $\Gamma$ is odd. Finally notice that, since $\Gamma \gg 1$ and minimum crossing weights through adjacent rectangles are super-additive, each of the four additional crossings will have expected weight
bounded by, say, $\frac{2.1}{\Gamma}d_{\gamma, \ell}$.
Using the fact that $\alpha \gg 1$, we then deduce that in this case
\begin{equation*}
d_{\gamma, \ell + 1} \leq (2 - \gamma^2/2) d_{\gamma, \ell} \leq 2\Gamma(2 - 0.2)^{\ell + 1}\,.
\end{equation*}

\begin{figure}[!htb]
 \centering
\begin{tikzpicture}[semithick, scale = 1.1]
\draw [dashed] (0, -1.9) -- (0, 1.9);
\draw [dashed] (-5.9, 0) -- (5.9, 0);
\draw (0.1, 0.1) rectangle (1.9, 1.9);

\draw (2.1, 0.1) rectangle (3.9, 1.9);

\draw (4.1, 0.1) rectangle (5.9, 1.9);

\draw (-0.1, 0.1) rectangle (-1.9, 1.9);

\draw (-2.1, 0.1) rectangle (-3.9, 1.9);

\draw (-4.1, 0.1) rectangle (-5.9, 1.9);

\draw (0.1, -0.1) rectangle (1.9, -1.9);

\draw (2.1, -0.1) rectangle (3.9, -1.9);

\draw (4.1, -0.1) rectangle (5.9, -1.9);

\draw (-0.1, -0.1) rectangle (-1.9, -1.9);

\draw (-2.1, -0.1) rectangle (-3.9, -1.9);

\draw (-4.1, -0.1) rectangle (-5.9, -1.9);

\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (-5.9, 1) -- (-4.1, 1);

\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (-3.9, 1) -- (-2.1, 1);

\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1.9, 1) -- (-1, 1);

\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1, 1) -- (-0.1, 1);

\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1, 1.9) -- (-1, 1);
\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1, 1) -- (-1, 0.1);
\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1, -0.1) -- (-1, -0.55);
\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1, -0.55) -- (-1, -1.9);
\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1.9, -0.55) -- (-1, -0.55);
\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (-1, -0.55) -- (-0.1, -0.55);
\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (0.1, -0.55) -- (1, -0.55);
\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (1, -0.55) -- (1.9, -0.55);
\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (1, -0.55) -- (1, -1);
\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (1, -0.55) -- (1, -0.1);
\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (1, -1) -- (1, -1.9);

\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (5.9, -1) -- (4.1, -1);

\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (3.9, -1) -- (2.1, -1);

\draw [red, style = {decorate, decoration = {snake, amplitude = 0.6}}] (1.9, -1) -- (1, -1);

\draw [style = {decorate, decoration = {snake, amplitude = 0.6}}] (1, -1) -- (0.1, -1);

\end{tikzpicture}
 \caption{{\bf The construction of $\pi^{*, \ell+1}$ for Case 1.} Here $\Gamma = 3$.
The segments of $\pi^{*, \ell+1}$ have been colored in red.}
 \label{fig:Case1}
\end{figure}
\noindent {\bf Case~2.} If $\ell < 60$, we use the strategy described in Case~1. It then follows from the previous discussions that
\begin{equation*}
d_{\gamma, \ell + 1} \leq (2 + 2\gamma^2)d_{\gamma, \ell}\,.
\end{equation*}
Since $d_{\gamma, 0} = \Gamma$, we get by repeated application of the previous inequality
\begin{equation*}
d_{\gamma, \ell + 1} \leq \Gamma(2 + 2\gamma^2)^{\ell + 1} \leq 2\Gamma(2 - 0.2\gamma^2)^{\ell + 1}\,.
\end{equation*}
\noindent {\bf Case~3.} $\sum_{j=1}^\Gamma d_{\gamma, \ell, j} \mathbf 1_{d_{\gamma, \ell, j} \geq
\Gamma^{-2/3} d_{\gamma, \ell}} \leq d_{\gamma, \ell, j} \Gamma^{-1/10}$ and $\ell \geq 60$. This is the main and
real case. We will also assume that
$$d_{\gamma, \ell} \geq 2\Gamma (2 - 0.2\gamma^2)^{\ell} (1 - \gamma^2)\,,$$
for otherwise we can simply follow the strategy in Case~1. In this case our switching strategy
will in fact be informed by the way we plan to contruct the $\mathcal L$-segments $v_r^{*, \ell +
1}$'s. So we first give a detailed description of the latter. In what follows, we may assume
without any loss of generality that $\delta L \geq 2$ for $\delta = 2^{-100}$. We refer the
reader to Fig~\ref{fig:vertical_embed} for an illustration of the construction.\\
For each $r \leq R$, let $V_{L;{U(r)}}$ and $V_{L; D(r)}$ respectively denote the $L \times L$
boxes in the top and bottom row of $V_{2L}^\Gamma$ containing the endpoints of $H_r$ and $H_{r +
1}$. Call the $L \times 2L$ rectangle formed by $V_{L; U(r)}$ and $V_{L; D(r)}$ as $V_{L; r}$.
Suppose the right boundary of $V_{L; r}$ lies along the line $\mathbb{L}_{j_rL}$ where $j_r \in
[2\Gamma - 1]$. It suffices to consider the case when $j_r \leq \Gamma$. Let $v_{j_r, L, \delta
L}^u$ and $v_{j_r, L, \delta L}^d$ respectively denote the last points along $\pi^{*, L,
\ell}_{\delta L, coarse}$ and $\pi^{*,\ell}_{\delta L, coarse}$ that lie to the left of $\mathbb
L_{j_rL}$. Then $v_{j_r, L, \delta L}^u$ and $v_{j_r, L, \delta L}^d$ are centers of two $\delta L
\times \delta L$ boxes, say, $A_{j_r, L}^u$ and $A_{j_r, L}^d$ respectively. Let $\tilde{A}_{j_r,
L}^u$ (respectively $\tilde{A}_{j_r, L}^d$) be the box around $A_{j_r, L}^u$ (respectively
$\tilde{A}_{j_r, L}^d$) of side length $2\delta L$. Then we construct light contours in each of the annuli $\tilde{A}_{j_r, L}^u \setminus A_{j_r, L}^u$ and $\tilde{A}_{j_r, L}^d \setminus A_{j_r, L}^d$.
These can easily be achieved by concatenating four
minimum weight crossings in four rectangles. In addition, we let $\tilde A_r$ be a rectangle whose left top corner is $v_{j_r, L, \delta L}^u + (0.5, \delta L - 0.5)$ and right bottom corner is $v_{j_r, L, \delta L}^d
+ (0.5, -\delta L + 0.5)$. We construct a minimum weight
vertical crossing through $\tilde A_r$. Finally we concatenate these light crossings into $v_r^{*, \ell
+ 1}$. The total expected weight of these crossings can
be bounded by our induction hypotheses. Notice that we can always concatenate these paths in a way so that our construction of $v_r^{*, \ell + 1}$ obeys induction hypothesis~\ref{induct:hypo4}.
\begin{figure}[!htb]
 \centering
 \begin{tikzpicture}[semithick, scale = 2.2]
 \draw (-3.1, 0) rectangle (-0.1, 3);
 \draw (0, 0) rectangle (3, 3);
 \draw (-3.1, 3.1) rectangle (-0.1, 6.1);

 \draw (-0.1, 0.8) -- (-0.9, 0.8) -- (-0.9, 1.6) -- (-0.1, 1.6);
 \fill [blue!10] (-0.11, 0.8) rectangle (-0.9, 1.59);
 \fill [blue!10] (-0.11, 4.4) rectangle (-0.9, 5.19);

 \fill [brown!20](-0.11, 1.6) rectangle (-1.2, 1.89);
 \fill [brown!20](-0.11, 0.5) rectangle (-1.2, 0.79);
 \fill [brown!20](-0.91, 0.79) rectangle (-1.2, 1.89);

 \fill [brown!20](-0.11, 5.2) rectangle (-1.2, 5.5);
 \fill [brown!20](-0.11, 4.1) rectangle (-1.2, 4.39);
 \fill [brown!20](-0.91, 4.39) rectangle (-1.2, 5.49);

 \draw (-0.1, 0.5) -- (-1.2, 0.5) -- (-1.2, 1.9) -- (-0.1, 1.9);
 \draw (0, 0.5) -- (0.3, 0.5) -- (0.3, 1.9) -- (0, 1.9);
 \fill [brown!20](0.29, 0.51) rectangle (0.01, 1.89);

 \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.1, 0.65) -- (-1.2, 0.65);
 \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-1.05, 0.5) -- (-1.05, 1.9);
 \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-1.2, 1.75) -- (-0.3, 1.75);
 \draw [red, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.3, 1.75) -- (-0.1, 1.75);
 \draw [red, style = {decorate, decoration = {snake , amplitude = 0.6}}] (0, 1.75) -- (0.15, 1.75) -- (0.15, 0.65) -- (0.05, 0.65);
 \draw [blue] (0, 0.65) -- (0.05, 0.65);
 \draw [green, style = {decorate, decoration = {snake , amplitude = 0.6}}] (0, 1) -- (0.1, 1) -- (0.05, 0.65) -- (0.2, 0.3) -- (0.9, 0.3);
 \draw [style = {decorate, decoration = {snake , amplitude = 0.4}}] (0.2, 1.75) -- (0.3, 1.75);
 \draw [style = {decorate, decoration = {snake , amplitude = 0.4}}] (0.2, 1.9) -- (0.2, 1.75);

 \fill (-0.5, 1.2) circle [radius = 0.04];
 \draw [thick, dashed] (-0.5, 0.5) -- (-0.5, 5.5);
 \fill (-0.5, 4.8) circle [radius = 0.04];

 \draw (-0.1, 4.4) -- (-0.9, 4.4) -- (-0.9, 5.2) -- (-0.1, 5.2);
 \draw (-0.1, 4.1) -- (-1.2, 4.1) -- (-1.2, 5.5) -- (-0.1, 5.5);
  \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.1, 4.25) -- (-1.2, 4.25);
  \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-1.05, 4.1) -- (-1.05, 5.5);
  \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-1.2, 5.35) -- (-0.3, 5.35);
  \draw [red, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.3, 5.35) -- (-0.2, 5.35);
  \draw [blue, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.2, 5.35) -- (-0.1, 5.35);
  \draw [green, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-1.6, 5.7) -- (-0.2, 5.7) -- (-0.2, 5) -- (-0.1, 5);
  \draw [style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.3, 5.5) -- (-0.3, 5.35);

  \draw [red, style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.3, 5.35) -- (-0.3, 1.75);
  \draw [style = {decorate, decoration = {snake , amplitude = 0.6}}] (-0.3, 1.75) -- (-0.3, 0.5);

 \end{tikzpicture}
\caption{{\bf The construction of $v_r^{*, \ell + 1}$.} The two big boxes on the left define the rectangle $V_{L; r}$, the top and bottom one being $V_{L;U(r)}$ and $V_{L;D(r)}$ respectively. The two boxes filled with light blue are $A_{j_r, L}^u$ (top) and $A_{j_r, L}^d$ (bottom). The annuli $\tilde A_{j_r, L}^u \setminus A_{j_r, L}^u$ and $\tilde A_{j_r, L}^d \setminus A_{j_r, L}^d$ (or portions thereof) are indicated by light brown.
The path colored in red is $v_r^{*, \ell + 1}$. The green paths represent $h_r^{*, \ell + 1}$ and $h_{r+1}^{*, \ell + 1}$.}
\label{fig:vertical_embed}
\end{figure}

Let us now try to bound the expected weight of $v_r^{*, \ell + 1}$ where we assign vertex weight
based on variables \emph{associated with boxes of side length $ \leq L$}. Notice that this does
not require the knowledge of particular switching strategy we employ. Denote by $D_r$
the total weight of the light crossings in the two annuli (where we assign vertex weight based
variables associated with boxes of side length $\leq \delta L/2$). Since the $\delta L$ coarsening
of $\pi^{*, u, \ell'}$ is independent of all the Gaussians at level lower than $\delta \ell$, we
see that
\begin{equation}\label{eq-weight-D-i}
\mathbb E D_{r} \leq \frac{33 d_{\gamma, \ell - \log_2(2/\delta)}}{\Gamma}\,.
\end{equation}
Let $\tilde D_r$ denote the weight of the vertical crossing through $\tilde{A}_r$ (where again, we
assign the vertex weight based on Gaussian variables associated with boxes of side length $\leq
\delta L/2$). Clearly we have
\begin{equation}\label{eq-weight-tilde-D-i}
\mathbb E \tilde{D}_r \leq \frac{\big( 3\delta L + \mathbb E |v_{j_r, L, \delta L}^u - v_{j_r, L, \delta L}^d|
\big)}{(\delta L / 2)\Gamma}d_{\gamma, \ell - \log_2(2/\delta)}\,.
\end{equation}
In both \eqref{eq-weight-D-i} and \eqref{eq-weight-tilde-D-i}, we have silently used the fact that
expected minimum weights of crossings through $V_{L'}^m$'s form a superadditive sequence in $m$. The following lemma says that we do not lose much in terms of the expected costs of these
crossings after we take into account the contributions from Gaussians appearing in higher levels.
\begin{lemma}\label{lem-max-exponential-Gaussian}
Let $X_1, \ldots, X_m$ i.i.d.\ standard Gaussian variables, where $m \leq (1/\gamma)^{100}$ and
$U$ be a nonnegative random variable bounded by $C$. Then
$$\mathbb E (U\max_i \mathrm{e}^{\gamma X_i}) \leq  \mathbb E U \big(1 + O(\gamma \log (1/\gamma))\big) +
C \mathrm{e}^{-\Omega(\log(1/\gamma))}\,.$$
\end{lemma}
\begin{proof}
Let $M = \max_i X_i$. It is clear that $\mathbb E M \leq \sqrt{200 \log (1/\gamma)}$, and also that
$\P(M\geq \mathbb E M+ u)\leq e^{-u}$. The conclusion follows by a straightforward computation.
\end{proof}
\noindent Combining \eqref{eq-weight-D-i}, \eqref{eq-weight-tilde-D-i}, Lemmas~\ref{lem:uniform}
and \ref{lem-max-exponential-Gaussian}, we get that the expected weight for each ``switching gadget''is bounded by $(1 + 10\delta)(2 - 0.2\gamma^2)^{\ell} \leq \frac{1 + 20\delta}{\Gamma}d_{\gamma, \ell}$, where we use the fact that $\gamma \ll 1$. Recall the expression for $\mathbb E\tilde{D}_{\gamma, \ell; k}$ as
given in \eqref{eq:weight_upper_bnd} and the fact that
$$Z_{\ell; i, k, j} = (-1)^{(k - 1)\Gamma + j}b_\ell a_{\tilde{R}_{\ell; k, j}} + (-1)^{i} c_\ell
a_{B_{\ell; 2, k, j}}\,.$$
It is then natural to consider switching strategies $\{i_{j, k}\}_{j \in [\Gamma], k \in [2]}$ that gives a small value of the following quantity:
\begin{equation}
\label{eq:optimize}
c_\ell\gamma\sum_{j = 1}^\Gamma d_{\gamma, \ell, j}(-1)^{i_{j, k}} a_{B_{\ell; 2, k,
j}} + R\frac{1 + 20\delta}{\Gamma}d_{\gamma, \ell}\,.
\end{equation}
To this end we consider a similar optimization problem for a process that is related to this
quantity. Denote by $J = \{j\in [\Gamma]: d_{\gamma, \ell, j} \leq \Gamma^{-2/3} d_{\gamma,
\ell}\}$. By Cauchy-Schwartz inequality, we see that
\begin{equation}\label{eq-var-lower-J}
\sum_{j\in J} d_{\gamma, \ell, j}^2 \geq \frac{(1 - \Gamma^{-1/5}) d_{\gamma, \ell}^2}{\Gamma}\,.
\end{equation}
Write $J = \{j_1, \ldots, j_{|J|}\}$ with $1 \leq j_1 < j_2 < \cdots < j_{|J|} \leq \Gamma$, and
define the sequence
$$g_{\gamma, i} =c_\ell^2\gamma^2\mbox{$\sum_{k \leq i}$}d_{\gamma, \ell, j_k}^2 \mbox{ for } 1
\leq i \leq 2|J|\,,$$
where $j_i = \Gamma + j_{i - |J|}$ and $d_{\gamma, \ell, j_i} = d_{\gamma, \ell, j_{i - |J|} }$ for $i > |J|$.
Consider the process
$$S_{t; \ell} = c_\ell \gamma \mbox{$\sum_{0 \leq k \leq i}$}  d_{\gamma, \ell, j_{k+1}} (W_{t_i
\wedge j_{k+1}}^{\ell} - W_{t_i \wedge (j_{k+1} - 1)}^{\ell}) \mbox{ for } g_{\gamma, i} \leq t
\leq g_{\gamma, i+1}\,,$$
where $t_i = j_{i+1} - 1 + \tfrac{t - g_{\gamma, i}}{c_\ell^2 \gamma^2 d_{\gamma, \ell, j_{i+1}}^2}$ and
$g_{2; \gamma, 0} = 0$. Thus $\{S_{t; \ell}\}_{0 \leq t \leq g_{\gamma, 2|J|}}$ is a standard Brownian motion whose increment between the time points $g_{\gamma, i-1}$ and $g_{\gamma, i}$ is given by $c_\ell \gamma d_{\gamma, \ell, j_i} a_{B_{\ell; 2, k, j_i}}$
for $1 \leq i \leq 2|J|$. Now suppose that $[T_0, T_1, \cdots, T_{R'+1}]$ is the partition of $[0, g_{\gamma, 2|J|}]$ given by Theorem~\ref{thm:brownian_switch} when applied to $\{S_{t; \ell}\}_{0 \leq t \leq g_{\gamma, 2|J|}}$ for the penalty term $\tfrac{(1+20\delta)d_{\gamma, \ell}}{\Gamma}$ (with appropriate change of scales).
Using this we will define $L$-coarsening of the path that we will construct. Since this has to be
simple by our induction hypothesis, it suffices to assign an element in $\{-1, 1\}^{2|J|}$ to
$j_1, j_2, \cdots, j_{2|J|}$. For this purpose define the random variables $\mathrm{sgn}_{1;
\ell}, \cdots, \mathrm{sgn}_{2|J|; \ell}$ as follows:
\begin{equation*}
    \mathrm{sgn}_{i; \ell} = \begin{cases}
               -1 & \text{if }T_r < g_{2; \gamma, i} \leq T_{r + 1} \mbox{ such that } S_{T_r;
               \ell} <
               S_{T_{r+1}; \ell}\\
                1 & \text{if }T_r < g_{2; \gamma, i} \leq T_{r + 1} \mbox{ such that } S_{T_r;
                \ell} >
               S_{T_{r+1}; \ell}\,.
           \end{cases}
\end{equation*}
It is clear that $(\mathrm{sgn}_{1; \ell}, \cdots, \mathrm{sgn}_{2|J|; \ell})$ is identically
distributed as $(-\mathrm{sgn}_{1; \ell}, \cdots, -\mathrm{sgn}_{2|J|; \ell})$. As a consequence
the distribution of $\pi^{*, \ell + 1}_{L, coarse}$ is invariant relative to reflection about the
line $y = L - 0.5$. \par

Now we collect the contributions to the weight of $\pi^{*, \ell+1}$ from all the sources. We use
crossings through $V_{\textrm{up}}, V_{\textrm{down}}, V_{\textrm{left}}$ and $V_{\textrm{right}}$, as in Case 1, to link the crossings $\pi^{*, \ell}_{i, 1}$ and
$\pi^{*, \ell}_{i, 1}$ for $i \in [2]$. As discussed in Case 1, this will not tamper with induction
hypothesis~\ref{induct:hypo4}. We also need a bound on the error we made by interpolating $(S_{t; \ell})$ as a
Brownian motion. Notice that for each $(T_r, T_{r + 1})$ that is contained in some interval $(g_{2; \gamma, i-1}, g_{2;\gamma, i})$, we lose at most $M_{\mathrm{dis}}$ due to ``discretization'' of the optimal partition obtained from Theorem~\ref{thm:brownian_switch}, where
$$M_{\mathrm{dis}} = \max_{1 \leq i \leq 2|J|}\max_{g_{2; \gamma, i-1} \leq s, t \leq g_{2;
\gamma, i}} |S_{t; \ell} - S_{s; \ell}|\,.$$
Thus the total loss we incur due to discretization is at most $M_{\mathrm{dis}}R'$. We use a
simple estimate for $M_{\mathrm{dis}}$ as follows:
\begin{equation}\label{eq-discretization}
\mathbb E M_{\mathrm{dis}}  \leq O(\gamma \Gamma^{-2/3} \log(\Gamma)) d_{\gamma, \ell}, \mbox{ and } \P(|M_{\mathrm{dis}} - \mathbb E M_{\mathrm{dis}}| \geq u\gamma \Gamma^{-2/3} \log(\Gamma)d_{\gamma, \ell} ) \leq e^{-\Omega(u^2)}\,.
\end{equation}
Finally we need to make amends for the fact that what we pass as ``penalty'' term to \eqref{eq:optimize} is
actually based on the Gaussians up to level $\ell$. But the multiplier at level $\ell + 1$ can be at most $\mathrm{e}^{M_{\ell + 1}}$, where $M_{\ell+1}$
is the maximum of all Gaussians added at level $\ell + 1$. Altogether, we deduce that
\begin{align}\label{eq-key-induction}
d_{\gamma, \ell+1} &\leq  2 d_{\gamma, \ell} (1 + \gamma^2/2 + O(\gamma^3) +
\frac{10}{\Gamma}) - \mathcal I_\ell + \sqrt{\mathbb E R'^2}\sqrt{\mathbb E M_{\mathrm{dis}}^2} \nonumber \\
& + \frac{(1+20\delta)d_{\gamma, \ell}}{\Gamma} \sqrt{\mathbb E R'^2}\sqrt{\mathbb E(\mathrm{e}^{\gamma M_{\ell+1}}- 1)^2}\,,
\end{align}
where
\begin{equation*}
\mathcal I_{\ell} = \mathbb E \bigg( \sum_{i=1}^{R'+1} |S_{T_i; \ell} - S_{T_{i-1}; \ell}| - R' \frac{(1+20\delta)d_{\gamma, \ell}}{\Gamma}\bigg)\,,
\end{equation*}
It follows from condition~(d) in Theorem~\ref{thm:brownian_switch} and a straightforward computation that
\begin{equation}
\label{eq:switch_bound}
\mathbb E R'^2 \leq O\bigg(\Big(\frac{\Gamma^2\gamma^2\sum_{j \in J}d_{\gamma, \ell, j}^2}{d_{\gamma, \ell}^2}\Big)^2\bigg)\,.
\end{equation}
In addition, by Theorem~\ref{thm:brownian_switch}, we deduce that for all $\alpha > 0$
$$\mathcal I_\ell \geq \big(\frac{4}{3} - \delta\big) \gamma^2 \Gamma \frac{\sum_{j\in J} d_{\gamma, \ell,
j}^2}{d_{\gamma, \ell}}\,.$$
Here we used the fact that $\ell \geq 60$. Plugging the bounds from \eqref{eq-discretization}, \eqref{eq:switch_bound} and Lemma~\ref{lem-max-exponential-Gaussian} into \eqref{eq-key-induction} we obtain
$$d_{\gamma, \ell+1} \leq (2 - 0.6\gamma^2)2\Gamma(2 - 0.2\gamma^2)^\ell \leq 2\Gamma(2 -
0.2\gamma^2)^{\ell + 1}\,.$$
This completes the proof of Theorem~\ref{thm-main}.


\section{Introduction}

\begin{figure}[!t]
\centering
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{270244_vgg16_bn_saliency.jpg}
    \caption{Class-guided explanation with class 'zebra'.}\label{fig:zebra1}
\end{subfigure}
    \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{270244_vgg16_bn_pointing_game_hit.jpg}
    \caption{Pointing Game with $acc=100\%$.}\label{fig:zebra2}
\end{subfigure}
   \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{270244_vgg16_bn_weighting_game_616.jpg}
    \caption{Weighting Game with $acc=61.6\%$.}\label{fig:zebra3}
\end{subfigure}

\bigskip
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{99182_swin_t_saliency.jpg}
    \caption{Class-guided explanation with class 'person'.}\label{fig:person1}
\end{subfigure}
    \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{99182_swin_t_pointing_game_hit.jpg}
    \caption{Pointing Game with $acc=100\%$.}\label{fig:person2}
\end{subfigure}
   \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{99182_swin_t_weighting_game_147.jpg}
    \caption{Weighting Game with $acc=14.7\%$.}\label{fig:person3}
\end{subfigure}
\caption{Differences of Weighting Game to Pointing Game. Top explanation is distributed better over the correct class, while bottom explanation is spread out more over the image. With Pointing Game, the accuracy results are the same for both explanations, as the highest magnitude pixel (visualized with green circle) is within the class segmentation mask. Weighting Game aims to better explain, how accurate explanations are by calculating the proportion of the magnitude of the saliency contained within the correct class mask. Class-guided explanations were calculated using ViT-B/32 \cite{ViT} and Layer-CAM \cite{LayerCam}}
\label{fig:pointing-game-hits}
\end{figure}

The rise of deep learning in the field of computer vision has seen models become increasingly powerful, however, this has been with the cost of interpretability. Due to this, the field of explainable AI has recently seen a rise into prominence. Several explainability methods have been proposed, which aim to explain what parts of an input image are most relevant to the given output of a model \cite{GuidedBackProp,GradCAM,GradCAMPLUS,AblationCam,LayerCam,XGradCam}, the output of an explainability method is usually called a \emph{saliency map}. These methods can be used to evaluate the quality of models' predictions, and especially can be useful when trying to understand incorrect predictions.

Previous works in evaluating saliency maps has found that many explainability methods are independent of the model parameters and labeling of training data, thus rendering them to be simple edge detectors \cite{SanityChecks}. While it is necessary for an explainability method to be independent of such factors, it does not imply a method is accurate in its explanations, as the sanity checks do not measure how explanations align with input images. Our goal in this paper is to build a framework for measuring the accuracy and stability of explanations. By accuracy, we refer to the ability of a class-guided explanation to correctly concentrate on the correct class' objects, while with stability we refer to the consistency of explanations when minor transformations are applied to the image.

For assessment of explainability method accuracy, mainly two types of metrics have been presented. Pointing Game \cite{ExcitationBP} measures the accuracy at which a saliency map's highest magnitude pixel is contained within the correct class's segmentation map, thus recording individual instances as either hits or misses. The problem with this method is that the binary nature of the metric does not allow for measuring the quality of hits and misses. An example of the limitations of Pointing Game is shown in Figure. \ref{fig:pointing-game-hits}, where both explanations are counted as equally good, while the one found in Figure \ref{fig:zebra1} seems more concentrated on the correct class compared to the one shown in Figure. \ref{fig:person1}.

The second types of methods for measuring explanation accuracy usually operate by removing patches or points from an image, to record how the output changes based on removal of different points \cite{LRP,Xrai,RISE}. However, removal based methods tend to introduce unnatural patches into images, which can cause undesired effects into the results, the methods also have the tendency of being more computationally intensive.

\begin{figure}[!t]
\centering
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{319721_vit_b_32_saliency.jpg}
    \caption{Class-guided explanation with class 'horse'.}\label{fig:horse1}
\end{subfigure}
    \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{319721_vit_b_32_pointing_game_miss.jpg}
    \caption{Pointing Game with $acc=0\%$.}\label{fig:horse2}
\end{subfigure}
   \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{319721_vit_b_32_weighting_game_646.jpg}
    \caption{Weighting Game with $acc=64.6\%$.}\label{fig:horse3}
\end{subfigure}

\bigskip
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{511321_vit_b_32_saliency.jpg}
    \caption{Class-guided explanation with class 'boat'.}\label{fig:boat1}
\end{subfigure}
    \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{511321_vit_b_32_pointing_game_miss.jpg}
    \caption{Pointing Game with $acc=0\%$.}\label{fig:boat2}
\end{subfigure}
   \hfill
\begin{subfigure}[t]{.3\linewidth}
    \centering
    \includegraphics[width=\textwidth]{511321_vit_b_32_weighting_game_526.jpg}
    \caption{Weighting Game with $acc=52.6\%$.}\label{fig:boat3}
\end{subfigure}
\caption{Two examples of explanations and their accuracies. In both class-guided explanations, Pointing Game counts the explanations as misses, due to barely missing the segmentation mask. Weighting Game is able to correctly identify both explanations as quite accurate. Class-guided explanations were calculated using ViT-B/32 \cite{ViT} and Layer-CAM \cite{LayerCam}}
\label{fig:pointing-game-misses}
\end{figure}

In this paper, we extend the idea of Pointing Game and introduce the Weighting Game, which measures how much a saliency map's \emph{mass} or \emph{magnitude} is contained within the correct class's segmentation map. The measurement of the distribution of the explanation map allows for more flexibility in assessing the quality of individual explanations, as it is not binary in its nature, such as the Pointing Game. Pointing Game can slightly miss a target and define an explanation as a miss, while Weighting Game is able to see the whole picture and give a better representation of the accuracy of the explanation, such as in Figure \ref{fig:pointing-game-misses}.

For assessing the stability of an explainability method, we introduce two separate methods. Firstly, we create 3D effect videos from an image, and observe the correlation between saliency maps from consecutive frames. With sufficiently small changes between frames, this grants us the ability to assess how minor changes can affect the resulting saliency maps. Secondly, we create a pre-defined zoom and pan transformation on an image, and process saliency maps for the original and transformed images. Then, we measure the correlations between the saliency map for the transformed image and a correspondingly transformed saliency map for the original image. These methods provide a way to assess how consistent explainability methods are, thus giving insights on how well they can function in real-world use cases

The main contributions of this work are: (\textbf{1}) The Weighting Game, a novel metric for measuring the accuracy of an explainability method; (\textbf{2}) two novel metrics for assessing the stability of an explainability method; and (\textbf{3}) comparison of several prominent CAM methods using proposed metrics with commonly used image classification architectures.

\section{Related Work}

\subsubsection{Backpropagation Explainability Methods}

First methods to visualize network attribution were based on gradients of a model found via backpropagation. These methods include DeConvNet \cite{DeConvNet}, Network Saliency \cite{SimonyanSaliency}, and Guided Backpropagation \cite{GuidedBackProp}. The methods have slight difference, however, all are based in the idea of visualizing gradients of a loss function from a forward propagated input image. Later introduced backpropagation methods include SmoothGrad \cite{SmoothGrad}, Integrated Gradients \cite{IntegratedGradients}, and Expected Gradients \cite{ExpectedGradients}.

\subsubsection{Class Activation Explainability Methods}

The use of class activation maps was first introduced with CAM as a way to visualize the attribution for an output of a network \cite{CAM}. This was later expanded by Grad-CAM, which used the backpropagated gradients to weight the class activation maps \cite{GradCAM}. After Grad-CAM, multiple different methodologies have been introduced to attempt to improve upon Grad-CAM. Notable later CAM methods include, Grad-CAM++ \cite{GradCAMPLUS}, Ablation-CAM \cite{AblationCam}, Layer-CAM \cite{LayerCam}, XGrad-CAM \cite{XGradCam}, and SmoothGrad-CAM++ \cite{SmoothGradCAMPLUS}.

\subsubsection{Evaluation of Explainability Methods}

In 2018, a study into explainability methods was conducted, which determined that most methods were acting as simple edge detectors \cite{SanityChecks}. The experiments showed that produced saliency maps were mostly independent of the model parameters and labels of the training data, except for notably the saliency maps created using Grad-CAM and DeConvNet. The Pointing Game was introduced as a way to measure the class-sensitivity and accuracy of class-guided explainability methods \cite{ExcitationBP}. Finally, ranking methods for pixel importance have been proposed via removal based methods \cite{LRP,Xrai,RISE}.

\section{Method}

We aim to generate methods for assessing how accurate and stable explanations are. For accuracy, the method should be class-specific and concentrated within the segmentation map of an object of the correct class. Stability should be quantified by the consistency of explanations when minor transformations are applied to an image.

\subsection{The Weighting Game}

\begin{figure}[!t]
    \centering
    \includegraphics[width=\textwidth]{weighting_game_diagram.png}
    \caption{Process of Weighting Game starting from an input image and a class. Firstly, we obtain the class mask for the image and calculate an explanation for the corresponding class. Then, we dilate the class segmentation mask with the produced explanation, which creates a masked explanation. Finally, the mass of the masked explanation is divided with the mass of the explanation, to obtain the accuracy of an explanation.}
    \label{fig:weighting-game-diagram}
\end{figure}

We defined the Weighting Game as a metric of class-guided explainability method accuracy. The prerequisite for Weighting Game is a dataset, which contains class segmentation masks for the classes that the given model is trained to classify. Thus, object detection datasets, such as Pascal Visual Object Classes 2007 \cite{PASCALVOC07} or MS COCO \cite{COCO}, function well for this task.

For calculating one instance of the Weighting Game, we begin with an image and a corresponding class segmentation mask for a class contained within the image. The class segmentation mask is the union of all corresponding class' object segmentation masks found within the image. After obtaining the class segmentation mask, it is dilated using a simple 2D convolution defined with the following equation:
\begin{equation}
\label{eqn:dilation}
    D = M \circledast K,
\end{equation}
where $M$ is the binary class segmentation mask to be dilated, $K$ is a $9\times9$ convolution kernel, and $D$ is the resulting mask. As the resulting mask $D$ is binary, all parts of the mask $M$ that overlap with the convolutional kernel $K$, will be included in the dilated mask $D$. The dilation provides the method with more robustness, as outer edges of the desired objects may also be relevant to the classification of the objects.

Then, using the class corresponding to the segmentation mask, the class-guided explanation is calculated for the image. The resulting explanation is a saliency map, which is either produced in the resolution of the input image or upscaled to it.

The dilated class segmentation mask is then used to measure, how much of the \emph{magnitude} or \emph{mass} of the saliency map is contained within the segmentation mask. This gives a measure of how much of the saliency map is contained within the relevant class' area in the image, thus providing a value for the accuracy of an explainability method. One class/image combination of Weighting Game can be formulated with the following equation:
\begin{equation}
\label{eqn:weighting}
    Accuracy = \frac{\sum_{i,j}S_{i,j}D_{i,j}}{\sum_{i,j}S_{i,j}},
\end{equation}
where $S$ is the class-guided saliency map and $D$ is the corresponding class' dilated segmentation mask. As Equation. \ref{eqn:weighting} provides the accuracy of one class within an image, it will be used to calculate the accuracy of all class/image combinations in the chosen dataset. The final result of Weighting Game is the mean accuracy of all class/image combinations. A diagram of the process of calculating Weighting Game accuracy is visualized in Figure. \ref{fig:weighting-game-diagram}.

\subsection{Explanation Stability}

Stability of an explainability method can be assessed via the consistency of saliency maps when minor transformations are applied to an image. Thus, this factor is important when, e.g. producing class-guided explanations for videos. Stability can characterize the reliability of an explainability method, as when given similar information to an explainability method, it should produce a similar saliency map.

Stability of an explanation method will be assessed via two separate metrics. Firstly, we simulate camera motion by creating 3D effect zoom and pan videos using 3D Ken Burns effect \cite{KenBurns}. Artificially created videos are used, as they can provide a diverse set of videos with a consistent level of motion between frames. This also provides the opportunity to stay within the same distribution of data as in training, as the validation split of the training dataset can be used to generate the videos. The created videos will have 150 frames, of which, the first half consists of a zoom in with ratio 1.5, with the second half consisting of a zoom back to the first frame. Using these videos, we calculate explanations for five pairs of consecutive frames in the video.

\begin{figure}[!t]
    \centering
    \includegraphics[width=\textwidth]{stability_diagram.png}
    \caption{Diagram of explanation stability using a zoom and pan transformation. First, a random resized crop is calculated, which is visualized as red borders in the original image. Then, a class-guided explanation is calculated for the original image, which is then cropped with the same calculated crop. After that, the original image is cropped and a class-guided explanation is calculated for the cropped image. Finally, the correlation between the resulting two saliency maps is calculated. Explanations were processed with class-guided Grad-CAM \cite{GradCAM} and ResNet50 \cite{ResNet} using the class 'cat'.}
    \label{fig:stability-diagram}
\end{figure}

The explanations will be processed using class-guided methods, with the chosen class being the highest probability class for the first frame of the video. Then, the correlation between these saliency maps is calculated, to produce a measure for the stability of an explanation method. If the difference between frames is small enough, where the placement of objects is practically the same, the correlation between the saliency maps should be high. The correlations between saliency maps are calculated using Spearman's rank-order correlation \cite{Spearman}, as in other literature where correlation between saliency maps is evaluated \cite{NormGrad,SaliencyEye}. Thus, the equation used to calculate the stability of a method will be as follows:
\begin{equation}
\label{eqn:stability-video}
    Stability = corr(e(M_i, c), e(M_{i+1}, c)),
\end{equation}
where $M_i$ is the $i$th frame of a video, $c$ is the highest probability class for frame $M_0$, $corr$ is the Spearman rank-order correlation, and $e$ is the class-guided explanation function that takes in an image and a class.

The second method for assessing explanation stability is by calculating a zoom, pan, and upscaling (back to original resolution) transformation on an image. After this, saliency maps are calculated for both images. Then, the saliency map for the original image is transformed in the same manner as the transformed image, as the transformation is greater than those seen with the earlier method using consecutive frames of a video. Finally, the correlations are calculated between the saliency map for the transformed image and the transformed saliency map for the original image. Using the same transformation on the original image and the corresponding saliency map, we can align the saliency maps for the original and transformed images. Therefore, the contents are should be the same, however, with different scale in the image. The stability can be expressed with:
\begin{equation}
\label{eqn:stability-transformation}
    Stability = corr(t(e(M, c)), e(t(M), c)),
\end{equation}
where $t$ is the zoom/pan transformation function, $M$ is the input image, $c$ is the highest probability class for $M$, $corr$ is the Spearman rank-order correlation, and $e$ is the class-guided explanation function. A diagram of the stability metric is shown in Figure. \ref{fig:stability-diagram}.

\section{Experiments}

In this section, we describe the evaluation of various class-guided explanation methods using our accuracy and stability frameworks. Code to reproduce the experiments is available at \url{https://github.com/lassiraa/weighting-game}.

\subsection{Models and Dataset}

The chosen model architectures for experiments are ResNet50 \cite{ResNet}, VGG16-BN \cite{VGG}, ViT-B/32 \cite{ViT}, and Swin-T \cite{Swin}. The models represent some of the most popular architectures from both convolutional and transformer based image classification architectures. The inference times of all models are also comparable, which is why they are desirable to compare to each other. Pre-trained versions of all models were obtained from Torchvision's available models \cite{PyTorch}.

As all the chosen models are trained on the ImageNet \cite{deng2009imagenet} dataset, they need to be fine-tuned in order to be used for the Weighting Game. Thus, the COCO dataset is chosen for fine-tuning the models to a multi-label classification task. The fine-tuning procedure is similar to that of the one implemented in the Pointing Game \cite{ExcitationBP}.

Only the weights of the last layer are adjusted in the fine-tuning procedure, using binary cross entropy loss. For the training procedure, random resized crop and horizontal flips are used for data augmentation. For validation, images are center cropped and resized to $224\times224$ resolution. The Adam optimizer \cite{AdamOpt} is used for optimization, using learning rates of 0.001 and 0.002 for the convolutional and transformer models respectively. Exponential learning rate scheduler was used with 0.95 gamma. All models were trained for 50 epochs, at which point the validation losses had plateaued for all models.

\begin{figure}[!t]
    \centering
    \includegraphics[width=\textwidth]{video_frames.png}
    \caption{Examples of consecutive frames from created 3D effect videos and their corresponding class-guided explanations. Displayed correlations values are mean correlations between consecutive frame pairs, as a measure of the explainability method's stability. Full videos can be found in the supplementary materials. First and third columns processed with ResNet50 \cite{ResNet}, Grad-CAM \cite{GradCAM}, and class 'person', while second and fourth columns were processed with ViT-B/32 \cite{ViT}, Layer-CAM \cite{LayerCam}, and class 'elephant'}
    \label{fig:video-frames}
\end{figure}

For consistency in models and datasets, the COCO validation set is used in all explanation accuracy and stability evaluations. This way, the distribution of the data remains similar in training and evaluation environments, as we are not attempting to measure the out-of-distribution performance of the models or explanation methods. Examples of produced 3D effect videos can be found in the supplementary materials with saliency maps overlaid on the video, such as in Figure \ref{fig:video-frames}.

\subsection{Saliency Methods}

The chosen saliency methods to assess in this study were Guided Backpropagation \cite{GuidedBackProp}, Grad-CAM \cite{GradCAM}, Grad-CAM++ \cite{GradCAMPLUS}, Layer-CAM \cite{LayerCam}, XGrad-CAM \cite{XGradCam}, and Ablation-CAM \cite{AblationCam}. All methods have implementations available in the Pytorch Grad-CAM package \cite{PytorchGradCam}, which provides consistence in implementations. All methods except Ablation-CAM only require one pass through a model, making them computationally efficient, and interesting to determine whether the added computational cost of Ablation-CAM is worth it. Guided Backpropagation provides a baseline for comparison. All methods also satisfy the requirement of being class-guided.

The target layers for the methods are chosen to be from either the last convolutional or transformer block. For ResNet50, the chosen target layers are the layers of the last identity block, while VGG16-BN has the last max pooling layer used as the target layer. Both the ViT-B/32 and Swin-T models have the last transformer block's first layer normalization as the target layer. All saliency maps are upscaled to the input image's resolution using bilinear interpolation. Transformer models' saliency maps are also reshaped from the flattened patches to the shape of the input image.

\subsection{Results}

\subsubsection{Weighting Game}

\begin{table}[t]
\caption{Explanation accuracy results of different saliency methods for different model architectures. For Weighting Game, the recorded values are the mean percentage of saliency map magnitude contained within the correct class segmentation mask. Weighting Game small is the same metric for objects under 10\% of the size of the image. Results are contrasted to the reproduced Pointing Game \cite{ExcitationBP} metric. Best and second best explainability method results for each model are bolded and underlined, respectively.}
\label{table:weighting-game}
\centering
\begin{tabularx}{\textwidth}{|X|p{0.22\textwidth}|X|X|X|X|}
\hline
 & & \multicolumn{4}{|c|}{Explanation accuracy (\%)} \\
\hline
 & & ResNet50 & VGG16-BN & ViT-B/32 & Swin-T \\
\hline
\multirow{3}{0.12\textwidth}{Ablation-CAM \cite{AblationCam}} & Weighting & 21.1 & 19.3 & 23.7 & \underline{22.2}\\
 & Pointing & 29.3 & 23.2 & 26.3 & \textbf{30.9}\\
 & Weighting small & 4.74 & 4.00 & 5.01 & \underline{4.93} \\
\hline
\multirow{3}{0.12\textwidth}{Grad-CAM \cite{GradCAM}} & Weighting & \textbf{30.5} & \textbf{29.2} & 26.7 & 14.6\\
 & Pointing & \textbf{38.0} & \textbf{36.6} & \underline{32.4} & 15.6\\
 & Weighting small & \underline{11.9} & \textbf{10.4} & 8.81 & 4.17 \\
\hline
\multirow{3}{0.12\textwidth}{Grad-CAM++ \cite{GradCAMPLUS}} & Weighting & 20.4 & 22.0 & \underline{27.7} & 13.2\\
 & Pointing & 29.8 & 33.7 & 32.2 & 13.7\\
 & Weighting small & 5.29 & 6.06 & \underline{9.55} & 4.10\\
\hline
\multirow{3}{0.12\textwidth}{Guided Backprop. \cite{GuidedBackProp}} & Weighting & 24.5 & 21.7 & 16.5 & 16.7\\
 & Pointing & 25.9 & 23.6 & 16.3 & 17.2 \\
 & Weighting small & 8.40 & 7.04 & 3.79 & 4.49 \\
\hline
\multirow{3}{0.12\textwidth}{Layer-CAM \cite{LayerCam}} & Weighting & 20.3 & 22.3 & \textbf{30.6} & \textbf{22.5}\\
 & Pointing & 28.8 & 34.5 & \textbf{34.2} & \underline{30.5}\\
 & Weighting small & 5.20 & 6.17 & \textbf{13.8} & \textbf{10.5} \\
\hline
\multirow{3}{0.12\textwidth}{XGrad-CAM \cite{XGradCam}} & Weighting & \underline{30.5} & \underline{29.0} & 16.2 & 16.1\\
 & Pointing & \textbf{38.0} & \underline{36.4} & 14.6 & 14.3\\
 & Weighting small & \textbf{11.9} & \underline{10.3} & 3.66 & 3.66 \\
\hline
\end{tabularx}
\end{table}

After calculating the results of the Weighting Game, we obtain mean accuracy results displayed in Table. \ref{table:weighting-game}. The table contains results for the Weighting Game using all objects and only objects that are under 10\% of the size of the image. Pointing Game \cite{ExcitationBP} scores are also included for comparison.

\begin{figure}[t]
     \centering
     \begin{subfigure}[t]{0.3\textwidth}
         \centering
         \includegraphics[width=\textwidth]{344059_swin_t_saliency.jpg}
         \caption{Class-guided explanation with class 'giraffe'.}
     \end{subfigure}
     \hfill
     \begin{subfigure}[t]{0.3\textwidth}
         \centering
         \includegraphics[width=\textwidth]{344059_swin_t_pointing_game_hit.jpg}
         \caption{Pointing Game with $acc=100\%$.}
     \end{subfigure}
     \hfill
     \begin{subfigure}[t]{0.3\textwidth}
         \centering
         \includegraphics[width=\textwidth]{344059_swin_t_weighting_game_192.jpg}
         \caption{Weighting Game with $acc=19.2\%$.}
     \end{subfigure}
        \caption{Example of Pointing Game's limitations, when saliency map is not concentrated around one central point. Explanation was processed with class-guided XGrad-CAM \cite{XGradCam} and Swin-T \cite{Swin}}
        \label{fig:weighting-game-superior}
\end{figure}

When results of Weighting Game are compared to the ones of Pointing Game, we can immediately see that the values of Pointing Game are higher, implying that the task of Pointing Game is generally easier. When relative values are compared, it appears that generally CNNs tend to perform better with the chosen explainability methods, when using Pointing Game as the metric. This could be due to the very low resolution of CAM methods when using late layers of CNNs. This is emphasized with for example Figure \ref{fig:video-frames}, where saliency maps produced with ResNet50 are much more spread out, when compared to the ones produced by ViT-B/32. Pointing Game fails to quantify this loss in fidelity of saliency maps, as the accuracy values are binary.

Pointing Game is more prone to errors in measuring the accuracy of a saliency map. In cases where the saliency map is mostly concentrated around one point, there is a possibility to narrowly miss the correct segmentation mask, such as in Figure \ref{fig:pointing-game-misses}. Simultaneously, it fails to quantify cases where the saliency map is distributed all over the image, such as in Figure \ref{fig:weighting-game-superior}. In these cases, the classification of an explanation as either a hit or a miss is reductive to the quality of the explanation. Additional Pointing Game and Weighting Game visualizations are provided in the supplementary materials.

From observing the results of the Weighting Game, it shows that model architecture affects the results significantly, with Grad-Cam/XGrad-CAM functioning the best with CNN-based architectures, while Layer-CAM/Grad-CAM++ show great results with ViT-B/32. Swin-T shows poor results with all saliency methods, except for Ablation-CAM and Layer-CAM. Grad-CAM and Grad-CAM++ show results worse than randomly distributed saliency using Swin-T, which shows that the selection of the target layer could be poor or that the saliency methods simply do not function well with the architecture.

While CAM methods were initially designed for CNNs, they do show promise with vision transformers, as ViT-B/32 has the best results with three out of the five CAM methods. The results highlight the need to consider the model architecture in the process of choosing a saliency method, as even within the same family of architectures (CNN/ViT), the results vary significantly.

When only small objects (>10\% of the size of the image) are considered, the results of the Weighting Game change significantly. Notably, the relative accuracy of Layer-CAM with transformer models improves significantly when compared to other methods. Results of Ablation-CAM can also be seen deteriorating significantly for all model architectures. CNNs perform comparatively better in the Pointing Game, which highlights the lack of the full picture of explanation accuracy when using the Pointing Game as a metric. It does not properly capture, how the explanations are distributed, which can lead to results favoring certain method/model combinations over others.

\subsubsection{Explanation Stability Using 3D-Effect Videos}

\begin{table}[t]
\caption{Results of explanation stability. Consecutive frames values are mean correlations between saliency maps of consecutive frames from artificially generated 3D effect videos from still images. Zoom and pan transformation values are mean correlations calculated with Equation. \ref{eqn:stability-transformation}}
\label{table:stability}
\centering
\begin{tabularx}{\textwidth}{|p{0.2\textwidth}|X|X|X|X|X|}
\hline
& & \multicolumn{4}{|c|}{Explanation stability} \\
\hline
& & ResNet50 & VGG16-BN & ViT-B/32 & Swin-T \\
\hline
\multirow{2}{0.2\textwidth}{Grad-CAM \cite{GradCAM}} & 3D-effect & 0.961 & 0.938 & \textbf{0.966} & \textbf{0.878}\\
& Crop & 0.878 & 0.782 & \textbf{0.669} & \textbf{0.593}\\
\hline
\multirow{2}{0.2\textwidth}{Grad-CAM++ \cite{GradCAMPLUS}} & 3D-effect & \underline{0.987} & \textbf{0.984} & \underline{0.943} & 0.747\\
& Crop & \textbf{0.908} & \textbf{0.886} & \underline{0.634} & 0.459\\
\hline
\multirow{2}{0.2\textwidth}{Guided Backprop. \cite{GuidedBackProp}} & 3D-effect & 0.592 & 0.599 & 0.743 & 0.444\\
& Crop & 0.473 & 0.572 & 0.132 & 0.107\\
\hline
\multirow{2}{0.2\textwidth}{Layer-CAM \cite{LayerCam}} & 3D-effect & \textbf{0.987} & \underline{0.983} & 0.928 & \underline{0.845}\\
& Crop & \underline{0.908} & \underline{0.882} & 0.505 & \underline{0.475} \\
\hline
\multirow{2}{0.2\textwidth}{XGrad-CAM \cite{XGradCam}} & 3D-effect & 0.961 & 0.938 & 0.0732 & 0.0718\\
& Crop & 0.878 & 0.777 & 0.0238 & 0.0252\\
\hline
\end{tabularx}
\end{table}

Table. \ref{table:stability} shows results of explanation stability using 3D-effect videos and zoom/pan transformations. Ablation-CAM was omitted from the stability experiments due to the high inference time of the method. The results of the video stability show that Grad-CAM++/Layer-CAM are most stable for CNNs, slightly ahead of Grad-CAM/ XGrad-CAM, which are the most accurate in the results of the Weighting Game. This shows that stability does not necessarily result in greater accuracy. However, high stability does seem like a prerequisite for good quality explanations, as all methods with clearly worse stability show poor results in explanation accuracy. Interestingly, ViT-B/32 also shows worse stability with Layer-CAM when contrasted to the CNN models, while the accuracy of the method is significantly superior with ViT-B/32.

\subsubsection{Explanation Stability Using Zoom and Pan Transformation}

Pan and zoom effect transformations were created using a random resized crop with crop scale 0.75-0.9, 1:1 aspect ratio, and bilinear interpolation. For every image in COCO validation split, a random resized crop is applied, and saliency maps for the highest probability class are processed for both images. Then, the same crop is applied for the source image's saliency map. Finally, the correlation is calculated between the cropped saliency map and the cropped image's saliency maps, as the saliency maps should align at this point. Table. \ref{table:stability} shows the mean correlations for the method/model combinations.

The mean correlations are significantly lower than those found in the consecutive frames approach, which is sensible as the step taken in this method is far greater. However, the ranks of results also change significantly. Notably, the relative stability of transformer models drops clearly. This effect is especially seen with Layer-CAM and Guided Backpropagation, however, it is present with all other methods. The reason for this behavior could be explained via how the receptive fields operate within transformer based models. This is due to transformer receptive fields being more concentrated locally, when contrasting ViT-B/32 and ResNet50 receptive fields at later layers \cite{DoViTsSeeLikeCNNs}. More concentrated receptive field could yield to results that are less stable, but perhaps more capable of correctly explaining smaller objects, as seen in results of the Weighting Game with small objects in Table. \ref{table:weighting-game}.

\section{Conclusion}

We introduced a new method for more comprehensively assessing the accuracy of a class-guided saliency method, which calculates how much of the weight is contained within the correct class' segmentation mask. We also introduced two ways of assessing an explanation method's stability via transforming an image, either by a sufficiently small transformation or by a slightly larger zoom and pan operation. Our experiments show that, generally, sufficient stability is a prerequisite for accurate saliency maps. However, better stability does not always yield better results in explanation accuracy, as especially small objects are not always detected well with highly stable methods. We also highlight the importance of considering model architecture when choosing a saliency method, as the choice of model architecture can significantly affect which saliency methods give the best results.

\bibliographystyle{splncs04}