\section{Introduction}
It is well known that the mapping class group $\M(S)$ of a closed orientable surface $S$ is generated by Dehn twists. This result was originally proved by Dehn \cite{Dehn} and rediscovered  by Lickorish \cite{Lick0}. On a closed nonorientable surface $N$, however, there are elements of the mapping class group $\M(N)$ which cannot be expressed as a product of Dehn twists. The first example of such element is the crosscap slide, also called Y-homeomorphism, introduced by Lickorish \cite{Lick1}. He proved \cite{Lick1,Lick2} that the twist subgroup $\T(N)$, generated by Dehn twists about all two-sided simple closed curves, has index 2 in $\M(N)$, and that $\M(N)$ is generated by Dehn twists and one crosscap slide. Later, Chillingworth \cite{Chill} found finite generating sets for $\M(N)$ and $\T(N)$. Korkmaz \cite{KorkHom,Kork} and Stukow \cite{Stu2,Stu3} extended Chillingworth's results to surfaces with punctures and boundary and also computed the abelianizations of $\M(N)$ and $\T(N)$.

In this paper we consider the subgroup $\Y(N)$ of $\M(N)$ generated by all crosscap slides.
Our main result is Theorem \ref{main}, which asserts that for a closed nonorientable surface $N$ of genus $g\ge 2$, $\Y(N)$ is equal to the level 2 subgroup $\Gamma_2(N)$ of $\M(N)$ consisting of those mapping classes which act trivially on $H_1(N;\Z_2)$. In particular $\Y(N)$ is a subgroup of finite index. We also prove that $\Y(N)$ is generated by involutions (Theorem \ref{invol}).

It is an interesting problem to find a finite generating set for $\Y(N)$ and also compute its abelianization. For an orientable surface $S$, the abelianization of the level 2 subgroup of $\M(S)$  was computed by Sato \cite{Sato}.

The paper is organised as follows. After Preliminaries, we prove in Section \ref{s3} that certain elements of $\M(N)$ belong to $\Y(N)$ and that the last group is generated by involutions. In Section \ref{s4} we study the level $m$ mapping class groups $\Gamma_m(S)$ of an orientable surface $S$. We use known results about generators of Torelli groups and the Bass-Milnor-Serre's solution to the congruence subgroup problem for the symplectic group, to obtain convenient generating sets of $\Gamma_m(S)$. In Section \ref{s5} we use lemmas proved in previous sections, as well as some results of McCarthy-Pinkall \cite{McCP} and
Korkmaz \cite{KorkHom}, to prove our main result, which identifies the group $\Y(N)$ with the level 2 mapping class group $\Gamma_2(N)$.

\medskip

\noindent{\bf Acknowledgment.} I was informed by Professor Susumu Hirose that he found a shorter proof of my Theorem \ref{main} using Nowik's result \cite[Theorem 3.6]{Now}. I thank him for the discussion and for the reference \cite{Now}.

\section{Preliminaries}
Let $F$ be a connected compact surface, possibly with
boundary. Define $\Dif(F)$ to be the group of all, orientation
preserving if $F$ is orientable, homeomorphisms $h\colon F\to F$
equal to the identity on the boundary $\bdr{F}$. The {\it mapping class
group} $\M(F)$ is the group of isotopy classes in $\Dif(F)$. By
abuse of notation we will use the same symbol to denote a
homeomorphism and its isotopy class. If $g$ and $h$ are two
homeomorphisms, then the composition $gh$ means that $h$ is applied
first.
A surface of genus $g$ with $n$ boundary components will be denoted by $S_{g,n}$ if it is orientable, or by $N_{g,n}$ if it is nonorientable. The genus of a nonorientable surface is the number of projective planes in the connected sum decomposition.

\subsection{Simple closed curves and Dehn twists.}
By a {\it simple closed curve} in $F$ we mean an embedding
$\gamma\colon S^1\to F\backslash\bdr{F}$. Note that $\gamma$ has an orientation; the
curve with the opposite orientation but same image will be denoted by
$\gamma^{-1}$.
By abuse of notation, we will often identify a simple closed curve with its
oriented image and also with its isotopy class.
According to whether a regular neighborhood of $\gamma$ is an annulus or a M\"obius strip, we call $\gamma$ respectively {\it two-} or {\it one-sided}.
We say that $\gamma$ is {\it nonseparating} if $F\backslash\gamma$
is connected and {\it separating} otherwise. If $\gamma=\bdr{M}$ for some subsurface $M\subset F$ then we call $\gamma$ {\it bounding}. If $n\le 1$ then every separating curve is bounding.

Given a two-sided simple closed curve $\gamma$, $T_\gamma$ denotes a Dehn
twist about $\gamma$. On a  nonorientable surface it is
impossible to distinguish between right and left twists, so the
direction of a twist $T_\gamma$ has to be specified for each curve
$\gamma$. Equivalently we may choose an orientation of a regular
neighborhood of $\gamma$. Then $T_\gamma$ denotes the right Dehn twist with
respect to the chosen orientation. Unless we specify which of the
two twists we mean, $T_\gamma$ denotes any of the
two possible twists. Recall that $T_\gamma$ does not depend on the orientation of $\gamma$.

{\it Bounding pair} is a pair $(\gamma,\gamma')$ of disjoint


\section{Introduction}
The faint and metal-poor satellites of massive galaxies are among the oldest structures of the Universe, whether they are dwarf galaxies or globular clusters (\citealt{white_rees_78}, \citealt{beasley02}, \citealt{mo10}). Because the faintest, low-mass galaxies are, in a $\Lambda$CDM context, dominated by dark matter (DM), they can be the key to constraining the nature of the DM particle and more broadly the properties of our Universe (\citealt{klypin99}, \citealt{geringer-sameth15a}, \citealt{bullock17}, \citealt{nadler19}). The old age of the dwarf galaxies further implies that they have hosted Population III stars (\citealt{ishiyama16}), while their low mass and metallicity suggest that the pollution from successive supernovae is limited in these systems (\citealt{dekel86}, \citealt{frebel_norris15}). As a consequence, they are unique laboratories to study and quantify the different evolutionary pathways of stellar formation (\citealt{roederer16}, \citealt{webster16}, \citealt{ji19}).

On the other hand, globular clusters are usually not thought to be DM-dominated \citep{moore96} and the faintest of them are often modelled as simpler systems with a single stellar population (\citealt{gratton07}, \citealt{carretta09}, \citealt{cohen10}, \citealt{willman_strader12}). They are important for various reasons. They give us insight in the mode of stellar formation in the early Universe and its similarities and differences with star formation today (\citealt{gieles18},\citealt{gratton19}). The close link of their properties to those of their host galaxies has become clear over the years: for example, the number of clusters are related to the properties of their host galaxies (\citealt{brodie_huchra91}, \citealt{blakeslee97}). They are thereby direct probes of their galaxy' properties and therefore indirect probes for cosmology (\citealt{cote02_cluster}, \citealt{villaume19}, \citealt{riley20}). Old globular clusters are also witnesses of the build-up of massive galaxies halos, which are thought to have formed by disrupting and accreting faint stellar systems (\citealt{beasley18}, \citealt{pfeffer18}). They can also offer a unique insight into the observational validation of stellar population models \citep{chantereau16}.

The diversity of the properties of the MW faint satellites has been progressively unveiled over the past decades. For dwarf galaxies, it all started with Sculptor and Fornax, the very first dwarf spheroidal galaxies discovered (\citealt{shapley38a}, \citealt{shapley38b}) to the most recent discoveries (\citealt{willman05b}, \citealt{zucker06}, \citealt{belokurov06}, \citealt{laevens15}, \citealt{drlica-wagner15}, \citealt{kim15b}, \citealt{homma16}, \citealt{torrealba18}, \citealt{homma19}). Their observed diversity in size, luminosity and mass is not a surprise as it naturally arises in various simulations trying to reproduce our Local Universe (\citealt{springel08}, \citealt{vogelsberger14}, \citet{eagle15}, \citealt{wheeler19}). For the Milky Way's globular clusters, \citet{harris10} has compiled the properties of 157 systems and summarized years of efforts of the community to characterize and understand them. That list will only grow longer in the future considering the recent discoveries of new clusters (\citealt{martin16c}, \citealt{koposov17}, \citealt{mau19}), especially with the advent of the Legacy Survey of Space and Time (LSST) in the incoming years (\citealt{ivezic08}, \citealt{simon19}) \\

As increasingly faint MW satellites are discovered, the difficulties in classifying them have grown. Systems fainter than $ M_V\sim -6$ mag are often challenging to classify as globular cluster or galaxy. This issue was notably addressed by \citet{cote02} and \citet{gilmore07} and is commonly known as the ``valley of ambiguity". Distinguishing faint satellite galaxies from globular clusters relies on two main diagnostics: (i) evidence for the presence of substantial amounts of dark matter, and (ii) the presence of a substantial dispersion in metallicity, a feature usually associated with recurrent star formation.  This requires combining deep photometry with extensive spectroscopic campaigns. Dynamical evidence for an excess of mass over that of the stellar component, would then be attributed, in the standard cosmological model, to the presence of a massive DM halo \citep{willman_strader12} and would favour a dwarf galaxy scenario The first measurement of the velocity dispersion of a non-classical MW dwarf galaxy, undertaken by \citet{kleyna05}, showed that Ursa Major~I is indeed DM-dominated. This effort was repeated for most systems known at the time, enriching our knowledge of the MW satellites' properties (\citealt{martin07}, \citealt{simon07}, \citealt{koposov11}, \citealt{simon11}, \citealt{martin16_tri}), an effort that would also rely on other tracers such as the metallicity and the metallicity dispersion of these faint satellites (\citealt{kirby08}, \citealt{koposov15}, \citealt{walker16}). However, in practice, the velocity and metallicity dispersions can be challenging to constrain due to the low number of member stars observed with spectroscopy for a significant fraction of the known faint MW satellites. This problem, that first arose with the study of the kinematics of Segue~2 \citep{kirby13a} still limits our understanding of some of them. The examples over the last few years are numerous, with Draco~II (\citealt{martin16_dra}, \citealt{longeard18}), Tucana~III \citep{simon17}, Colomba~I or Horologium~II \citep{fritz19} that are still not well understood, despite extensive studies with photometry and spectroscopy.

In that context, we present a new spectroscopic study of the faint MW satellite Sagittarius~II (Sgr~II) discovered by \citet{laevens15}, and studied in depth by \citet[L20]{longeard20}. L20 used deep photometry, multi-object spectroscopy and the metallicity-sensitive, narrow-band photometry provided by the Pristine survey \citep{starkenburg17} to refine the structural properties of Sgr~II and infer its dynamical and metallicity properties for the first time. Its velocity dispersion ($\sigma^\mathrm{L20}_\mathrm{vr} = 2.7^{+1.3}_{-1.0}$ km s$^{-1}$) favoured the existence of a low-mass DM halo and therefore a dwarf galaxy scenario. The metallicity of the satellite was found to be in good agreement with the metallicity-luminosity relation of dwarf galaxies of \citet{kirby13}, while its metallicity dispersion was deemed small ($\sigma^\mathrm{L20}_\mathrm{[Fe/H]} = 0.10^{+0.06}_{-0.04}$) but resolved. This last result, however, could be inflated by systematics, an underestimation of the individual uncertainties on the spectroscopic metallicities, or just by the small sample of six stars used to perform the metallicity analysis. If L20 marginally favour Sgr~II to be a very low-mass galaxy, an independent study of the satellite presented in an American Astronomical Society meeting \citep{fu_simon_19}, yet to be published, unequivocally found that Sgr~II is a globular cluster, based on an extremely low metallicity dispersion measurement from their own spectroscopic sample ($< 0.08$ at the 95\% confidence level). Furthermore, if the satellite was found to be physically compact by L20 (r$^\mathrm{L20}_h = 35.5 ^{+1.4}_{-1.2}$ pc), but still possibly within the realm of dwarf galaxies, an estimation of the heliocentric distance of the satellite based on the identification of five RR Lyrae stars potentially members of Sgr~II \citep[V20]{vivas20} would place the system closer than the estimate of L20 ($m-M = 18.97 \pm 0.20$ vs. $19.32^{+0.03}_{-0.02}$ mag, i.e. a difference of $\sim 10$ kpc). This would result in a slight overestimation of its physical size (r$^\mathrm{V20}_h = 30.7 ^{+2.7}_{-2.9}$ pc) and bring the system closer to the realm of old, metal-poor globular clusters. For the rest of this paper, the results of L20 will be used. In many ways, Sgr~II is shrouded in mystery, and the following work will attempt to close the case on the nature of this faint and elusive satellite.

\section{Data selection and acquisition}

\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{spectrum.pdf}}
\caption{Non sky-substracted spectrum of a Sgr~II member star with a S/N of $\sim 65$ focusing on the Calcium II triplet lines. This star has a metallicity of [Fe/H] $= -2.32 \pm 0.18$ using the calibration of \citet{starkenburg10} as detailed in section 3.2.}
\label{spectrum}
\end{center}
\end{figure}

\begin{figure*}
\begin{center}
\centerline{\includegraphics[width=\hsize]{CMDs_field.pdf}}
\caption{{\textit{Left panel: }}  Spatial distribution of Sgr~II-like stars, i.e. stars with a CMD probability membership of 1 per cent or higher. The field is centered on ($\alpha_0 = 298.16628^\circ, \delta_0 = -22.89633^\circ$). The red contour defines two half-light radii ($r_h \sim 1.7'$) of the satellite, as inferred by L20. The locations of all stars in the spectroscopic dataset are indicated as large, coloured markers. Squares represent the DEIMOS observations from L20, while the circles stand for the new FLAMES data presented in this work. All stars observed spectroscopically are colour-coded according to their heliocentric velocities. Cyan stars have a radial velocity corresponding to Sgr~II at around $-177$ km s$^{-1}$. Filled squares and circles are the member stars of the satellite. {\textit{Right panel: }} CMD of Sgr~II within two half-light radii (grey) superimposed with the entire spectroscopic dataset. The best-fitting Darmouth isochrone from L20 ($12$ Gyr, [Fe/H] $= -2.35$, $[\alpha$/Fe] $= 0$, $m-M = 19.32$ mag) is shown as a black dashed line and is perfectly compatible with the identified members of Sgr~II.}
\label{cmd}
\end{center}
\end{figure*}

\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{CaHK_FLAMES.pdf}}
\caption{Pristine colour-colour diagram of Sgr~II showing field stars (grey) and the spectroscopic dataset with large, coloured markers. Member stars of Sgr~II are shown with filled markers. While the x-axis is a temperature proxy, the y-axis contains the Pristine metallicity-sensitive, narrow-band photometry denoted $c_0$, and therefore the metallicity information. As a result, stars are distributed according to their metallicity: the ones around solar metallicity form the stellar locus where most of the grey field stars are located. As a star goes upwards in the diagram (i.e. lower y-axis value) for a fixed temperature, its metallicity decreases. This is represented more visually with the three iso-metallicity sequences in black dashed lines, showing where stars with a [Fe/H] of $-4.0$, $-3.0$ and $-2.0$ should be located. This plot shows the efficiency of Pristine in identifying Sgr~II members, as almost all DEIMOS stars dynamically compatible with the satellite come out as metal-poor.}
\label{cahk}
\end{center}
\end{figure}

The spectroscopy used in this work is a combination of two datasets. The first data were observed with the DEep Imaging Multi-Object Spectrograph (DEIMOS, \citealt{faber03}), already detailed and analysed in L20. The second dataset is composed of new observations performed with the Fibre Large Array Multi Element Spectrograph \citep[FLAMES]{pasquini02} mounted on the Very Large Telescope (VLT). The \mbox{GIRAFFE}/HR21 grating, with a resolution of R $\sim 18000$, a central wavelength of $8757$ $\; \buildrel \circ \over {\rm A}$ $\; $and a bandwidth between 8484 and 9001$\; \buildrel \circ \over {\rm A}$$\;$was used to resolve the Calcium II triplet infrared lines. These FLAMES observations were conducted during three different nights to attempt to detect binary stars: on the 22.06.16 ($1 \times 2610$s of integration), the 21.07.16 ($2 \times 2610$s) and the 30.07.16 ($2 \times 2610$s).

The data were reduced by the European Southern Observatory team using their pipeline specifically tailored to FLAMES data \citep{melo09}. The spectrum of a Sgr~II member star is displayed in Figure \ref{spectrum} to illustrate the quality of the data. In order to infer the radial velocities, the equivalent widths and their respective uncertainties, the method already used in L20 was used and consists of determining the combination of Gaussian profiles that best fits a continuum composed of the Calcium II triplet lines at rest as the unique spectral features. It also includes a correction to account for the non-gaussianity of the wings' lines. The typical uncertainty on the radial velocities for the FLAMES dataset is of the order of $1$ km s$^{-1}$, and all the stars with signal-to-noise (S/N) ratio below 3 per resolution element are discarded from the sample. In the same fashion as in \citet{simon_geha07} and L20, the systematic error on the velocities is determined from stars observed more than once in the FLAMES sample. The resulting systematics are a bias of $ b = 0.4 \pm 0.3$ km s$^{-1}$ and a standard deviation of $ \delta_\mathrm{thr} = 0.8 \pm 0.1$ km s$^{-1}$. \\

To build our list of FLAMES targets, we used the locations of stars in the colour-magnitude diagram (CMD) of Sgr II built from deep MegaCam photometry and shown in the right panel of Figure \ref{cmd}, along with their spatial distribution in the left panel. More importantly, this spectroscopic study benefits from the use of the narrow-band, metallicity-sensitive photometry provided by the Pristine survey \citep{starkenburg17}.  All stars observed in Pristine have a photometric metallicity measurement, as illustrated in the colour-colour diagram shown in Figure \ref{cahk}. In this diagram, stars are distributed according to their metallicities, from the most metal-rich at the bottom of the plot, to the most metal-poor at the top, down to a metallicity of [Fe/H] $\sim -4.0$. Using the prior knowledge that Sgr~II is indeed a very metal-poor satellite ([Fe/H]$_\mathrm{SgrII} \sim -2.28$, L20), the target list was built so that the stars observed are confirmed to be very metal-poor according to the Pristine survey. In doing so, we increase the probability of finding new members (\citealt{youakim17}, \citealt{aguado19}). The DEIMOS spectroscopy supplementing this dataset consists only of the DEIMOS stars identified as non variables, non binaries and metal-poor according to Pristine ([Fe/H]$_\mathrm{CaHK} < -1.6$) by L20.\\

In order to clean the FLAMES data, we check for the existence of binaries in the sample. For a given star, each measurement from the $k$-th epoch is assumed to be reasonably well-described by a Gaussian distribution centered on the radial velocity measurement and a standard deviation corresponding to its uncertainty. Then, the probability that the measurement $k$ is discrepant from that of the same star at another epoch $j$ at the $2\sigma$ level is computed. This probability corresponds to the star being variable between epochs $k$ and $j$. The same procedure is used to compute the probability that each velocity measurement of a given epoch is compatible with all the others within $2\sigma$, i.e. the probability that it is not varying. If the product of the probabilities representing the hypothesis "variable" is greater than the hypothesis "non-variable", then the star is considered a binary star and discarded from the main sample so that their variability does not affect the dynamical analysis of Sgr~II. We find a total of 6 potential binaries in the sample, however, none have the right velocity to be a member of Sgr~II.

The final spectroscopic sample consists of 113 stars, with 47 new stars observed with FLAMES as detailed in Table \ref{table1} with four stars out of those 47 in common with L20's sample. As a consistency check, we compare the velocities obtained with FLAMES and DEIMOS for these four stars and find no statistical differences between the two samples.

\section{Results}

In this section, we constrain the systemic heliocentric velocity of Sgr~II and its associated velocity dispersion.

\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{v_vs_r.pdf}}
\caption{Heliocentric velocities vs. radial distance to Sgr~II's centroid for FLAMES (blue circles) and DEIMOS (grey squares) datasets. The addition of the FLAMES data doubles the number of identified members for Sgr~II. All error bars not appearing in this plot are smaller than the size of the circles/squares.}
\label{histos_vel}
\end{center}
\end{figure}

\subsection{Dynamical properties}

\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{pdfs_vel.pdf}}
\caption{1D marginalised PDFs of the systemic radial velocity (top left panel) and velocity dispersion (top right panel) of Sgr~II from this work as black solid lines, and L20 in grey dashed lines using only DEIMOS data. The expected velocity dispersion for a Sgr~II-like typical MW globular cluster ($1.1 \pm 0.1$ km s$^{-1}$) is indicated as the red shaded region. The bottom panel shows the observed velocity dispersions of most globular clusters (black) and confirmed dwarf galaxies (blue) of the MW as a function of their absolute magnitudes. The properties of the globular clusters were taken from the \citet{harris10} catalog and references therein. The dwarf galaxy measurements come from \citet{simon19} and references therein. In this plot, the location of Sgr~II is indicated by the red diamond.}
\label{pdfs_vel}
\end{center}
\end{figure}

All stars in our spectroscopic sample with a heliocentric velocity are shown in Figure \ref{histos_vel}, along with a more detailed view. As already detailed in section 2, the final sample consists of all non-binary, non-HB stars compatible with Sgr~II's sequence in the CMD and that appear as very metal-poor using the CaHK photometry. Stars with mediocre CaHK photometry from the DEIMOS catalog are also included as their photometric metallicities are not reliable. Thanks to the photometry provided by the Pristine survey, the FLAMES data are already quite clean with only a few clear contaminants discrepant from the Sgr~II population at around $-180$ km s$^{-1}$. When available, the proper motions provided by the Gaia Data Release 2 \citep{brown18} are also used to filter out contaminants. This procedure allows us to get rid of one star with CMD and metallicity properties strongly compatible with Sgr~II, but a proper motion unequivocally discrepant from that of the satellite (\citealt{massari18}, L20).

In order to derive the dynamical properties of Sgr~II, the velocity distribution of our dataset and shown in Figure \ref{histos_vel} is assumed to be the sum of two Gaussian distributions, one standing for Sgr~II's population and the other for the MW contamination. The favoured dynamical model is obtained through a Monte Carlo Markov Chain \citep[MCMC]{hastings70} algorithm. \\

The Probability Distribution Functions (PDFs) of the systemic heliocentric velocity and the velocity dispersion of Sgr~II are displayed in the top panels of Figure \ref{pdfs_vel}. The resulting systemic velocity $\langle \mathrm{v}_\mathrm{SgrII} \rangle$ is $-177.2_{-0.6}^{+0.5}$ km s$^{-1}$ and is perfectly compatible with the one of L20. The velocity dispersion found in this work is also compatible with L20 ($\sigma^\mathrm{SgrII}_\mathrm{v} = 2.7_{-1.0}^{+1.3}$ km s$^{-1}$), but with much tighter constraints, it is measured to be $\sigma^\mathrm{SgrII}_\mathrm{v} = 1.7 \pm 0.5$ km s$^{-1}$. Using the formalism of \citet{wolf10} which assumes dynamical equilibrium and a flat velocity dispersion profile, this results into a mass-to-light (M/L) ratio of $3.5^{+2.7}_{-1.6}$ M$_{\odot}$ L$_{\odot}^{-1}$. Sgr~II shows no sign of a velocity dispersion gradient as the determination of its dynamical properties by selecting stars inside and outside 1 arcminute yields no statistical difference. Furthermore, adding another contaminating population to the likelihood model does not change our results.

We classify as a member of Sgr~II any star with  line-of-sight velocity and proper motion membership probabilities (when available) above 50\%. We identify $22$ member stars in the FLAMES data. 3 of them were already identified in the L20 dataset, leading to the identification of 19 new members. Combined with the spectroscopically confirmed members of L20, 43 stars are confirmed to belong to Sgr~II, including binaries and horizontal branch (HB) stars. This underlines the importance of the Pristine photometry. To estimate the success rate of the FLAMES data in identifying new members, we do not simply take the fraction of confirmed members over the overall number of stars observed since the sample of Sgr~II candidates identified before observation was not large enough to fill all the FLAMES fibers. Among the 47 new stars observed, only $32$ were considered as promising Sgr~II candidates. Therefore, it yields a success rate of $\sim 60 \%$ for the FLAMES data sample based on the Pristine selection. In comparison, the DEIMOS dataset of L20, solely based on a CMD-based selection, had a 20\% success rate. The contrast is even more striking when considering that the DEIMOS selection focuses on the central region of the satellite and is therefore more likely to find Sgr~II stars, while the FLAMES observations had a much wider field of view and aim to find stars located much further away than Sgr~II's centroid.

\subsection{Metallicity properties}

\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{FeH_individual_metallicities.pdf}}
\caption{Individual metallicities of all bright stars in the spectroscopic sample used to derive the metallicity properties of Sgr~II using the calibration of \citet{starkenburg10}. Each measurement is modelled by a Gaussian with a mean corresponding to the favoured metallicity of the star, and a standard deviation equal to its uncertainty. Stars from the L20 sample are shown as blue dots, while the new FLAMES stars are represented in red.}
\label{pdfs_feh}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\centerline{\includegraphics[width=\hsize]{pdfs_metal.pdf}}
\caption{One-dimensional PDFs of the systemic metallicity (left panel) and metallicity dispersion (right panel) of Sgr~II with spectroscopy. The 95\% C.L. on the metallicity dipersion is hown as a black dashed line. }
\label{metallicities}
\end{center}
\end{figure}

In the following section, the mean stellar metallicity of Sgr~II and its metallicity dispersion are derived using our new FLAMES data. To do so, we select only non-binary, non-HB stars with a spectral signal-to-noise ratio greater than 12 and with $g_0 < 20.5$ (i.e. $\sim$ 1 mag below the HB of Sgr~II). This yields a subsample of 17 stars to extract a measurement of the spectroscopic metallicity. This procedure is performed through the use of the empirical calibration of \citet{starkenburg10} that uses the equivalent widths (EWs) of the Calcium II triplet lines to deduce a measurement of the metallicity of a star. The uncertainties on the coefficients defining the polynomials to transform a set of EWs into a [Fe/H]$_\mathrm{spectro}$ are folded in with the uncertainties on the EWs through a Monte Carlo procedure to obtain the final uncertainties $\delta_\mathrm{[Fe/H]}$ on each individual spectroscopic metallicity measurement. We take the value of 8\% reported by \citet{starkenburg10} to be the uncertainty on each coefficients. We note that these uncertainties do not include several systematics (due to, for instance, the continuum placement, or the detailed chemical abundance pattern of the star when transforming a Ca II measurement into [Fe/H]). Figure \ref{pdfs_feh} shows the spectroscopic metallicities of the 17 stars and their uncertainties assuming each measurement can be modelled by a Gaussian centered on [Fe/H]$_\mathrm{spectro}$ and with a standard deviation corresponding to $\delta_\mathrm{[Fe/H]}$. For stars observed more than once, a metallicity measurement is derived for each epoch and we verify that each epoch yields a metallicity compatible with the others. This procedure highlights two stars with discrepant metallicity measurements at different epochs. We decide to discard those stars for the rest of the analysis, however, including them has a very limited impact on the results. None of the 4 stars in common between the FLAMES and DEIMOS samples have a spectroscopic metallicity measurement.

Since all 15 stars in the subsample are likely Sgr~II members, their metallicity distribution is assumed to be only reflective of Sgr~II population, which is therefore modelled with a Gaussian distribution weighted with the membership probability of each star according to L20. We find a spectroscopic systemic metallicity of $\langle$[Fe/H]$^\mathrm{SgrII}_\mathrm{spectro} \rangle = -2.23 \pm 0.06$, and an unresolved metallicity dispersion with $\sigma^\mathrm{SgrII}_\mathrm{spectro} < 0.20$ at the 95\% confidence limit. The PDFs corresponding to these results are shown in Figure \ref{metallicities}. The metallicity dispersion is unresolved, and is constrained to be below 0.20 dex at the 95\% confidence limit (C.L.).

\section{Discussion}
Throughout this paper, we analyse the dynamical and metallicity properties of the faint MW satellite Sgr~II. To this end, we combine the Keck II/DEIMOS spectroscopic dataset of our previous analysis of the satellite with new VLT/FLAMES data. These FLAMES observations are the first ones to be carried out by selecting \textit{a priori} the interesting candidates using the narrow-band, metallicity sensitive photometry of the Pristine survey for a faint satellite of the MW.  The spectroscopic observations presented in this work are decisive as they double the overall sample of members stars known. Nine new stars are bright enough to estimate their spectroscopic metallicities with the empirical Calcium II triplet calibration of \citet{starkenburg10}, while the sample of L20 only consists of 6 stars. Therefore, this study considerably enlarges the statistics available for the satellite and allows us to put much tighter constraints on the dynamical and metallicity properties of SgrII.

The systemic radial velocity of Sgr~II is found to be $\langle \mathrm{v}_\mathrm{SgrII} \rangle = -177.2_{-0.6}^{+0.5}$ km s$^{-1}$. The velocity dispersion of $\sigma_\mathrm{v}^\mathrm{SgrII} = 1.7 \pm 0.5$ km s$^{-1}$ is well among the range observed for the MW globular clusters at the same luminosity, as shown in Figure \ref{pdfs_vel}. It translates into a M/L ratio of $3.5^{+2.7}_{-1.6}$ M$_{\odot}$ L$^{-1}_{\odot}$, thus suggesting that the dynamics of the satellite is not mainly driven by a DM halo. We identify 19 new members in the faint system, for a total of 43 confirmed spectroscopic Sgr~II members. With a fraction of member identified over the entirety of the FLAMES sample of $60$\%, the \textit{a priori} selection using Pristine performs three times better than the simple CMD-based selection of L20, despite a riskier observational strategy through the search for stars located beyond two half-light radii of the satellite. Four of these stars are identified, with the outermost one lying at $6.2$r$_{h}$. 15 out of the 43 members have good enough quality spectra to measure the EWs of their Calcium II triplet and measure their metallicity using the empirical calibration of \citet{starkenburg10}. We find a systemic metallicity of $-2.23 \pm 0.07$ and constrain the metallicity dispersion of the satellite to be less than $0.20$ at the $95$\% confidence limit.

 The velocity dispersion as inferred by L20 is larger than the one found in the present analysis. Furthermore, L20's metallicity analysis indicates that there are multiple stellar populations in the system, as the metallicity dispersion is resolved both with CaHK photometry and spectroscopy. This is not the case anymore in this work as the spectroscopic metallicity distribution of our dataset is well-described with only one stellar population despite the addition of 9 new stars with spectroscopic metallicites. Moreover, Figure \ref{pdfs_feh} shows that no star with a metallicity below $-2.5$ is confidently detected, a limit commonly attributed to the lowest metallicity achievable by globular clusters (\citealt{harris10} and references therein, \citealt{beasley19}). Therefore, we are able to conclude that Sgr~II is a globular cluster.

Nonetheless, the satellite is still quite extended when compared to other MW globular clusters of the same luminosity. Within a range of one magnitude around Sgr~II's absolute magnitude, the largest MW cluster is Pal~5 (M$_V \sim -5.2 $), a fairly metal-poor system ([Fe/H] $\sim -1.4$) with a size of $\sim 19.2$ pc, while most of the others have a size below $10$ pc \citep{harris10}. Whether we consider the distance of L20 or \citet{vivas20}, Sgr~II is still at least 1.5 times more extended than Pal~5. Sgr~II is a case in point of the ambiguity surrounding the faint MW satellites discovered in recent years, with a somewhat larger half-light radius than expected from GCs at that luminosity, a metallicity perfectly compatible with the luminosity-metallicity relation of dwarf galaxies \citep{kirby13}, and velocity and metallicity dispersions challenging to resolve. Though we conclude that Sgr~II is an old and metal-poor globular cluster, the system remains interesting in many aspects, from its possible association to the Sgr stream to an even deeper understanding of the unusually large cluster, which could build a bridge to the still exclusive definitions of clusters and galaxies of the MW.

\section*{Acknowledgments}
We gratefully thank the CFHT staff for performing the observations in queue mode, for their reactivity in adapting the schedule, and for answering our questions during the data-reduction process.

NFM, RI, and NL gratefully acknowledge support from the French National Research Agency (ANR) funded project ``Pristine'' (ANR-18-CE31-0017) along with funding from CNRS/INSU through the Programme National Galaxies et Cosmologie and through the CNRS grant PICS07708. ES gratefully acknowledges funding by the Emmy Noether program from the Deutsche Forschungsgemeinschaft (DFG). This work has been published under the framework of the IdEx Unistra and benefits from a funding from the state managed by the French National Research Agency as part of the investments for the future program. The authors thank the International Space Science Institute, Berne, Switzerland for providing financial support and meeting facilities to the international team ``Pristine''.

Based on observations collected at the European Southern Observatory under ESO programme(s) 099.B-0690(A).

Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made pos- sible by the generous financial support of the W. M. Keck Foundation. Furthermore, the authors wish to recognize and acknowledge the very significant cultural role and rever- ence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.

Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.

This work has made use of data from the European Space Agency (ESA)
mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by
the {\it Gaia} Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding
for the DPAC has been provided by national institutions, in particular
the institutions participating in the {\it Gaia} Multilateral Agreement.

\clearpage

\newpage

\begin{table*}

\caption{Properties of the new FLAMES spectroscopic sample. The Pristine metallicity of a given star is indicated only if [Fe/H]$_\mathrm{CaHK} < -1.0$. The individual spectroscopic metallicity is reported for stars with S/N $>= 12$ and $g_0 <= 20.5$ mag. The systematic threshold $\delta_{\mathrm{thr}}$ is not included in the velocity uncertainties presented in this table.
\label{table1}}

\setlength{\tabcolsep}{2.5pt}
\renewcommand{\arraystretch}{0.3}
\begin{sideways}
\begin{tabular}{cccccccccccccc}
\hline
RA (deg) & DEC (deg) & $g_0$ & $i_0$ & $CaHK_0$ & $\mathrm{v}_{r} ({\rm\,km \; s^{-1}})$ & $\mu_{\alpha}^{*}$ (mas.yr$^{-1}$) & $\mu_{\delta}$ (mas.yr$^{-1}$) &  S/N & [Fe/H]$_{\mathrm{CaHK}}$ & [Fe/H]$^\mathrm{S10}_\mathrm{spectro}$ & Member\\
\hline

298.37718 & $-$21.98293 & 21.05 $\pm$ 0.01 & 20.42 $\pm$ 0.01 & 21.74 $\pm$ 0.04 & -96.5 $\pm$ 1.4 & 5.0 $\pm$ 4.0 & -5.1 $\pm$ 2.1 & 3.5 & $-$2.50 $\pm$ 0.12 & --- &  N  \\ \\
298.28281 & $-$21.92996 & 19.57 $\pm$ 0.01 & 18.80 $\pm$ 0.01 & 20.45 $\pm$ 0.02 & -177.8 $\pm$ 0.6 & -0.2 $\pm$ 0.5 & -1.1 $\pm$ 0.3 & 24.4 & $-$2.58 $\pm$ 0.12 & -2.2 $\pm$ 0.11 &  Y  \\ \\
298.24280 & $-$21.97804 & 19.75 $\pm$ 0.01 & 19.05 $\pm$ 0.01 & 20.67 $\pm$ 0.02 & 34.1 $\pm$ 2.4 & -1.9 $\pm$ 2.7 & -1.9 $\pm$ 1.3 & 12.7 & $-$1.90 $\pm$ 0.12 & -1.5 $\pm$ 0.21 &  N  \\ \\
298.20794 & $-$21.86545 & 19.18 $\pm$ 0.01 & 18.21 $\pm$ 0.01 & 20.64 $\pm$ 0.02 & -58.3 $\pm$ 0.5 & -0.6 $\pm$ 0.2 & -1.0 $\pm$ 0.1 & 48.9 & $-$1.25 $\pm$ 0.12 & -1.26 $\pm$ 0.16 &  N  \\ \\
298.18438 & $-$21.93702 & 19.93 $\pm$ 0.01 & 19.29 $\pm$ 0.01 & 20.80 $\pm$ 0.02 & -195.5 $\pm$ 3.3 & --- $\pm$ --- & --- $\pm$ --- & 6.3 & $-$1.72 $\pm$ 0.12 & --- &  N  \\ \\
298.18958 & $-$21.84866 & 21.18 $\pm$ 0.01 & 20.66 $\pm$ 0.01 & 21.84 $\pm$ 0.04 & -268.8 $\pm$ 16.9 & 1.7 $\pm$ 0.2 & -2.0 $\pm$ 0.1 & 3.1 & $-$2.01 $\pm$ 0.12 & --- &  N  \\ \\
298.05410 & $-$21.87831 & 18.29 $\pm$ 0.01 & 17.48 $\pm$ 0.01 & 19.51 $\pm$ 0.01 & 91.0 $\pm$ 0.3 & --- $\pm$ --- & --- $\pm$ --- & 79.8 & $-$1.34 $\pm$ 0.12 & -1.43 $\pm$ 0.16 &  N  \\ \\
298.08415 & $-$21.96661 & 18.93 $\pm$ 0.01 & 18.12 $\pm$ 0.01 & 20.03 $\pm$ 0.01 & -181.7 $\pm$ 0.4 & -12.8 $\pm$ 0.7 & -18.6 $\pm$ 0.4 & 60.6 & $-$1.99 $\pm$ 0.12 & -1.8 $\pm$ 0.13 &  N  \\ \\
298.00202 & $-$21.92952 & 19.33 $\pm$ 0.01 & 18.50 $\pm$ 0.01 & 20.37 $\pm$ 0.02 & 139.8 $\pm$ 0.4 & --- $\pm$ --- & --- $\pm$ --- & 31.3 & $-$2.28 $\pm$ 0.12 & -2.12 $\pm$ 0.11 &  N  \\ \\
298.01422 & $-$21.89008 & 19.93 $\pm$ 0.01 & 19.30 $\pm$ 0.01 & 20.73 $\pm$ 0.02 & -208.9 $\pm$ 1.0 & -4.7 $\pm$ 0.2 & -6.0 $\pm$ 0.1 & 16.4 & $-$2.00 $\pm$ 0.12 & -2.24 $\pm$ 0.09 &  N  \\ \\
298.01803 & $-$21.92193 & 20.17 $\pm$ 0.01 & 19.47 $\pm$ 0.01 & 21.06 $\pm$ 0.03 & 145.7 $\pm$ 0.9 & 4.2 $\pm$ 1.2 & -6.5 $\pm$ 0.9 & 14.6 & $-$1.95 $\pm$ 0.12 & -1.13 $\pm$ 0.22 &  N  \\ \\
298.36550 & $-$22.00547 & 20.22 $\pm$ 0.01 & 19.60 $\pm$ 0.01 & 20.98 $\pm$ 0.02 & 112.9 $\pm$ 2.1 & -0.6 $\pm$ 0.6 & -1.4 $\pm$ 0.4 & 15.3 & $-$2.10 $\pm$ 0.12 & -2.36 $\pm$ 0.12 &  N  \\ \\
298.36878 & $-$22.01959 & 21.36 $\pm$ 0.01 & 20.76 $\pm$ 0.01 & 21.95 $\pm$ 0.04 & -43.6 $\pm$ 3.0 & 4.8 $\pm$ 3.1 & 2.9 $\pm$ 1.8 & 3.6 & $-$3.11 $\pm$ 0.12 & --- &  N  \\ \\
298.27562 & $-$22.01614 & 20.23 $\pm$ 0.01 & 19.42 $\pm$ 0.01 & 21.40 $\pm$ 0.03 & -82.3 $\pm$ 2.8 & -0.6 $\pm$ 1.3 & -11.3 $\pm$ 0.8 & 4.1 & $-$1.58 $\pm$ 0.12 & --- &  N  \\ \\
298.27457 & $-$22.00481 & 20.67 $\pm$ 0.01 & 20.10 $\pm$ 0.01 & 21.34 $\pm$ 0.03 & 54.2 $\pm$ 1.5 & --- $\pm$ --- & --- $\pm$ --- & 10.1 & $-$2.23 $\pm$ 0.12 & --- &  N  \\ \\
298.23845 & $-$22.11143 & 20.69 $\pm$ 0.01 & 20.06 $\pm$ 0.01 & 21.40 $\pm$ 0.03 & -175.5 $\pm$ 0.9 & -0.6 $\pm$ 1.3 & -0.0 $\pm$ 0.8 & 10.8 & $-$2.41 $\pm$ 0.12 & --- &  Y  \\ \\
298.25661 & $-$22.0404 & 20.73 $\pm$ 0.01 & 20.19 $\pm$ 0.01 & 21.42 $\pm$ 0.03 & -34.7 $\pm$ 1.4 & 0.4 $\pm$ 1.1 & 0.0 $\pm$ 0.7 & 6.1 & $-$2.03 $\pm$ 0.12 & --- &  N  \\ \\
298.23141 & $-$22.10773 & 20.75 $\pm$ 0.01 & 20.12 $\pm$ 0.01 & 21.53 $\pm$ 0.03 & -69.0 $\pm$ 1.7 & -1.1 $\pm$ 1.1 & -1.8 $\pm$ 0.6 & 4.4 & $-$2.13 $\pm$ 0.12 & --- &  N  \\ \\
298.19512 & $-$22.09244 & 18.18 $\pm$ 0.01 & 17.21 $\pm$ 0.01 & 19.37 $\pm$ 0.01 & -175.9 $\pm$ 0.3 & -13.5 $\pm$ 0.6 & -25.9 $\pm$ 0.4 & 95.7 & $-$2.31 $\pm$ 0.12 & -2.15 $\pm$ 0.12 &  Y  \\ \\
298.18145 & $-$22.07735 & 18.72 $\pm$ 0.01 & 17.92 $\pm$ 0.01 & 19.56 $\pm$ 0.01 & -177.7 $\pm$ 0.8 & -7.7 $\pm$ 1.3 & -9.1 $\pm$ 0.7 & 18.3 & $-$2.97 $\pm$ 0.12 & -2.18 $\pm$ 0.11 &  N  \\ \\
298.11270 & $-$22.04758 & 18.97 $\pm$ 0.01 & 18.23 $\pm$ 0.01 & 19.74 $\pm$ 0.01 & -178.0 $\pm$ 0.9 & -1.3 $\pm$ 1.1 & -8.7 $\pm$ 0.6 & 11.9 & $-$2.91 $\pm$ 0.12 & -2.13 $\pm$ 0.18 &  N  \\ \\
298.19043 & $-$22.03676 & 19.14 $\pm$ 0.01 & 18.32 $\pm$ 0.01 & 20.03 $\pm$ 0.01 & -180.2 $\pm$ 0.3 & -1.7 $\pm$ 1.5 & -1.9 $\pm$ 0.9 & 53.5 & $-$2.87 $\pm$ 0.12 & -2.23 $\pm$ 0.11 &  Y  \\ \\
298.15270 & $-$22.08463 & 19.25 $\pm$ 0.01 & 18.46 $\pm$ 0.01 & 20.11 $\pm$ 0.01 & -176.2 $\pm$ 0.5 & 12.3 $\pm$ 0.5 & -27.2 $\pm$ 0.3 & 27.6 & $-$2.83 $\pm$ 0.12 & -2.23 $\pm$ 0.11 &  Y  \\ \\
298.15185 & $-$22.09654 & 19.29 $\pm$ 0.01 & 18.52 $\pm$ 0.01 & 20.18 $\pm$ 0.01 & -177.6 $\pm$ 0.4 & 6.8 $\pm$ 3.3 & 0.1 $\pm$ 2.1 & 29.0 & $-$2.56 $\pm$ 0.12 & -2.25 $\pm$ 0.12 &  Y  \\ \\
298.18099 & $-$22.20728 & 19.58 $\pm$ 0.01 & 18.99 $\pm$ 0.01 & 20.55 $\pm$ 0.02 & 35.3 $\pm$ 0.8 & --- $\pm$ --- & --- $\pm$ --- & 11.7 & $-$1.14 $\pm$ 0.12 & --- &  N  \\ \\
298.21784 & $-$22.06409 & 19.66 $\pm$ 0.01 & 18.90 $\pm$ 0.01 & 20.47 $\pm$ 0.02 & -175.1 $\pm$ 0.4 & -2.4 $\pm$ 0.7 & -5.2 $\pm$ 0.4 & 34.3 & $-$99.0 $\pm$ 0.00 & -2.36 $\pm$ 0.1 &  Y  \\ \\
298.16019 & $-$22.09027 & 19.94 $\pm$ 0.01 & 19.34 $\pm$ 0.01 & 20.77 $\pm$ 0.02 & 93.9 $\pm$ 2.8 & -0.3 $\pm$ 0.8 & -1.3 $\pm$ 0.4 & 3.6 & $-$1.70 $\pm$ 0.12 & --- &  N  \\ \\
298.19280 & $-$22.05979 & 19.99 $\pm$ 0.01 & 19.28 $\pm$ 0.01 & 20.72 $\pm$ 0.02 & -174.3 $\pm$ 0.6 & -0.8 $\pm$ 0.3 & -1.0 $\pm$ 0.1 & 22.6 & $-$3.07 $\pm$ 0.12 & -2.3 $\pm$ 0.11 &  Y  \\ \\
298.19388 & $-$22.07918 & 20.01 $\pm$ 0.01 & 19.28 $\pm$ 0.01 & 20.79 $\pm$ 0.02 & -176.2 $\pm$ 0.6 & -7.1 $\pm$ 1.0 & -8.9 $\pm$ 0.6 & 17.7 & $-$2.67 $\pm$ 0.12 & -2.15 $\pm$ 0.13 &  Y  \\ \\
298.20960 & $-$22.07638 & 20.03 $\pm$ 0.01 & 19.33 $\pm$ 0.01 & 20.87 $\pm$ 0.02 & 121.4 $\pm$ 1.3 & 0.0 $\pm$ 0.5 & -0.4 $\pm$ 0.3 & 22.5 & $-$2.26 $\pm$ 0.12 & -1.78 $\pm$ 0.13 &  N  \\ \\
298.16641 & $-$22.06769 & 20.21 $\pm$ 0.01 & 19.51 $\pm$ 0.01 & 20.95 $\pm$ 0.02 & -172.2 $\pm$ 1.5 & -3.2 $\pm$ 0.6 & -4.5 $\pm$ 0.3 & 10.8 & $-$2.73 $\pm$ 0.12 & -2.72 $\pm$ 0.1 &  Y  \\ \\
298.13014 & $-$22.14273 & 20.24 $\pm$ 0.01 & 19.56 $\pm$ 0.01 & 21.01 $\pm$ 0.02 & -178.5 $\pm$ 1.0 & --- $\pm$ --- & --- $\pm$ --- & 14.5 & $-$2.40 $\pm$ 0.12 & -2.06 $\pm$ 0.14 &  Y  \\ \\
298.14121 & $-$22.06475 & 20.28 $\pm$ 0.01 & 19.62 $\pm$ 0.01 & 21.05 $\pm$ 0.02 & -175.2 $\pm$ 1.2 & -14.8 $\pm$ 1.5 & -20.4 $\pm$ 0.8 & 15.1 & $-$99.0 $\pm$ 0.00 & -1.89 $\pm$ 0.16 &  Y  \\ \\
298.19401 & $-$22.08638 & 20.38 $\pm$ 0.01 & 19.70 $\pm$ 0.01 & 21.09 $\pm$ 0.03 & -175.7 $\pm$ 1.2 & -1.2 $\pm$ 0.9 & -1.1 $\pm$ 0.5 & 7.7 & $-$2.78 $\pm$ 0.12 & -2.17 $\pm$ 0.11 &  Y  \\ \\
298.21019 & $-$22.05475 & 20.51 $\pm$ 0.01 & 19.82 $\pm$ 0.01 & 21.20 $\pm$ 0.03 & -188.7 $\pm$ 5.8 & -1.5 $\pm$ 0.5 & -0.8 $\pm$ 0.3 & 3.8 & $-$3.06 $\pm$ 0.12 & --- &  Y  \\ \\
298.14290 & $-$22.07784 & 20.53 $\pm$ 0.01 & 19.87 $\pm$ 0.01 & 21.23 $\pm$ 0.03 & -177.4 $\pm$ 1.3 & -1.4 $\pm$ 1.5 & -9.8 $\pm$ 0.7 & 14.3 & $-$2.68 $\pm$ 0.12 & --- &  Y  \\ \\
298.17751 & $-$22.16337 & 20.60 $\pm$ 0.01 & 19.93 $\pm$ 0.01 & 21.36 $\pm$ 0.03 & 89.3 $\pm$ 0.8 & -7.3 $\pm$ 0.4 & -10.1 $\pm$ 0.2 & 13.4 & $-$2.38 $\pm$ 0.12 & --- &  N  \\ \\
298.21550 & $-$22.08104 & 20.65 $\pm$ 0.01 & 20.00 $\pm$ 0.01 & 21.36 $\pm$ 0.03 & -177.4 $\pm$ 1.0 & -1.4 $\pm$ 1.0 & -1.2 $\pm$ 0.6 & 12.1 & $-$2.60 $\pm$ 0.12 & --- &  Y  \\ \\
298.14837 & $-$22.04551 & 20.64 $\pm$ 0.01 & 19.99 $\pm$ 0.01 & 21.26 $\pm$ 0.03 & -176.7 $\pm$ 1.3 & 1.0 $\pm$ 1.3 & 0.6 $\pm$ 0.7 & 12.6 & $-$99.0 $\pm$ 0.00 & --- &  Y  \\ \\
298.13415 & $-$22.04367 & 21.03 $\pm$ 0.01 & 20.46 $\pm$ 0.01 & 21.58 $\pm$ 0.04 & -267.1 $\pm$ 2.5 & -1.0 $\pm$ 1.5 & -2.2 $\pm$ 0.9 & 5.3 & $-$2.99 $\pm$ 0.12 & --- &  N  \\ \\
298.11950 & $-$22.0253 & 21.19 $\pm$ 0.01 & 20.56 $\pm$ 0.01 & 21.89 $\pm$ 0.05 & -279.0 $\pm$ 2.8 & -2.0 $\pm$ 2.4 & -7.7 $\pm$ 1.4 & 6.7 & $-$2.44 $\pm$ 0.12 & --- &  N  \\ \\
298.14955 & $-$22.10703 & 21.24 $\pm$ 0.01 & 20.62 $\pm$ 0.01 & 21.89 $\pm$ 0.05 & -174.5 $\pm$ 11.9 & 4.8 $\pm$ 3.1 & 2.9 $\pm$ 1.8 & 3.4 & $-$2.69 $\pm$ 0.12 & --- &  Y  \\ \\
298.03665 & $-$22.04132 & 18.13 $\pm$ 0.01 & 17.23 $\pm$ 0.01 & 19.54 $\pm$ 0.01 & 127.2 $\pm$ 0.5 & 0.2 $\pm$ 3.5 & -1.4 $\pm$ 1.8 & 105.2 & $-$1.23 $\pm$ 0.12 & -1.19 $\pm$ 0.17 &  N  \\ \\
298.09544 & $-$22.14968 & 19.71 $\pm$ 0.01 & 18.89 $\pm$ 0.01 & 20.98 $\pm$ 0.02 & -2.7 $\pm$ 0.9 & 0.1 $\pm$ 1.4 & -1.9 $\pm$ 0.7 & 31.0 & $-$1.25 $\pm$ 0.12 & -1.36 $\pm$ 0.15 &  N  \\ \\
298.06312 & $-$22.11093 & 20.13 $\pm$ 0.01 & 19.44 $\pm$ 0.01 & 21.05 $\pm$ 0.02 & -29.5 $\pm$ 1.1 & -0.3 $\pm$ 0.8 & -1.3 $\pm$ 0.4 & 12.8 & $-$99.0 $\pm$ 0.00 & -1.37 $\pm$ 0.17 &  N  \\ \\
298.05710 & $-$22.02942 & 20.59 $\pm$ 0.01 & 20.02 $\pm$ 0.01 & 21.35 $\pm$ 0.03 & -63.9 $\pm$ 0.8 & 2.7 $\pm$ 1.8 & 0.4 $\pm$ 1.0 & 10.9 & $-$1.89 $\pm$ 0.12 & --- &  N  \\ \\
297.96338 & $-$22.04654 & 20.35 $\pm$ 0.01 & 19.76 $\pm$ 0.01 & 21.11 $\pm$ 0.02 & -18.8 $\pm$ 1.5 & --- $\pm$ --- & --- $\pm$ --- & 12.1 & $-$1.99 $\pm$ 0.12 & -1.61 $\pm$ 0.16 &  N  \\ \\
298.15344 & $-$22.04959 & 17.76 $\pm$ 0.01 & 16.74 $\pm$ 0.01 & 18.98 $\pm$ 0.01 & -175.5 $\pm$ 0.3 & -0.9 $\pm$ 0.4 & -1.0 $\pm$ 0.2 & 125.2 & $-$2.31 $\pm$ 0.12 & -2.23 $\pm$ 0.12 &  Y  \\ \\
298.28018 & $-$22.12346 & 17.18 $\pm$ 0.01 & 16.40 $\pm$ 0.01 & 18.60 $\pm$ 0.01 & -14.2 $\pm$ 0.4 & -0.3 $\pm$ 1.3 & 0.8 $\pm$ 0.8 & 131.5 & --- & -1.19 $\pm$ 0.18 &  N  \\ \\
298.01474 & $-$21.91537 & 17.29 $\pm$ 0.01 & 16.29 $\pm$ 0.01 & 18.90 $\pm$ 0.01 & -48.0 $\pm$ 0.4 & -4.0 $\pm$ 1.7 & -8.8 $\pm$ 1.0 & 131.5 & --- & -1.23 $\pm$ 0.18 &  N  \\ \\

\end{tabular}
\end{sideways}
\end{table*}

\newcommand{MNRAS}{MNRAS}
\newcommand{PASA}{PASA}
\newcommand{Nature}{Nature}
\newcommand{ARAA}{ARAA}
\newcommand{AJ}{AJ}
\newcommand{ApJ}{ApJ}
\newcommand{ApJ}{ApJ}
\newcommand{ApJSupp}{ApJSupp}
\newcommand{A\&A}{A\&A}
\newcommand{A\&ASupp}{A\&ASupp}
\newcommand{PASP}{PASP}
\newcommand{PASJ}{PASJ}


\section{Introduction}
Self-supervised contrastive learning has been shown great potential in extracting general visual  representations recently \cite{chen2020simple}. Models pre-trained in such a manner achieve better results in downstream tasks like object detection and segmentation, comparing to fully-supervised pre-trained models \cite{he2020momentum}. Besides, the gap of performance in the basic image classification is also narrowing. However, problems with self-supervised learning are discovered recently, including inductive bias and sensitivity to data augmentation. These are mainly caused by the weak-supervising signal and image pairs used in the contrastive mechanism. The goal of learning general representations has not been achieved yet.

We propose a novel framework FLAGS  with a custom sampling strategy that incorporate features from different latent spaces. Our motivation is to combine the strengths of unsupervised contrastive learning and fully-supervised learning to generate richer representations. We argue that there are two levels of semantics in images, global semantics and local semantics. Global semantics refer to all entities in the image including the background and the foreground. Local semantics refer to the entity that is defined by the label of the image. Self-supervised contrastive learning methods perform instance discrimination which can be seen as learning global semantics. While the supervised learning methods on the other side, learn the local semantics. As shown in  Table 1, We find a trend that tasks like object detection and segmentation benefit from global semantics, whereas image classification benefits more from local semantics. In details, for image classification, the supervised method which learns local semantics achieves much better accuracy than self-supervised methods which learn global semantics. On the other hand, for detection and segmentation which require more information besides the semantics of the entity of interest, self-supervised methods perform close to or even better than the supervised method. Thus intuitively a method learns both global and local semantics should be a better pre-training for larger variety of tasks, performing better than the methods learn either one. The authors in \cite{wei2020semantic} also argue that self-supervised methods and fully-supervised methods learn different kinds of representations. They try to alleviate the conflict between the fully-supervised and unsupervised learning objectives. Comparing to their method, our method aims to learn different level of semantics features thoroughly with equal attention on global and local semantics. Our contributions are summarized below:

\begin{enumerate}
\item	We design a novel architecture which can effectively guide feature extractors to learn both global and local semantic features.
\item	We avoid the conflict between the learning objectives, by using a sampling strategy and splitting the learning of two semantic features by projecting them into separate hyperspheres.
\item	Our method of guiding contrastive networks opens a new direction of research for general representation learning.
\end{enumerate}

\begin{table*}
\begin{center}
\centering
\label{Table_1}
\caption{ Comparison of different pre-training methods on three downstream tasks. The standard backbone is ResNet-50. Linear evaluation protocol \cite{he2020momentum} is applied for image classification. For object detection and semantic segmentation, models are fine-tuned on the PASCAL VOC dataset \cite{pascal}.}
\begin{tabular}{c|c|c|c|c}
    \hline
    Type of semantics &pre-train &  Classification (Accuracy\%) &  Detection ($AP_{50}$)  & Segmentation (mIOU)  \\ \hline
    \multirow{4}{*}{global} &Colorization \cite{zhang2016colorful}  & 39.6    &-       &-   \\ \cline{2-5}
    & Jigsaw \cite{noroozi2017unsupervised}  & 44.6    &61.4    &-   \\ \cline{2-5}
     &SimCLR \cite{chen2020simple}           & 69.3    & 79.4 \cite{wang2021dense}  & 64.3 \cite{wang2021dense}   \\ \cline{2-5}
     &MoCo-v2 \cite{chen2020improved}        & 71.1    & 82.5       & 67.5 \cite{wang2021dense}  \\ \hline
    local &supervised \cite{he2015deep}           & 76.5    & 81.3       & 67.7 \cite{wang2021dense}    \\ \hline
\end{tabular}
\end{center}
\end{table*}

\section{Related Work}
In this section, we summarize the current development in representation learning. The objective of representation learning is to abstract and disentangle underlying factors of variation existing in raw data  \cite{bengio2014representation}. A robust representation contains unique properties that can be easily distinguished from others. To achieve this goal, different learning strategies are explored.

\subsection{Supervised Learning}
Fully-supervised methods heavily rely on labels as strong supervising signals when training. The objective is to learn a representation that maximizes the probability of finding the correct label of images. In terms of network architectures, different convolutional neural networks \cite{simonyan2015deep, he2015deep, tan2020efficientnet} are commonly used to encode input images and generate feature maps for classification and other tasks. Recently, transformer-based models have been shown to better encode visual information and achieved promising results in downstream tasks \cite{dosovitskiy2021image, carion2020endtoend, zheng2021rethinking}. Commonly, these methods train on large datasets such as ImageNet \cite{imagenet_cvpr09} and JFT-300M \cite{sun2017revisiting} which capture a great variation of objects. The extracted representations primarily encode specific features biased to the learning objective. Geirhos et al. \cite{geirhos2019imagenettrained} has shown through experiments that CNN models trained on ImageNet are biased to texture features that are beneficial to object recognition tasks. This indicates the learned representation only captures partial invariance of objects.

\subsection{Pretext Learning}
Learning to perform a predefined pretext task is a discriminative approach in self-supervised learning. Pretext tasks are hand-designed to learn desired features using self-generated pseudo labels. Specifically, image inpainting \cite{pathak2016context} and colorization \cite{zhang2016colorful} are generation-based tasks where the objective is to guide models to recover incomplete images. To achieve these tasks, both semantic and context information of scenes are necessary to be learned. Solving image jigsaw puzzle \cite{noroozi2017unsupervised, kim2018learning} and image rotation \cite{gidaris2018unsupervised} guide networks to encode positional and global context information of images. However, experiments from Chen et al. \cite{chen2020simple} have demonstrated the gap between the model performance on downstream tasks trained using pretext task methods and fully-supervised methods as shown in Table 1. The common problem is that a single pretext task cannot provide enough supervision signal when training and it limits networks to learn biased representations.

\subsection{Contrastive Learning}
Contrastive learning methods utilize instance discrimination as the pretext task and it shows the potential to better extract useful representation that transfers well in downstream tasks. In the beginning, image features are extracted by an encoder and they are projected into a latent space. Contrastive loss \cite{1640964} is then used to measure the similarity(distance) between those projected features. For supervised contrastive learning,  SCL \cite{khosla2021supervised} utilizes images with the same label as positive samples to contrast which achieves competitive results as traditional supervised methods. In unsupervised contrastive learning,  MoCo \cite{he2020momentum} introduces a dynamic dictionary with the momentum update mechanism which extracts representation by comparing with a diverse set of negative samples. SimCLR \cite{chen2020simple} proves the importance of data augmentation and large batch size in a simple contrastive setting. However, data augmentation introduces inductive bias that restricts the generalization of representations. Experiments show that both strength and selection of data augmentation affect the performance of pre-trained models. As shown by Purushwalkam and Gupta \cite{purushwalkam2020demystifying}, data augmentations like random cropping learn occlusion invariance. Besides, instead of contrasting with different images within the same class, such approaches do not use class labels and can only contrast with augmented images of themselves which fail to explore object invariance.

\begin{figure*}
\begin{center}
\includegraphics[width=0.64\linewidth]{positive_pair_example.jpg}
\end{center}
   \caption{Illustration of the sorted images in the same class based on cosine similarity.
   In this example, the top-2 images are selected as global positive pairs since they have similar background as the query image. While the two images in the middle of the list, do not have the similar background as the query image. So they are selected as local positive pairs.}

\label{fig:short}
\end{figure*}

\section{Method}
\subsection{Preliminary Knowledge}
\textbf{Supervision Signals} To learn a desired representation, specific supervision signals need to be supplied to guide networks during training. However, direct combination of supervision signals injects confusion into learning objectives. In self-supervised contrastive learning, methods are sensitive to data augmentation which serves as a weak supervision signal. To tackle this problem, LooC \cite{xiao2021contrastive} projects features into different latent subspaces for each augmentation so the invariance of raw data and the variance of augmentation are both preserved in the extracted representations. We argue that this concept of disentangling features into separate latent spaces can help to learn a more robust representation with stronger guidance.

\textbf{Adjustment In Positive Pair Selection} The selection of positive pairs plays an important role in contrastive learning. In self-supervised settings, augmented images of themselves are treated as positive sample whereas other images including those from the same classes are treated as negatives. In supervised settings, SCL \cite{khosla2021supervised} treats all samples from the same class as positives and other remaining samples in the batch as negative samples, which networks learn a richer representation within local regions. Recent development of SCAN \cite{wei2020semantic} first reveals the conflicting objectives in supervised and self-supervised learning and shows task-agnostic appearance information are learned in common self-supervised methods. To alleviate such conflict in contrastive learning, SCAN selects the top-k nearest neighbors of query image from the same class as positives in the feature space formulated by pre-trained MoCo-v2, which the pre-trained model has been proved to generate more robust representations.

\begin{figure}[t]
\label{figure_2}
\begin{center}
\includegraphics[width=0.97\linewidth]{model_structure_new.png}
\end{center}
   \caption{\textbf{The main architecture of FLAGS} At each training step, a query image X is processed by the encoder. A local key pair \{$X_{l0}, X_{l1}$\}, a global key pair \{$X_{g0}, X_{g1}$\}, and an augmented version of query $X_{aug0}$ are fed into the momentum encoder. The output features \{$k_{l0}, k_{l1}, k_{g0}, k_{g1}, k_{aug0}$\} are then projected to corresponding local and global subspace via mlp\_l\_k and mlp\_g\_k. The encoded query q is projected into local and global subspace via mlp\_l\_q and mlp\_g\_q. Losses for each subspace are calculated separately and combined to form the total loss as shown in the loss function section. Two queues are maintained for local and global semantic pairs. }
\label{fig:long}
\label{fig:onecol}
\end{figure}

\subsection{Our Approach}
Our goal is to enable networks to learn rich representations that can smoothly adapt to various downstream tasks. We first define that self-supervised contrastive methods learn \textbf{global semantics} and fully-supervised methods learn \textbf{local semantics}. Specifically, global semantics encode rough context and semantic information of all objects throughout the region. Local semantics focus on extracting detailed representations of specific objects located at certain regions. Our intuitive is to guide models to encode both local and global semantics by fusing the learning objectives of fully-supervised learning and contrastive learning. Inspired by LooC \cite{xiao2021contrastive}, we propose a novel supervised contrastive framework FLAGS as shown in Figure 2. The core idea of FLAGS is to project pairs of features into either global or local subspace. Specifically, a query image with a local and a global positive pair (keys) are fed into the encoders, then the output features are projected into corresponding subspaces using two separate sets of MLPs. We maintain two queues to store local and global keys that are used to contrast during training. Losses are computed independently for each subspace and combined in the end.

\newpage
\begin{widetext}
\begin{equation}
\label{equation 1}
\mathcal{L}_{ALL} = \sum_{i\in I} \mathcal{L}_{global}(i) + \sum_{i\in I}\mathcal{L}_{local}(i)
\end{equation}

\begin{equation}
\label{equation 2}
\mathcal{L}_{global}(i) =  \frac{-1}{|P_g(i)|}
                \sum_{p\in P_g(i)}
                    \log \frac{\exp(z_g(i) \cdot z_g(p) / \tau )}
                             {\sum\limits_{a\in Q_g}\exp(z_g(i) \cdot z_g(a) / \tau) + {\exp(z_g(i) \cdot z_g(p) / \tau )}}
\end{equation}
\begin{equation}
\label{equation 3}
\mathcal{L}_{local}(i) = \frac{-1}{|P_l(i)|}
                \sum_{p\in P_l(i)}
                    \log \,
                        \frac{\exp(z_l(i) \cdot z_l(p) / \tau )}
                             {\sum\limits_{a\in Q_l}\exp(z_l(i) \cdot z_l(a) / {\tau)+\exp(z_l(i) \cdot z_l(p) / \tau )}}
\end{equation}
\end{widetext}

\textbf{Image Pairs Generation} To form image pairs for each query image that will be fed into different subspaces, we adapt the image pair selection strategy in SCAN \cite{wei2020semantic} and modify it accordingly. In details, images are fed into the ResNet pre-trained in MoCo-v2 to generate features \(p\) with the size of [1, 1024]. The cosine similarity between each image within the same class is calculated. Then images are sorted using the similarity and stored into a list  as shown in Figure 1. We select the top-2 similar images as a pair to project on global-semantic subspace. To alleviate the conflict between global and local learning objectives,  two images at middle of lists are chosen to form a pair for local-semantic subspace. Since their extent of global semantics similarity with the query image is moderate.

\textbf{Loss Function}
Our loss function is based on the common contrastive loss \cite{1640964}. We call it combined contrastive loss \autoref{equation 1}. It is the summation of contrastive loss at the global branch and the contrastive loss at the local branch. \autoref{equation 2} shows the loss for one query image at the global branch. \autoref{equation 3} shows the loss for one query image at the local branch. The form of equation is inspired by the supervised contrastive loss introduced in SCL \cite{khosla2021supervised}. It allows the contrastive loss function to generalize to an arbitrary number of positives. In details, there are $N$ query images in one batch. Let $i$ be the index of images in the batch. For the $i$-th query image, it has global positive keys and its augmentation $P_g(i)$. It also has local positive keys $P_l(i)$. $P_g(i)$ has dimension $M+1$ where $M$ is the number of positive keys and one represents the augmentation. $P_l(i)$ has dimension $M$. $Q_g$ is the queue that contains negative images for the global semantic branch and $Q_l$ is the queue for the local semantic branch. $Q_g$ and $Q_l$ both has dimension of $K$. $Z_g(i)$ is the normalized projected features of image $i$ in the global subspace. Accordingly, $Z_l(i)$ is in the local subspace. $\tau \in \mathbb R$ is temperature parameter.

\section{Experiments}
\subsection{Pre-training}
\textbf{Dataset Preparation}
We use ImageNet-1M \cite{imagenet_cvpr09} for pre-training and validation. We use the proposed image pair generation strategy to prepare the global and local positive pairs for each image in the ImageNet-1M \cite{imagenet_cvpr09} train set. These positive pairs are keys for the corresponding query image during training.

\textbf{Training}
We perform three different pre-training methods. The first pre-training method is FLAGS with only the global branch. The second pre-training method is FLAGS with both the global and the local branch. The last is MoCo\_v2 \cite{chen2020improved} which using the same loss as FLAGS. The hyper parameters are basically the same as those used in MoCo \cite{he2020momentum}. We use SGD as the optimizer with a momentum of 0.9. The learning rate is 0.03. The batch size is 256. We use the loss function shown in Section 3.2. The checkpoint at the 200th epoch of each pre-training is used for the rest of experiments. Our train set contains about 1.23 million images rather than normal 1.28 million images. 50,000 images are evenly taken out of the train set from each class.

\subsection{Linear Evaluation}
We follow the common linear evaluation protocol where we freeze the weights of the pre-trained model except the fully-connected layers. Then, we train a supervised classifier using the ImageNet-1M \cite{imagenet_cvpr09}. The hyper parameters are the same as those used in MoCo \cite{he2020momentum}. The top-1 accuracy is recorded during training after each epoch and the top accuracy is shown in  Table 2. As we can see, FLAGS models achieve much higher accuracy than MoCo \cite{he2020momentum}. This is expected because FLAGS models used more supervised signals during pre-training. At the same time, FLAGS with a local branch has a lower accuracy than FLAGS without a local branch. This might be due to the noise added when we select positive local pairs. The selection strategy for local positive pair should vary for different query images or different classes. Since currently, the local key might be too different from the global key which creates a conflicting signal that makes the model confused. In other words, we tell the model to encode two very different images in the same way. This may explains the accuracy difference between FLAGS with and without the local branch.

\begin{table}[t]
\begin{center}
\centering
\label{Table_2}
\caption{Linear evaluation accuracy of ResNet models trained using different methods}
\begin{tabular}{c|c}
    \hline
    $\ $ & Accuracy\%\\
    \hline
    MoCo                   & 67.16  \\ \hline
    FLAGS aug+global        & 78.45  \\ \hline
    FLAGS aug+global+local  & 77.75  \\
    \hline
\end{tabular}
\end{center}
\end{table}

\subsection{Object Detection and Segmentation}
We use object detection and segmentation as downstream tasks to evaluate how good the features are transferable. We perform the experiments with two datasets, PASCAL\_VOC\_2007 \cite{pascal} and COCO\_2017 \cite{coco}. In all experiments, we train detectors with the ResNet R50-C4 \cite{he2015deep} as the feature encoder, where the weights of encoder are initialized from the three pre-trained models, two of them using FLAGS and one using MoCo. We then fine-tune the whole detectors end-to-end. The same hyper parameters settings are used for all experiments.

\textbf{COCO Object Detection and Instance Segmentation}
We use Mask R-CNN \cite{mask} as the detector with the backbone of ResNet R50-C4 \cite{he2015deep}. The batch size is 8 and iteration is 180,000. These mean that the total epoch is about the same as 1*schedule defined in detectron2 \cite{detectron2}. The learning rate is 0.01. We fine-tune models using COCO train\_2017 and evaluate using COCO val\_2017. The results are shown in  Table 3 and  4. As we can see, initialized with weights of FLAGS with global and local branch surpass with weights of a supervised pre-training in both detection and segmentation. This shows the effectiveness of our proposed method.

\begin{table}[t]
\begin{center}
\centering
\label{Table_3}
\caption{Object detection results of Mask R-CNN on COCO dataset with different pre-trained ResNet R50-C4 models.}
\begin{tabular}{c|c|c|c}
    \hline
    pre-train                & AP        & $AP_{50}$  & $AP_{75}$  \\ \hline
    supervised 1*schedule    & 38.200    & 58.200     & 41.200     \\ \hline
    MoCo batch size 8        & 37.879    & 57.201     & 40.903     \\ \hline
    FLAGS aug+global          & 38.344    & 57.852     & 41.357     \\ \hline
    FLAGS aug+global + local  & 38.197    & 58.053     & 40.994     \\
    \hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[t]
\begin{center}
\centering
\label{Table_4}
\caption{Instance segmentation results of Mask R-CNN on COCO dataset with different pre-trained ResNet R50-C4 models.}
\begin{tabular}{c|c|c|c}
    \hline
    pre-train                  & AP          & $AP_{50}$    & $AP_{75}$  \\ \hline
    supervised 1*schedule      & 33.300      & 54.700       & 35.200     \\ \hline
    MoCo batch size 8          & 33.312      & 54.055       & 35.598     \\ \hline
    FLAGS aug + global          & 33.511      & 54.461       & 35.505     \\ \hline
    FLAGS aug + global + local  & 33.431      & 54.552       & 35.460     \\
    \hline
\end{tabular}
\end{center}
\end{table}

\textbf{PASCAL VOC Object Detection}
The detector used is Faster R-CNN \cite{faster} with R50-C4 \cite{he2015deep} as backbone. The batch size is 4 and the number of iteration is 96,000. The learning rate is 0.005. The same feature normalization (sync norm) described in MoCo \cite{he2020momentum} is used. Other parameters are defaults defined in detectron2 \cite{detectron2}. We fine-tune models using VOC trainval\_2012 and VOC trainval\_2007. Evaluation is conducted on VOC test\_2007. The results are shown in Table 5. FLAGS with only global branch gets similar performance as MoCo \cite{he2020momentum}. When the local branch is added, the performance decreases. This is opposite to the results with COCO \cite{coco} where FLAGS with local branch achieves the best overall scores. We think this phenomenon is due to the difference between two datasets. As survey shows, in PASCAL\_VOC \cite{pascal} more than 40\% instances take above 50\% of whole image size. On the other side, in COCO \cite{coco} only 1\% instances take more than 50\% of whole image size. In addition, COCO \cite{coco} has 7.3 objects per image and PASCAL\_VOC \cite{pascal}  has 2.3 objects per image. These differences mean that detection and segmentation requires more local semantics to get into details of part of images in COCO \cite{coco}. FLAGS’s local branch successfully captures local semantics, improving the detector’s performance on COCO \cite{coco}.

\begin{table}[t]
\begin{center}
\centering
\label{Table_5}
\caption{Object detection results of Mask R-CNN on  PASCAL\_VOC dataset with different pre-trained ResNet R50-C4 models.}
\begin{tabular}{c|c|c|c}
    \hline
    pre-train               & AP      & $AP_{50}$  & $AP_{75}$  \\ \hline
    MoCo                    & 50.571  & 78.879     & 55.193     \\ \hline
    FLAGS aug+global         & 50.701  & 79.328     & 54.310     \\ \hline
    FLAGS aug+global+local   & 48.141  & 78.348     & 51.034     \\
    \hline
\end{tabular}
\end{center}
\end{table}

\section{Conclusion}
In this paper, we propose FLAGS that extracts rich and transferable representations for various downstream tasks. We suggest representations have two levels of semantics: global semantics and local semantics. By contrasting with local and global image pairs in different subspaces, models benefit from both learning objectives. Through a few experiments, we demonstrate the learned representations based on FLAGS improves the performance in image classification, object detection and segmentation. It has been shown that this direction is promising and we hope others can expand the research. The positive pair sampling strategy can be optimized to get more precise keys. In addition, more visualizations of learned representations are beneficial.

\newpage
\section*{Acknowledgement}
We would like to express our great appreciation to Dr. Hailin Hu for providing tremendous support and constructive suggestions. We also want to thank Xiangqian Wang and Lin Du for facilitating the publishing. Last but not least, thanks Ce Wang for the great effort in validating experiment results.

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\bibliographystyle{ieee}