Posted here too: <https://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition>


Let $q\_0, q\_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are $L^2[0,1]$, i.e. $q\_0, q\_1 \in L^2([0,1],\mathbb{R}^n)$. Denote by $||.||$ the Euclidean distance in $\mathbb{R}^n$. Consider the following two minimization problems:


1) minimize $\int\_{0}^{1} || q\_0(t) - A q\_1(t) ||^2 dt $ over $A \in SO(n)$.


2) Consider $U \subset L^2([0,1],\mathbb{R}^n)$ to be the unit sphere in $L^2([0,1],\mathbb{R}^n)$. Let $q\_0, q\_1 \in U$ now. Then $Aq\_1 \in U$ as well for any $A \in SO(n)$. Consider the geodesic $c(A)$ in $U$ joining $q\_0, Aq\_1$. 
Minimize the length of $c(A)$.


How can we solve these two problems with/without, preferably with, using **singular value decomposition**? It was mentioned in the paper <http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5601739>


But I'd appreciate a detailed explanation for solving these two problems.



Two minimization problems using singular value decomposition
Problem **1** is essentially finite dimensional and quite geometric in nature.


For a real square matrix $M$ of order $n$ and $1\le p\le\infty$ denote $\big| M\big|\_p$ its [$p$-trace norm](https://en.wikipedia.org/wiki/Schatten_norm).


Consider the $n\times n$ matrix $Q$ with entries $Q\_{ij}:=(q\_0^i, q\_1^j)\_{ 2}:=\int\_0^1q\_0^i(x)q\_1^j(x)dx$. Then, for any $A\in SO(n)$


$$\|q\_0 - Aq\_1\|\_2^2:=\int\_0^1\big|q\_0(x) - Aq\_1(x)\big|^2dx=\|q\_0\|\_2^2 +\|q\_1\|\_2^2- 2\int\_0^1\sum\_{ij} A\_{ij}q\_0^i(x) q\_1^j(x)dx $$


$$=\|q\_0\|\_2^2 +\|q\_1\|\_2^2- 2 \sum\_{ij} A\_{ij}Q\_{ij}=\|q\_0\|\_2^2 +\|q\_1\|\_2^2- 2 \operatorname{tr}(A^TQ). $$


Therefore, the initial minimization reduces (up to an additive constant) to maximizing the trace of a matrix in the $SO(n)$-orbit of $Q$ by (left or right) multiplication. Consider first the case $\operatorname{det}(Q)\ge0$. For any $A\in SO(n)$ we have, by duality,


$$ \operatorname{tr}(A^TQ)\le |A|\_\infty |Q|\_1= |Q|\_1 .$$


The nuclear norm of $Q$ on the RHS is indeed attained by a suitable $A\in SO(n)$ on the LHS, and here the SVD of $Q$ may help. Write $Q=UDV^T$ with $U, V$ in $SO(n)$ and $D$ non-negative and diagonal, and choose $A:=UV^T$; then $A^TQ=VDV^T$ and by invariance of the trace $\operatorname{tr}(A^TQ)=\operatorname{tr}(D)= |Q|\_1 .$
$$\*$$


Note that the latter optimization problem is also equivalent to a point-set distance minimization, that is, finding the closest rotation to $Q$ in the Frobenius distance. Indeed, for any rotation $A$, since $\big|A\big|\_2^2=n$ we have


$$ \big| A-Q\big|\_2^2 = n + \big|Q\big|\_2^2 -2\operatorname{tr}(A^TQ)$$ so that maximizing $\operatorname{tr}(A^TQ)$ is again equivalent to minimizing a distance.


