Datasets:

zai-org
/

RPC-Bench

+# On the Fundamental Trade-offs in Learning Invariant Representations
+Anonymous Author(s)
+Affiliation
+Address
+email
+# Abstract
+1 Many applications of representation learning, such as privacy-preservation, al
+2 gorithmic fairness and domain adaptation, desire explicit control over semantic
+3 information being discarded. This goal is often formulated as satisfying two po
+4 tentially competing objectives: maximizing utility for predicting a target attribute
+5 while simultaneously being independent or invariant with respect to a known seman
+6 tic attribute. In this paper, we identify and determine two fundamental trade-offs
+7 between utility and semantic dependence induced by the statistical dependencies
+8 between the data and its corresponding target and semantic attributes. We derive
+9 closed-form solutions for the global optima of the underlying optimization prob
+10 lems under mild assumptions, which in turn yields closed formulae for the exact
+11 trade-offs. We also derive empirical estimates of the trade-offs and show their
+12 convergence to the corresponding population counterparts. Finally, we numeri
+13 cally quantify the trade-offs on representative problems and compare the solutions
+14 achieved by baseline representation learning algorithms.
+# 15 1 Introduction
+16 Real-world applications of representation learning algorithms often have to contend with objectives
+17 beyond predictive performance. These include cost functions pertaining to, invariance (e.g., to
+18 photometric or geometric variations), semantic independence (e.g., w.r.t to age or race for face
+19 recognition systems), privacy (e.g., mitigating leakage of sensitive information [1]), algorithmic
+20 fairness (e.g., demographic parity [2]), and generalization across multiple domains [3], to name a few.
+21 At its core, the underlying goal of the aforementioned formulations of representation learning is to
+22 satisfy two competing objectives, extracting as much information necessary to predict a target label
+23 $\textbf {  { y } }$ (e.g., face identity) while intentionally and permanently suppressing information pertaining to a
+24 desired semantic attribute $\pmb { s }$ (e.g., age, gender or race). When $\textbf {  { y } }$ is independent of $\pmb { s }$ , one can learn a
+25 representation that is independent of $\pmb { s }$ with no loss of performance, i.e., no trade-off exists between
+26 the two objectives. However, when the two attributes $\textbf {  { y } }$ and $\pmb { s }$ are correlated, attaining semantic
+27 independence will necessarily reduce the performance of the target predictor, i.e., there is a trade-off
+28 between the two objectives. The trade-off is unknown yet is important for understanding the limits of
+29 existing and future representation learning algorithms that involve semantic independence constraints.
+30 Let $z = f ( { \pmb x } )$ be a representation of input data $_ { \textbf { \em x } }$ , and $f ( \cdot )$ be the encoder (see Fig 1(a)). Invariant
+31 learning requires that prediction of the target label, ${ \widehat { \pmb { y } } } = g _ { Y } ( z )$ be independent of a semantic attribute
+32 $\pmb { s }$ i.e., $\boldsymbol { \widehat { y } } \perp \perp \boldsymbol { s }$ for all possible downstream target predictors $\overset { \cdot } { g _ { Y } ( \cdot ) }$ . This independence condition is
+33 satisfied if and only if (iff), the representation $_ z$ is independent of $\pmb { s }$ i.e., $z \perp \perp s$ . Therefore, Invariant
+34 representation learning (IRL) seeks to optimize two objectives: i) the degree of dependence between
+35 data representation $_ z$ and semantic attribute $\pmb { s }$ , and ii) target task utility. These two objectives can be
+36 combined into one, with a parameter $\tau$ controlling the trade-off.
+37 In this paper, we identify and analytically determine two fundamental trade-offs in the invariant
+38 representation learning setting introduced above, namely Data Space Trade-Off and Label Space
+39 Trade-Off. These trade-offs are illustrated in Figure 1 (b) and formally defined next.
+40 Definition 1. Data Space Trade-Off arises from the statistical dependence between the target attribute
+41 $\textbf {  { y } }$ and the semantic attribute $\pmb { s }$ conditioned on the given input data $_ { \textbf { \em x } }$ . When the learner’s hypothesis
+42 class contains all Borel-measurable functions1 we have:
+![](images/4b41f619b913d0b024a06e9ffd146496a768341a1e6fc6bf1f8d3e1e0ad4349a.jpg)
+Figure 1: (a): Generic frame work of invariant representation learning (IRL) where attributes $\pmb { s }$ and $\textbf {  { y } }$ are caused by a latent factor $\textbf { \em a }$ and are not marginally independent. Under this setting, IRL seeks a representation $z = f ( { \pmb x } )$ that contains enough information for downstream target predictor $g _ { Y } ( \cdot )$ while being independent of the semantic attribute $\pmb { s }$ . Consequently, the prediction ${ \widehat { \pmb { y } } } = g _ { Y } ( z )$ will also be independent of $\pmb { s }$ for any downstream predictor $g _ { Y } ( \bar { \cdot } )$ . (b): We identify and determine two different fundamental trade-offs between utility (i.e., the performance of target task predictor) and dependence measure $\deg ( z , s )$ by an optimal learner in the hypothesis class of Borel-measurable functions. Trade-off $\mathbf { L }$ is induced by the joint distribution of the labels $p _ { y s }$ . Trade-off $\mathbf { D }$ is induced by the joint distribution of the data $p _ { { \pmb x } { \pmb y } { \pmb s } }$ . Trade-off $\mathbf { F }$ is a relaxed version of trade-off $\mathbf { D }$ obtained by either using a surrogate measure of dependence, e.g., adversarial learning [3] or from a constrained hypothesis class [4], or from using sub-optimal optimization algorithms.
+$$
+\operatorname* { i n f } _ { f ( \cdot ) \mathrm { ~ m e a s u r a b l e } } \Big \{ ( 1 - \tau ) \operatorname* { i n f } _ { g _ { Y } ( \cdot ) \mathrm { ~ m e a s u r a b l e } } \mathbb { E } _ { x , y } \Big [ \mathcal { L } _ { Y } \Big ( g _ { Y } \big ( f ( x ) ) , y \Big ) \Big ] + \tau \mathrm { d e p } ( f ( x ) , s ) \Big \} .
+$$
+43 where $f ( \cdot )$ is the encoder that extracts representation $_ z$ from $_ { \textbf { \em x } }$ , $g _ { Y } ( \cdot )$ predicts $\widehat { \pmb { y } }$ from the repre
+44 sentation $_ z$ , $\mathcal { L } _ { Y } ( \cdot , \cdot )$ is the loss for the desired task of predicting the task label $\textbf {  { y } }$ . The function
+45 $\mathrm { d e p } ( \cdot , \cdot ) \geq 0$ is a parametric or non-parametric measure of statistical dependence i.e., $\mathrm { d e p } ( q , r ) = 0$
+46 means $\pmb q$ and $\mathbfit { \Delta } \mathbf { r }$ are independent, and $\mathrm { d e p } ( q , r ) > 0$ means $\pmb q$ and $\pmb { r }$ are dependent with larger values
+47 indicating greater degrees of dependence. The scalar $\tau \in [ 0 , 1 )$ is a hyper-parameter that controls
+48 the trade-off between the two objectives, with $\tau = 0$ being the standard approach that enforces no
+49 independence to the attribute $\pmb { s }$ , while $\tau  1$ enforces representation $_ z$ to be independent of $\pmb { s }$ .
+50 Including all measurable functions in the hypothesis class of the encoder $f ( \cdot )$ and target predic
+51 tor $g _ { Y } ( \cdot )$ ensures that the best possible trade-off is included within the feasible solution space.
+52 For example, when $\tau = 0$ and $\bar { \mathcal { L } } _ { Y } ( \cdot , \cdot )$ is the mean-squared error, the optimal Bayes estimator,
+53 $g _ { Y } ( f ( \pmb { x } ) ) = \mathbb { E } _ { \pmb { y } } [ \pmb { y } | \pmb { x } ]$ is reachable. This definition corresponds to the trade-off $\mathbf { D }$ in Figure 1 (b).
+54 Definition 2. Label Space Trade- $O f f$ arises by ignoring the data $_ { \textbf { \em x } }$ and is purely determined by the
+55 statistical dependence between the target feature $\textbf {  { y } }$ and the semantic attribute $\pmb { s }$ . Such a trade-off can
+56 be defined as:
+$$
+\operatorname* { i n f } _ { z \in L ^ { 2 } } \Big \{ ( 1 - \tau ) \operatorname* { i n f } _ { g _ { Y } ( \cdot ) \mathrm { ~ m e a s u r a b l e } } \mathbb { E } _ { x , y } \Big [ \mathcal { L } _ { Y } \big ( g _ { Y } ( z ) , y \big ) \Big ] + \tau \deg ( z , s \big ) \Big \} ,
+$$
+where 57 $L ^ { 2 }$ is the space of all random vectors with finite second-order moment (i.e., $\mathbb { E } _ { z } [ \| z \| ^ { 2 } ] < \infty$ ) 58 on the same probability space in which the joint variable $( s , y )$ comes from.
+59 This definition corresponds to the optimal trade-off obtained by an ideal representation $_ { z }$ that is not
+60 constrained by the learnability of the encoder $f ( \cdot )$ . For example, if $\tau = 0$ , the ideal representation
+61 $_ z$ is perfectly aligned with the target label $\textbf {  { y } }$ i.e., $z = y$ and $g _ { Y } ( \cdot )$ is the identity function, perfect
+62 prediction of target attribute is feasible. Therefore, this trade-off corresponds to the best trade-off that
+63 any combination of data $_ { \textbf { \em x } }$ and learnable encoder $f ( \cdot )$ can aspire to. This definition corresponds to
+64 the trade-off $\mathbf { L }$ in Figure 1 (b), and it necessarily dominates the Data Space Trade-Off D.
+65 Contributions: i) Identify two fundamental trade-offs in invariant representation learning. ii) Obtain
+66 closed-form solution for the corresponding optimization problems, and consequently determine the
+67 trade-offs exactly. iii) Provide consistent empirical closed-form solution for the representations that
+68 achieve optimal trade-offs. iv) Numerically quantify the trade-offs defined here and compare them to
+69 those obtained by existing solutions.
+70 Implications: i) Our closed-form empirical estimators for the optimal representations lend themselves
+71 to practical invariant representation learning algorithms. ii) Theoretically elucidating and empirically
+72 quantifying the intrinsic limits of invariant representations will enable researchers and practitioners
+73 alike to identify the feasible and infeasible solution space for the trade-offs and lead to informed
+74 development and deployment of optimal IRL methods. iii) Our theoretical analysis sheds light on the
+75 utility-semantic independence trade-off, the role of statistical dependency between target label $\textbf {  { y } }$ , the
+76 semantic attribute $\pmb { s }$ , and the input data $_ { \textbf { \em x } }$ , and the hypothesis class adopted for the learners.
+# 77 2 Related Work
+78 Trade-Offs in Representation Learning: While there are abundant empirical approaches for the
+79 representation learning applications considered in this paper, to the best of our knowledge, there
+80 is no prior work that exactly characterizes and empirically quantifies the trade-offs inherent to
+81 representation learning with semantic independence constraints.
+82 Prior work primarily sought to either obtain lower or upper bounds or characterize the extreme
+83 points of the trade-off in specific contexts such as fair representation learning. For instance, [5]
+84 uses information theoretic tools and characterizes the utility-fairness trade-off in terms of a lower
+85 bounds when both $\textbf {  { y } }$ and $\pmb { s }$ are binary labels. Later [6] provided both upper and lower bound for the
+86 binary labels. By leveraging Chernoff bound [7] proposed a construction method to generate an ideal
+87 representation beyond input data to achieve perfect fairness while maintaining the best performance
+88 on target task for equalized odds. In the case of categorical features, a lower bound on utility-fairness
+89 trade-off has been provided by [8]. The notion of Pareto optimality was used by [9] to minimize
+90 the maximum possible error among sensitive attributes where both target and sensitive features are
+91 categorical. In contrast to this body of work, our trade-off analysis is applicable to multi-dimensional
+92 discrete and/or continuous attributes where we find the exact optimal trade-offs.
+93 The only prior work that investigates fundamental trade-offs in a general setting where both $\textbf {  { y } }$ and $\pmb { s }$
+94 can be continuous or discrete features, are [4] and [10]. [4] considers only linear dependence between
+95 the representation and semantic attribute and proposed a closed-form solution for the utility-fairness
+96 trade-off. Even though [10] considers non-linear dependencies, optimal losses have been derived only
+97 for the extremes of the trade-off (i.e., $\tau  0$ and $\tau  1$ ). In a more general setting where $0 < \tau < 1$
+98 [10] only provides a lower bound on utility-invariance trade-off through information plane analysis.
+99 In contrast to the foregoing, we take a functional analysis approach and utilize covariance operator
+100 based measures of dependence that account for all non-linear dependence relations. We exactly
+101 characterize and quantify the utility-invariance trade-offs, while also providing a means to empirically
+102 estimate the encoder that achieves said optimal trade-off. Lastly, in addition to the Data Space
+103 Trade-Off, we also introduce and determine the Label Space Trade-Off which is the ideal trade-off
+104 that any unrestricted learning algorithm can aspire to.
+105 Invariant, Fair, Privacy-Preserving Representation Learning: The basic idea of representation
+106 learning that discards unwanted semantic information has been explored under different contexts like
+107 invariant, fair, or privacy-preserving learning. In domain adaptation [11, 12, 13], the goal is to learn
+108 features that are independent of the data domain. In fair learning [14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
+109 2, 24, 25, 26, 27, 4], the goal is to discard the demographic information that leads to unfair outcomes.
+110 Similarly, there is a growing interest in mitigating unintended leakage of private information from
+111 data representations [28, 29, 1, 30, 31]. A vast majority of this body of work is empirical in nature.
+112 These methods implicitly look for a single or more points in the trade-off between utility and fairness
+113 and do not explicitly seek to characterize the whole trade-off front. Overall, these approaches are
+114 not concerned (or aware) about the feasibility and limitations on the utility-invariance trade-off. In
+115 contrast, this paper determines the fundamental theoretical limits of controlling independence to
+116 semantic attributes, and proposes practical learning algorithms that achieve this limit.
+117 Adversarial Representation Learning: Most practical approaches for learning fair, invariant, do
+118 main adaptive or privacy-preserving representations discussed above are based on adversarial repre
+119 sentation learning (ARL). This learning problem is typically formulated as,
+$$
+\operatorname* { i n f } _ { f \in \mathcal { H } _ { x } } \Big \{ ( 1 - \tau ) \operatorname* { i n f } _ { g \mathrm { y } \in \mathcal { H } _ { y } } \mathbb { E } _ { \mathbf { x } , y } \Big [ \mathcal { L } _ { Y } \Big ( g _ { Y } \big ( f ( x ) \big ) , y \Big ) \Big ] - \tau \operatorname* { i n f } _ { g _ { S } \in \mathcal { H } _ { s } } \mathbb { E } _ { \mathbf { x } , s } \Big [ \mathcal { L } _ { S } \Big ( g _ { S } \big ( f ( x ) \big ) , s \Big ) \Big ] \Big \} ,
+$$
+120 where $\mathcal { L } _ { S } ( \cdot , \cdot )$ is the loss function of a hypothetical adversary $g _ { S } ( \cdot )$ who intends to extract the semantic
+121 attribute $\pmb { s }$ through the best predictor within the hypothesis class $\mathcal { H } _ { s }$ . ARL is a special case of the Data
+122 Space Trade-Off in (1) where the negative loss of the adversary, $- \operatorname* { i n f } _ { g _ { S } \in \mathcal { H } _ { s } } \mathbb { E } _ { { \pmb x } , s } \Big [ { \mathscr L } _ { S } \Big ( g _ { S } \big ( f ( { \pmb x } ) \big ) , s \Big ) \Big ]$
+123 plays the role of $\mathrm { d e p } ( f ( \pmb { x } ) , \pmb { s } )$ . However, this form of adversarial learning suffers from a fundamental
+124 drawback as also noted in [32, 33]. The measure of dependence induced by ARL does not account
+125 for all modes of non-linear dependence between $\pmb { s }$ and the representation $_ z$ . The next theorem states
+126 this observation precisely,
+127 Theorem 1. 2 Let $\mathcal { H } _ { s }$ contain all Borel-measurable functions and $\mathcal { L } _ { S } ( \cdot , \cdot )$ be mean squared error
+128 (MSE) loss. Then,
+$$
+z \in \arg \operatorname* { s u p } \left\{ \operatorname* { i n f } _ { \substack { g _ { S } \in \mathcal { H } _ { s } } } \mathbb { E } _ { \pmb { x } , s } \Big [ \mathcal { L } _ { S } \Big ( g _ { S } ( z ) , \pmb { s } \Big ) \Big ] \right\} \Leftrightarrow \mathbb { E } [ \pmb { s } | z ] = \mathbb { E } [ \pmb { s } ] .
+$$
+129 This theorem implies that an optimal adversary does not necessarily lead to a representation $_ { z }$ that
+130 is statistically independent of $\pmb { s }$ (i.e., $p ( s | z ) \stackrel { } { = } p ( s ) \rangle$ ), but rather leads to $\pmb { s }$ being mean independent
+131 of representation $_ z$ i.e., independence with respect to first order moment only. In other words,
+132 adversarially learned measure of dependence is not a complete measure of dependence and hence
+133 does not account for all modes of non-linear dependence between two random variables. As such, ARL
+134 is inherently incapable of attaining the trade-offs achievable by complete measures of dependence.
+# 35 3 Theoretical Results
+# 3.1 Problem Setting
+137 Consider the probability space $( \Omega , \mathcal { F } , \mathbb { P } )$ , where $\Omega$ is the sample space, $\mathcal { F }$ is a $\sigma -$ algebra on $\Omega$ , and
+138 $\mathbb { P }$ is a probability measure on $\mathcal { F }$ . We assume that the joint random vector $( { \pmb x } , { \pmb y } , { \pmb s } )$ , containing the
+139 input data $\pmb { x } \in \mathbb { R } ^ { d _ { x } }$ , the target label $\boldsymbol { y } \in \mathbb { R } ^ { d _ { \boldsymbol { y } } }$ and the sensitive attribute $\boldsymbol { s } \in \mathbb { R } ^ { d _ { s } }$ , is a random vector
+140 on $( \Omega , { \mathcal { F } } )$ with joint distribution $\mathbf { \nabla } _ { p _ { x y s } }$ .
+141 Assumption 1. We assume that the encoder consists of $r$ functions in an $L _ { 2 }$ -universal RKHS
+142 $( \mathcal { H } _ { \pmb { x } } , k _ { \pmb { x } } ( \cdot , \cdot ) )$ (e.g., Gaussian kernel), where $L _ { 2 }$ −universality guarantees that $\mathcal { H } _ { x }$ can approximate
+143 any Borel-measurable function with arbitrary precision [34].
+144 Now, the representation vector $_ z$ can be expressed as
+$$
+\begin{array} { r } { z = f ( \pmb { x } ) : = \left[ f _ { 1 } ( \pmb { x } ) , \cdots , f _ { r } ( \pmb { x } ) \right] ^ { T } \in \mathbb { R } ^ { r } , \quad f _ { j } ( \cdot ) \in \mathcal { H } _ { \pmb { x } } \forall j = 1 , \ldots , r . } \end{array}
+$$
+145 where $r$ is the dimensionality of the representation $_ z$ . As discussed in Corollary 5.1, unlike common
+146 practice where it is chosen arbitrarily, $r$ itself is an object of interest for optimization. We consider a
+147 general scenario where both $\textbf {  { y } }$ and $\pmb { s }$ can be continuous or discrete, or one of $\textbf {  { y } }$ or $\pmb { s }$ is continuous
+148 while the other is discrete. To do this, we substitute3 the target loss, inf $\mathbb { E } _ { { \pmb x } , { \pmb y } } [ { \mathcal { L } } _ { Y } ( g _ { Y } ( { \pmb z } ) , { \pmb y } ) ]$ in (1)
+gY
+149 with the negative of a non-parametric measure of dependence i.e., $- \mathrm { d e p } ( z , y )$ . Furthermore, in
+$$
+\pmb { x }  \overbrace { ( \underbrace { f ( \cdot ) } ) } ^ { \iint ( \underbrace { \longrightarrow ( \mathbb { C } \mathrm { o v } ( f ( \pmb { x } ) , \beta _ { y } ( \pmb { y } ) ) ) } _ { \beta } + \underbrace { ( \beta _ { y } ( \cdot ) ) } _ { \beta } + \textbf { { y } } ) } ^ { \iint ( \mathbb { C } \mathrm { o v } ( f ( \pmb { x } ) , \beta _ { y } ( \pmb { y } ) ) ) + ( \beta _ { y } ( \cdot ) ) + \textbf { \delta } ^ { y } }
+$$
+150 unsupervised settings, when there is no target attribute $\textbf {  { y } }$ , the target dependence $\mathrm { d e p } ( z , y )$ can be
+151 replaced with $\mathrm { d e p } ( z , \boldsymbol { x } )$ , which implicitly forces the representation $_ { z }$ to retain as much information
+152 as is necessary for reconstructing the input data $_ { \textbf { \em x } }$ . This scenario is of practical interest when a data
+153 producer aims to provide a representation of data that is independent of a desired semantic attribute
+154 for any arbitrary downstream task.
+155 We start by designing $\deg ( z , s )$ , and $\mathrm { d e p } ( z , y )$ follows similarly. A key desiderata of dependence
+156 measures is that they should be able to account for all possible non-linear dependence relations
+157 between the random variables (or vectors). Examples of such measures include information theoretic
+158 measures such as mutual information (e.g., MINE [36]) or covariance operator based measures such
+159 as Hilbert-Schmidt Independence Criterion [37], Constrained Covariance [38] and Kernel Canonical
+160 Correlation [39]. The underlying principle behind the latter class of dependence measures is that
+161 finite dimensional spaces with non-linear dependencies behave as linearly dependent spaces when
+162 mapped appropriately to higher dimensional spaces. In this paper we adopt the covariance operator
+163 based measures as our choice of dependence measure for analytical tractability.
+164 Principally, $_ { z }$ and $\pmb { s }$ are independent iff $\mathbb { C } \mathrm { o v } ( \alpha ( \pmb { z } ) , \beta _ { s } ( \pmb { s } ) )$ is zero for all $\alpha ( \cdot )$ and $\beta _ { s } ( \cdot )$ belong
+165 ing to some universal RKHSs [38]. Since $z ~ = ~ f ( x )$ and $f ( \cdot ) \ \in \ \mathcal { H } _ { x }$ , $\mathbb { C } \mathrm { o v } ( \alpha ( \pmb { z } ) , \beta _ { s } ( \pmb { s } ) ) \ =$
+166 $\mathbb { C } \mathrm { { \bar { o v } } } ( \alpha ( \pmb { f } ( \pmb { x } ) ) , \beta _ { s } ( \pmb { s } ) )$ , which necessitates application of a kernel on top of another kernel. This
+167 limits the analytical tractability of our solution. However, as we argue below, it is almost sufficient to
+168 consider transformation on $\pmb { s }$ , only, in which case it reduces to $\mathbb { C } \mathrm { o } \bar { \mathbf { v } } ( \pmb { f } ( \pmb { x } ) , \beta _ { s } ( \pmb { s } ) )$ . Let $( \mathcal { H } _ { s } , k _ { s } ( \cdot , \cdot ) )$
+169 and $( \mathcal { H } _ { \boldsymbol { y } } , k _ { \boldsymbol { y } } ( \cdot , \cdot ) )$ be separable4 RKHSs of functions defined on $\mathbb { R } ^ { d _ { s } }$ and $\mathbb { R } ^ { d _ { y } }$ , respectively. Consider
+170 the bi-linear functional,
+$$
+h ( \cdot , \cdot ) : \mathscr { H } _ { \pmb { x } } \times \mathscr { H } _ { s }  \mathbb { R } , h _ { j } ( f _ { j } , \beta _ { s } ) : = \mathbb { C } \mathrm { o v } _ { \pmb { x } , s } ( f _ { j } ( \pmb { x } ) , \beta _ { s } ( \pmb { s } ) ) .
+$$
+171 Assumption 2. We assume in the rest of this paper that the positive definite kernel functions are
+172 bounded, i.e.,
+$$
+\mathbb { E } _ { x } [ k _ { x } ( x , x ) ] < \infty , \quad \mathbb { E } _ { s } [ k _ { s } ( s , s ) ] < \infty , \quad \mathrm { a n d } \quad \mathbb { E } _ { y } [ k _ { y } ( y , y ) ] < \infty .
+$$
+173 The assumptions in (6) guarantee that $h ( \cdot , \cdot )$ in (5) is bounded [40] and therefore, invoking Riesz
+174 representation theorem [41], there exists a unique and bounded linear operator $\Sigma _ { s x }$ , such that
+$$
+h ( f , \beta _ { s } ) = \mathbb { C } \mathsf { o v } _ { \pmb { x } , s } ( f ( \pmb { x } ) , \beta _ { s } ( \pmb { s } ) ) = \langle \beta _ { s } , \Sigma _ { s \pmb { x } } f \rangle _ { \mathscr { H } _ { s } } \quad \forall f \in \mathscr { H } _ { \pmb { x } } , \forall \beta _ { s } \in \mathscr { H } _ { s } .
+$$
+Based on 175 $h ( \cdot , \cdot )$ , we define the linear operator $h _ { f , s } : { \mathcal { H } } _ { s } \to { \mathbb { R } } ^ { r }$ as
+$$
+\begin{array} { r } { h _ { \pmb { f } , \mathscr { s } } ( \beta _ { \mathscr { s } } ) : = \left[ \begin{array} { c } { \mathbb { C } \mathrm { o v } _ { \pmb { x } , \mathscr { s } } ( f _ { 1 } ( \pmb { x } ) , \beta _ { \mathscr { s } } ( \pmb { s } ) ) } \\ { \vdots } \\ { \mathbb { C } \mathrm { o v } _ { \pmb { x } , \mathscr { s } } ( f _ { r } ( \pmb { x } ) , \beta _ { \mathscr { s } } ( \pmb { s } ) ) } \end{array} \right] = \left[ \begin{array} { c } { \langle \beta _ { \mathscr { s } } , \sum _ { \pmb { s } \pmb { x } } f _ { 1 } \rangle _ { \mathscr { H } _ { s } } } \\ { \vdots } \\ { \langle \beta _ { \mathscr { s } } , \sum _ { \pmb { s } \pmb { x } } f _ { r } \rangle _ { \mathscr { H } _ { s } } } \end{array} \right] . } \end{array}
+$$
+176 The operator $h _ { f , s }$ captures all modes of non-linear dependence, since the distribution of a low
+177 dimensional projection of high-dimensional data is approximately normal [42], [43]. In other words,
+178 we assume that $\bar { ( } f ( \pmb { x } ) , \beta _ { s } ( \pmb { s } ) \bar { ) }$ is an approximately Gaussian random vector.
+179 Among the different dependence measures that have been defined through the covariance operator
+180 we adopt the Hilbert-Schmidt Independence Criterion (HSIC) [37] which is defined as the Hilbert
+181 Schmidt norm (HS-norm) of the covariance operator,
+$$
+\mathrm { d e p } ( z , s ) : = \| h _ { f , s } \| _ { \mathrm { H S } } ^ { 2 } = \sum _ { \beta _ { s } \in \mathcal { U } _ { s } } \| h _ { f , s } ( \beta _ { s } ) \| _ { 2 } ^ { 2 } { = } \sum _ { \beta _ { s } \in \mathcal { U } _ { s } } \sum _ { j = 1 } ^ { r } h ^ { 2 } ( f _ { j } , \beta _ { s } )
+$$
+182 where $\mathcal { U } _ { s }$ is a countable orthonormal basis set for $\mathcal { H } _ { s }$ . Note that, based on this definition, if the
+183 distribution $( f ( \pmb { x } ) , \beta _ { s } ( \pmb { s } ) )$ fails to be a normal distribution, we end up measuring mean dependency
+184 of $z = f ( x )$ from $\pmb { s }$ which is still much stronger than the linear dependency between $_ z$ and $\pmb { s }$ [44].
+185 Even under this assumption, empirically (Section 4) we observe that trade-offs we obtain significantly
+186 dominate those from existing invariant representation learning algorithms.
+187 The following Lemma introduces a well-defined population expression for $\deg ( z , s )$ in (8).
+# Lemma 2.
+$$
+\begin{array} { r l r } { \mathrm { d e p } ( z , s ) } & { = } & { \displaystyle \sum _ { j = 1 } ^ { r } \Big \lbrace \mathbb { E } _ { \alpha , s , \alpha ^ { \prime } , s ^ { \prime } } \Big [ f _ { j } ( \alpha ) f _ { j } ( \pmb { x } ^ { \prime } ) k _ { s } ( s , s ^ { \prime } ) \Big ] + \mathbb { E } _ { \alpha } \big [ f _ { j } ( \pmb { x } ) \big ] \mathbb { E } _ { \pmb { x } ^ { \prime } } \big [ f _ { j } ( \pmb { x } ^ { \prime } ) \big ] \mathbb { E } _ { s , s ^ { \prime } } \big [ k _ { s } ( s , s ^ { \prime } ) \big ] } \\ & { } & { - 2 \mathbb { E } _ { \alpha , s } \Big [ f _ { j } ( \pmb { x } ) \mathbb { E } _ { \pmb { x } ^ { \prime } } \big [ f _ { j } ( \pmb { x } ^ { \prime } ) \big ] \mathbb { E } _ { \pmb { y } ^ { \prime } } \big [ k _ { s } ( s , s ^ { \prime } ) \big ] \Big ] \Big \rbrace } \end{array}
+$$
+where 188 $( { \pmb x } , { \pmb s } )$ and $( { \pmb x } ^ { \prime } , s ^ { \prime } )$ are independently drawn from the joint distribution $p _ { x s }$ .
+89 In practice, it is necessary to empirically estimate $\deg ( z , s )$ , since the population distributions are
+90 typically unknown in most real-world scenarios.
+191 Definition 3. Let $D = \{ ( \pmb { x } _ { 1 } , \pmb { s } _ { 1 } , \pmb { y } _ { 1 } ) , \cdot \cdot \cdot , ( \pmb { x } _ { n } , \pmb { s } _ { n } , \pmb { y } _ { n } ) \}$ be the training data, containing $n$ i.i.d.
+192 realizations from the joint distribution $p _ { x s y }$ . Using, the representer theorem [45], it follows that
+193 $\pmb { f } ( \pmb { x } ) = \Theta _ { E } { [ k _ { x } ( x _ { 1 } , \pmb { x } ) , \allowbreak \cdot \cdot \cdot , k _ { x } ( x _ { n } , \pmb { x } ) ] } ^ { T }$ , where $\boldsymbol { \Theta } \in \mathbb { R } ^ { r \times n }$ is a free parameter matrix.
+194 Lemma 3. Let an empirical estimation of covariance be
+$$
+\mathbb { C } \mathrm { o v } _ { \boldsymbol { x } , s } ( f _ { j } ( \boldsymbol { x } ) , \beta _ { s } ( \boldsymbol { s } ) ) \approx \frac { 1 } { n } \sum _ { i = 1 } ^ { n } f _ { j } ( \boldsymbol { x } _ { i } ) \beta _ { s } ( \boldsymbol { s } _ { i } ) - \frac { 1 } { n ^ { 2 } } \sum _ { i = 1 } ^ { n } \sum _ { k = 1 } ^ { n } f _ { j } ( \boldsymbol { x } _ { i } ) \beta _ { s } ( \boldsymbol { s } _ { k } ) .
+$$
+195 Then, the empirical estimator of $\deg ( z , s )$ is given by
+$$
+\mathrm { d e p } ^ { \mathrm { e m p } } ( z , s ) \quad : = \quad \frac { 1 } { n ^ { 2 } } \big \| \Theta K _ { x } H L _ { s } \big \| _ { F } ^ { 2 } ,
+$$
+196 where $K _ { x } , K _ { s } \in \mathbb { R } ^ { n \times n }$ are Gram matrices corresponding to $\mathcal { H } _ { x }$ and $\mathcal { H } _ { s }$ , respectively, ${ \textbf { \em H } } =$
+197 $\textstyle I - { \frac { 1 } { n } } \mathbf { 1 } \mathbf { 1 } ^ { T }$ , and $\mathbf { \boldsymbol { L _ { s } } }$ is a full column-rank matrix in which $\bar { \pmb { L } } _ { s } \pmb { L } _ { s } ^ { T } = \pmb { K } _ { s }$ (Cholesky factorization).
+198 This empirical estimator in (9) has a bias of $\mathcal { O } ( n ^ { - 1 } )$ and a convergence rate of $\mathcal { O } ( n ^ { - 1 / 2 } )$ .
+The population and empirical dependence measures between $_ z$ and $\textbf {  { y } }$ i.e., $\mathrm { d e p } ( z , y )$ and $\mathrm { d e p } ^ { \mathrm { e m p } } ( z , y )$ , respectively, can be defined and obtained similarly.
+# 3.2 Trade-Off D
+202 We now turn to the the optimization problem corresponding to the trade-off $\mathbf { D }$ in (1). Recall that
+203 $z = f ( x )$ is $r$ -dimensional, where the dimensionality $r$ is a free variable. A common desiderata of
+204 learned representations is that of compactness [46] in order to avoid learning representations with
+205 redundant information where different dimensions are highly correlated with each other. Therefore,
+206 going beyond the assumption that each component of $f ( \cdot )$ (i.e., $f _ { j } ( \cdot ) ) ,$ ) belongs to a $L _ { 2 }$ −universal
+207 RKHS $\mathcal { H } _ { x }$ , we impose additional constraints on the representation. Specifically, we constrain the
+208 search space of the encoder $f ( \cdot )$ to learn a disentangled representation [46] as follows,
+$$
+\begin{array} { r } { \mathcal { A } _ { r } : = \Big \{ \Big ( f _ { 1 } ( \cdot ) , \cdot \cdot \cdot , f _ { r } ( \cdot ) \Big ) \Big | f _ { i } , f _ { j } \in \mathcal { H } _ { \pmb { x } } , \mathbb { C } \mathrm { o v } _ { \pmb { x } } ( f _ { i } ( \pmb { x } ) , f _ { j } ( \pmb { x } ) ) + \gamma \langle f _ { i } , f _ { j } \rangle _ { \mathcal { H } _ { \pmb { x } } } = \delta _ { i , j } \Big \} , } \end{array}
+$$
+209 where the regularization term $\gamma \langle f _ { i } , f _ { j } \rangle _ { \mathcal { H } _ { x } }$ , encourages orthogonality and boundedness, which in turn
+210 forces the representation to be compact or non-redundant. Such disentangled representations have
+211 been studied in the context of independent component analysis (ICA) [39]. Now, the optimization
+212 problem in (1) reduces to,
+$$
+\operatorname* { s u p } _ { f \in \mathcal { A } _ { r } } \Big \{ J ( f ( x ) ) : = ( 1 - \tau ) \deg ( f ( x ) , y ) - \tau \deg ( f ( x ) , s ) \Big \} , \quad 0 \leq \tau < 1 ,
+$$
+where as justified earlier the target loss function in213 f $_ { Y } \mathbb { E } _ { { \pmb x } , { \pmb y } } [ { \mathcal { L } } _ { Y } ( f _ { T } ( { \pmb f } ( { \pmb x } ) ) , { \pmb y } ) ]$ is substituted by 214 $- \mathrm { d e p } ( f ( \pmb { x } ) , \pmb { y } )$ . Fortunately, the above optimization problem lends itself to a closed-form solu215 tion as given by the next theorem.
+216 Theorem 4. A solution5 to the optimization problem in (11) is the eigenfunctions corresponding to $r$
+217 largest eigenvalues of the following generalized problem
+$$
+\begin{array} { r } { \left( ( 1 - \tau ) \Sigma _ { y x } ^ { * } \Sigma _ { y x } - \tau \Sigma _ { s x } ^ { * } \Sigma _ { s x } \right) f = \lambda \Sigma _ { x x } f , } \end{array}
+$$
+where 218 $\Sigma _ { s x }$ and $\Sigma _ { y x }$ are the covariance operators defined in (7), and $\Sigma _ { s x } ^ { * }$ and $\Sigma _ { y x } ^ { * }$ are the adjoint 19 operators of $\Sigma _ { s x }$ and $\Sigma _ { y x }$ , respectively.
+Remark. If the trade-off parameter $\tau = 0$ (i.e., no semantic independence constraint is imposed), the solution in Theorem 4 resembles a supervised version of ICA in [39] which is essentially a kernelized dimensionality reduction supervised by the target attribute $\textbf {  { y } }$ . On the other hand, if $\tau  1$ (i.e., utility is ignored and only semantic independence is considered), the solution in Theorem 4 is the eigenfunctions corresponding to the negative eigenvalues of $\Sigma _ { s x } ^ { * } \Sigma _ { s x }$ , which are the directions that are least explanatory of the semantic attribute $\pmb { s }$ .
+226 An empirical version of (11) is the following optimization problem
+$$
+\operatorname* { s u p } _ { f \in { \mathcal A } _ { r } } \Big \{ J ^ { \mathrm { e m p } } ( f ( x ) ) : = ( 1 - \tau ) { \mathrm { d e p } } ^ { \mathrm { e m p } } ( f ( x ) , y ) - \tau { \mathrm { d e p } } ^ { \mathrm { e m p } } ( f ( x ) , s ) \Big \} , \quad 0 \leq \tau < 1
+$$
+where de ${ \mathfrak { p } } ^ { \mathrm { e m p } } ( f ( { \pmb x } ) , { \pmb s } )$ and $\log ^ { \mathrm { e m p } } ( f ( x ) , y )$ are given in (9).
+228 Theorem 5. Consider the Cholesky factorization $K _ { x } = L _ { x } L _ { x } ^ { T }$ , where $\scriptstyle L _ { x }$ is a full column-rank
+229 matrix. A solution to (13) is
+$$
+\pmb { f } ^ { \mathrm { o p t } } = \Theta ^ { \mathrm { o p t } } \Big [ k _ { x } ( x _ { 1 } , \cdot ) , \cdot \cdot \cdot , k _ { x } ( x _ { n } , \cdot ) \Big ] ^ { T }
+$$
+where 230 $\Theta ^ { \mathrm { o p t } } = U ^ { T } ( L _ { x } ) ^ { \dag }$ and the columns of $U$ are eigenvectors corresponding to $r$ largest eigenval231 ues, $\lambda _ { 1 } , \cdots , \lambda _ { r }$ of the following generalized problem,
+$$
+\Big ( L _ { x } ^ { T } \big ( ( 1 - \tau ) \tilde { K } _ { y } - \tau \tilde { K } _ { s } \big ) L _ { x } \Big ) u = \lambda \Big ( L _ { x } ^ { T } H L _ { x } + n \gamma I \Big ) u
+$$
+where 32 $\gamma$ is the regularization parameter from (10) and the supremum value of (13) is $\textstyle \sum _ { j = 1 } ^ { r } \lambda _ { j }$
+233 Corollary 5.1. Embedding Dimensionality: A useful corollary of Theorem 5 is optimal embedding
+234 dimensionality:
+$$
+\arg \operatorname* { s u p } _ { r } \left\{ \operatorname* { s u p } _ { f \in A _ { r } } \Big \{ J ^ { \mathrm { e m p } } ( f ( x ) ) : = ( 1 - \tau ) \mathrm { d e p } ^ { \mathrm { e m p } } ( f ( x ) , y ) - \tau \mathrm { d e p } ^ { \mathrm { e m p } } ( f ( x ) , s ) \Big \} \right\} ,
+$$
+235 which is the number of positive eigenvalues of the generalized eigenvalue problem in (14). To
+236 intuitively examine this result, consider two extreme cases: i) If there is no semantic independence
+237 constraint (i.e., $\tau = 0$ ), adding more dimensions to the optimum $r$ will not harm the representation
+238 power of $_ z$ . ii) If we only care about semantic independence and ignore the target task (i.e., $\tau  1$ ),
+239 the optimal $r$ would be equal to zero, indicating that a null representation is the best for discarding all
+240 semantic information. In this case, adding more dimension to $_ { z }$ will necessarily violate the semantic
+241 independence constraint. More discussion can be found in the supplementary material.
+242 In the following Theorem, we prove that the empirical solution converges to its population counterpart.
+243 Theorem 6. Assume that $k _ { s } ( \cdot , \cdot )$ and $k _ { y } ( \cdot , \cdot )$ are bounded by one and $f _ { k } ^ { 2 } ( { \pmb x } _ { i } )$ is bounded by $M$ for
+244 any $k = 1 , \ldots , r$ and $i = 1 , \ldots , n$ for which $\pmb { f } = ( f _ { 1 } , \dots , f _ { r } ) \in \mathcal { A } _ { r }$ . For any $n > 1$ and $0 < \delta < 1$
+245 with probability at least $1 - \delta$ , we have
+$$
+\Big | \operatorname* { s u p } _ { f \in A _ { r } } J ( f ( { \pmb x } ) ) - \operatorname* { s u p } _ { { \pmb f } \in A _ { r } } J ^ { \mathrm { e m p } } ( { \pmb f } ( { \pmb x } ) ) \Big | \leq r M \sqrt { \frac { \log ( 6 / \delta ) } { a ^ { 2 } n } } + \mathcal { O } \left( \frac { 1 } { n } \right) ,
+$$
+246 where $0 . 2 2 \leq a \leq 1$ is a constant.
+5The term ’solution’ in any optimization problem in this paper refers to a global optima.
+248 We recall that label space trade-off arises when the representation $_ { z }$ is ideal and is free to be designed
+249 optimally i.e., it does not necessarily depend on the input data $_ { \textbf { \em x } }$ or the encoder’s hypothesis class.
+250 However, we assume that the representation $_ z$ is a direct effect of the target and sensitive variables $_ y$
+251 and $\pmb { s }$ ). Following [47], we use an additive noise model as
+$$
+\pmb { z } = \pmb { f } _ { L } ( \pmb { y } , \pmb { s } ) + \pmb { e } , \quad \pmb { e } \perp \pmb { y } , \pmb { e } \perp \pmb { s }
+$$
+252 where $\pmb { f } _ { L } ( \cdot , \cdot ) \ : \ \mathbb { R } ^ { d _ { y } } \times \mathbb { R } ^ { d _ { s } }  \ \mathbb { R } ^ { r }$ is a Borel-measurable function. Following Section 3.1,
+253 we deploy $- \mathrm { d e p } ( z , y )$ , defined similar to $\deg ( z , s )$ in (8), as a proxy for the loss function
+254 $\operatorname* { i n f } _ { g _ { Y } \in \mathcal { H } _ { y } } \mathbb { E } _ { { \pmb x } , { \pmb y } } [ \mathcal { L } _ { T } ( g _ { T } ( { \pmb z } ) , { \pmb y } ) ]$ . Recall that, the desired optimization problem is given in (2). Instead
+255 of directly optimizing over $z \in L ^ { 2 }$ , we optimize over all Borel-measurable functions $f _ { L } ( \cdot , \cdot )$ by
+256 ignoring $e$ since it is independent of both $\textbf {  { y } }$ and $\pmb { s }$ :
+$$
+\operatorname* { s u p } _ { \pmb { f } _ { L } \in A _ { r } ( \pmb { y } , s ) } \Big \{ ( 1 - \tau ) \mathrm { d e p } ( \pmb { f } _ { L } ( \pmb { y } , s ) , \pmb { y } ) - \tau \mathrm { d e p } ( \pmb { f } _ { L } ( \pmb { y } , s ) , \pmb { s } ) \Big \} ,
+$$
+where $\scriptstyle A _ { r } ( y , s )$ is defined similar to $\mathcal { A } _ { r }$ in (10) by using $( y , s )$ instead of $_ { \textbf { \em x } }$ in the definition. Recall that $\scriptstyle A _ { r } ( y , s )$ ensures that $_ { z }$ will not contain highly correlated (entangled) dimensions, and thus be minimally redundant or maximally compact.
+Remark. The optimization problem in (16) and its empirical counterpart can be solved similar to that of trade-off $\mathbf { D }$ in Theorems 5 and 6 where $_ { \textbf { \em x } }$ is replaced with $( y , s )$ .
+# 3.4 Trade-Off F
+Here we define and discuss the trade-off achievable by practical realizations of representation learning algorithms with either fairness, invariance or semantic independence constraints.
+Definition 4. Feasible Space Trade-Off arises from the statistical dependence between the target feature $\textbf {  { y } }$ and the sensitive attribute $\pmb { s }$ conditioned on the given input data $_ { \textbf { \em x } }$ , the choice of hypothesis class for the learners involved, and the choice of dependence measure adopted. This setting can be formalized as,
+$$
+\operatorname* { i n f } _ { f \in \mathcal { H } _ { x } } \Big \{ ( 1 - \tau ) \operatorname* { i n f } _ { g _ { Y } \in \mathcal { H } _ { y } } \mathbb { E } _ { \mathbf { x } , y } \Big [ \mathcal { L } _ { Y } \Big ( g _ { Y } \big ( f ( x ) ) , y \Big ) \Big ] + \tau \widetilde { \deg } ( f ( x ) , s ) \Big \} , \quad 0 \leq \tau < 1 ,
+$$
+269 where $\mathcal { H } _ { x }$ and $\mathcal { H } _ { y }$ are the hypothesis class for the encoder network and target predictor, respectively,
+270 $\mathcal { L } _ { Y } ( \cdot , \cdot )$ denotes the loss function of target task, and $\widetilde { \mathrm { d e p } } ( f ( \pmb { x } ) , s )$ is a parametric or non-parametric
+271 surrogate measure of dependency quantifying the dependency between representation vector $z =$
+272 $f ( { \pmb x } )$ and the sensitive attribute $\pmb { s }$ .
+273 This setting corresponds to the trade-off $\mathbf { F }$ in Figure 1(b), and is necessarily dominated by the
+274 Data Space Trade-Off D. Multiple factors may lead to such sub-optimal trade-offs. These include,
+275 hypothesis classes that are not universal RKHSs (e.g., [4] considered the case where $\mathcal { H } _ { x }$ is universal,
+276 but $\mathcal { H } _ { s }$ and $\mathcal { H } _ { y }$ are linear RKHSs), the surrogate dependence measure $\widetilde { \mathrm { d e p } } ( f ( \pmb { x } ) , s )$ does not account
+277 for all non-linear dependencies (e.g., [3, 2, 21, 4] which consider adversarially learned dependence
+278 measures), sub-optimal optimization of (17) in terms of achieving only local optima but not the
+279 global optima (e.g., when the hypothesis class is deep neural networks that are optimized through
+280 stochastic gradient descent, or through stochastic gradient descent-ascent in the case of adversarial
+281 representation learning[3, 21, 2]), and combinations thereof.
+# 282 4 Numerical Estimation of Trade-Offs
+283 In this section, we demonstrate the practical utility of the analytical results developed in the paper
+284 and validate our theoretical insights. For this purpose, we design an illustrative toy example that
+285 conforms to the setting studied in the paper and numerically quantify the trade-offs that we introduced.
+286 Experimental validation on more tasks can be found in the supplementary material.
+287 Consider the following Gaussian mixture model from which we generate 4000, 2000, and 2000
+$$
+v = [ v _ { 1 } , v _ { 2 } ] \sim \frac { 1 } { 2 } \Big ( X ( m , \Sigma ) + \mathcal { N } ( m ^ { \prime } , \Sigma ) \Big ) , \quad m = [ 0 , 1 ] , m ^ { \prime } = [ 1 , 1 ] , \Sigma = \mathrm { d i a g } ( 0 . 1 ^ { 2 } , 0 . 1 ^ { 2 } )
+$$
+![](images/cb476a91659ba082677b8fbf209c2a8ac5d8549e7de8aa8c20de3ea4b19a3a9b.jpg)
+Figure 3: (a): A mixture of two Gaussians which generates the input data as ${ \mathbf { \boldsymbol { x } } } = \boldsymbol { v } _ { 1 }$ , the sensitive attribute as $\begin{array} { r } { \pmb { s } = v _ { 1 } ^ { 3 } } \end{array}$ , and the target attribute as $\pmb { y } = [ \widetilde { v } _ { 1 } , v _ { 2 } ^ { 3 } ]$ . (b): Two fundamental trade-offs, $\mathbf { L }$ and $\mathbf { D }$ , together with two baseline feasible trade-offs $\mathbf { F }$ , ARL optimized with SGDA [21] and global optima of ARL with a linear RKHS [4]. (c), (d): The learned embedding for $\tau = 0$ and $\tau = 0 . 5$ , respectively. An invariant representation should collapse $v _ { 1 }$ i.e., the two colors should fully overlap with each other in the embedding. The overlap is partial for $\tau = 0 . 5$ and as $\tau  1$ , the optimal representation is zero.
+288 independent samples for training, validation and testing, respectively. Figure 3(a) shows the test
+289 samples where the samples generated with $_ { m }$ and $m ^ { \prime }$ are in blue and red, respectively. The input data
+290 $_ { \textbf { \em x } }$ is set to $v _ { 1 }$ (the first entry of $\textbf {  { v } }$ ), the sensitive attribute $\pmb { s }$ is $v _ { 1 } ^ { 3 }$ , and the target attribute $\textbf {  { y } }$ is $[ v _ { 1 } , v _ { 2 } ^ { 3 } ]$
+291 In this problem both input data and target attribute are dependent on the sensitive attribute. We choose
+292 all three RKHS $\mathcal { H } _ { x }$ , $\mathcal { H } _ { y }$ , and $\mathcal { H } _ { s }$ to be Gaussian, which is a universal RKHS. The optimal $_ z$ is learned
+293 for the trade-off $\mathbf { D }$ through the closed-form solution in Theorem 5 for different invariance parameter
+294 values $\tau$ in $[ 0 , 1 )$ . Then, this optimal embedding is fed to a target task predictor which is a multi-layer
+295 perceptron (MLP) with two hidden layers, and 4, 8 neurons and optimize the mean-squared error
+296 (MSE). The $\mathbf { X }$ -axis is a normalized version of the dependence measure used in our optimization, while
+297 the y-axis quantifies utility normalized to $[ 0 , 1 ]$ as $\exp ( - \mathrm { \mathbf { M } S E ) }$ . The same procedure is implemented
+298 for trade-off $\mathbf { L }$ , except that the input data is $\textbf {  { v } }$ , instead of $_ { \textbf { \em x } }$ . These trade-offs are shown in Figure 3(b).
+299 We choose the input data to be $\textbf {  { v } }$ instead of $( y , s )$ for trade-off $\mathbf { L }$ since $( y , s )$ is fully generated from
+300 $\textbf {  { v } }$ and therefore, $\textbf {  { v } }$ perfectly explains $( y , s )$ . For $\tau = 0$ and $\tau = 0 . 5$ , the optimal embeddings are
+301 illustrated in Figure 3, (c) and (d), respectively. Since the sensitive attribute is only related to $v _ { 1 }$ ,
+302 an invariant embedding should collapse the corresponding dimension and cause the two colors to
+303 overlap with each other.
+304 We make the following observations, (a) Trade-off $\mathbf { L }$ dominates trade-off $\mathbf { D }$ as expected. (b) The
+305 trade-offs $\mathbf { F }$ obtained by the baselines are dominated by trade-off $\mathbf { D }$ . Adversarial representation
+306 learning [3, 21, 2] uses sub-optimal optimization (SGDA), while Spectral-ARL [4] uses a global
+307 optimum solution but restricts the hypothesis class in (3) to linear RKHS. As such, the baselines are
+308 unable to match the global optimal solution of (13), and (c) At $\tau = 0 . 5$ the embedding does indeed
+309 collapse $v _ { 1 }$ to an extent leading to partial overlap between the two mixtures.
+# 310 5 Conclusions and Societal Impact
+311 This paper developed the theoretical underpinnings for identifying and determining the fundamental
+312 trade-offs and limits of representation learning under competing objectives. These trade-offs included
+313 i) label space trade-off which is solely induced by the statistical relation between target task and
+314 semantic attribute; ii) data space trade-off which is due to the statistical dependence between the
+315 input data and both target and semantic attributes. Further, we found closed-from solutions for the
+316 global optima, both the population and empirical versions, for the underlying optimization problems,
+317 and thus quantify the trade-offs exactly. Our results shed light on the regions of the trade-off that are
+318 feasible or impossible to achieve by learning algorithms. Numerical results suggest that commonly
+319 used adversarial representation learning based techniques are unable to reach the optimal trade-offs.
+320 The theoretical results in this paper are useful for algorithmic fairness, privacy-preservation, and
+321 domain generalization applications of representation learning. Such systems are being widely
+322 deployed in a variety of practical applications: search engines, social media, law enforcement,
+323 healthcare, consumer devices, financial and judicial risk assessments, face analysis, and many more.
+324 Therefore, providing theoretical limits of performance is critically important for informed framing
+325 of regulatory policies, deployment of such solutions, and gaining societal trust. As such, we do not
+326 anticipate any adverse societal impacts from this work.
+[1] P. Roy and V. N. Boddeti, “Mitigating information leakage in image representations: A maximum entropy approach,” in IEEE Conference on Computer Vision and Pattern Recognition, 2019.
+[2] D. Madras, E. Creager, T. Pitassi, and R. Zemel, “Learning adversarially fair and transferable representations,” arXiv preprint arXiv:1802.06309, 2018.
+[3] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky, “Domain-adversarial training of neural networks,” The Journal of Machine Learning Research, vol. 17, no. 1, pp. 2096–2030, 2016.
+[4] B. Sadeghi, R. Yu, and V. Boddeti, “On the global optima of kernelized adversarial representation learning,” in IEEE International Conference on Computer Vision, pp. 7971–7979, 2019.
+[5] H. Zhao and G. J. Gordon, “Inherent tradeoffs in learning fair representations,” arXiv preprint arXiv:1906.08386, 2019.
+[6] D. McNamara, C. S. Ong, and R. C. Williamson, “Costs and benefits of fair representation learning,” in AAAI/ACM Conference on AI, Ethics, and Society, pp. 263–270, 2019.
+[7] S. Dutta, D. Wei, H. Yueksel, P.-Y. Chen, S. Liu, and K. Varshney, “Is there a trade-off between fairness and accuracy? a perspective using mismatched hypothesis testing,” in International Conference on Machine Learning, pp. 2803–2813, PMLR, 2020.
+[8] H. Zhao, J. Chi, Y. Tian, and G. J. Gordon, “Trade-offs and guarantees of adversarial representation learning for information obfuscation,” arXiv preprint arXiv:1906.07902, 2019.
+[9] N. Martinez, M. Bertran, and G. Sapiro, “Minimax pareto fairness: A multi objective perspective,” in International Conference on Machine Learning, pp. 6755–6764, PMLR, 2020.
+[10] H. Zhao, C. Dan, B. Aragam, T. S. Jaakkola, G. J. Gordon, and P. Ravikumar, “Fundamental limits and tradeoffs in invariant representation learning,” arXiv preprint arXiv:2012.10713, 2020.
+[11] Y. Ganin and V. Lempitsky, “Unsupervised domain adaptation by backpropagation,” in International Conference on Machine Learning, 2015.
+[12] E. Tzeng, J. Hoffman, K. Saenko, and T. Darrell, “Adversarial discriminative domain adaptation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2017.
+[13] H. Zhao, S. Zhang, G. Wu, J. M. Moura, J. P. Costeira, and G. J. Gordon, “Adversarial multiple source domain adaptation,” Advances in Neural Information Processing Systems, vol. 31, pp. 8559–8570, 2018.
+[14] C. Dwork, M. Hardt, T. Pitassi, O. Reingold, and R. Zemel, “Fairness through awareness,” in Innovations in Theoretical Computer Science Conference, pp. 214–226, 2012.
+[15] S. Ruggieri, “Using t-closeness anonymity to control for non-discrimination.,” Trans. Data Priv., vol. 7, no. 2, pp. 99–129, 2014.
+[16] M. Feldman, S. A. Friedler, J. Moeller, C. Scheidegger, and S. Venkatasubramanian, “Certifying and removing disparate impact,” in ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 259–268, 2015.
+[17] F. Calmon, D. Wei, B. Vinzamuri, K. N. Ramamurthy, and K. R. Varshney, “Optimized preprocessing for discrimination prevention,” in Advances in Neural Information Processing Systems, pp. 3992–4001, 2017.
+[18] R. Zemel, Y. Wu, K. Swersky, T. Pitassi, and C. Dwork, “Learning fair representations,” in International Conference on Machine Learning, pp. 325–333, 2013.
+[19] H. Edwards and A. Storkey, “Censoring representations with an adversary,” arXiv preprint arXiv:1511.05897, 2015.
+[20] A. Beutel, J. Chen, Z. Zhao, and E. H. Chi, “Data decisions and theoretical implications when adversarially learning fair representations,” arXiv preprint arXiv:1707.00075, 2017.
+[21] Q. Xie, Z. Dai, Y. Du, E. Hovy, and G. Neubig, “Controllable invariance through adversarial feature learning,” in Advances in Neural Information Processing Systems, pp. 585–596, 2017.
+[22] B. H. Zhang, B. Lemoine, and M. Mitchell, “Mitigating unwanted biases with adversarial learning,” in AAAI/ACM Conference on AI, Ethics, and Society, 2018.
+[23] J. Song, P. Kalluri, A. Grover, S. Zhao, and S. Ermon, “Learning controllable fair representations,” in International Conference on Artificial Intelligence and Statistics (AISTATS), 2019.
+[24] M. Bertran, N. Martinez, A. Papadaki, Q. Qiu, M. Rodrigues, G. Reeves, and G. Sapiro, “Adversarially learned representations for information obfuscation and inference,” in International Conference on Machine Learning, 2019.
+[25] E. Creager, D. Madras, J.-H. Jacobsen, M. A. Weis, K. Swersky, T. Pitassi, and R. Zemel, “Flexibly fair representation learning by disentanglement,” in International Conference on Machine Learning (ICML), 2019.
+[26] F. Locatello, G. Abbati, T. Rainforth, S. Bauer, B. Schölkopf, and O. Bachem, “On the fairness of disentangled representations,” in Advances in Neural Information Processing Systems, pp. 14611–14624, 2019.
+[27] J. Mary, C. Calauzenes, and N. El Karoui, “Fairness-aware learning for continuous attributes and treatments,” in International Conference on Machine Learning, pp. 4382–4391, PMLR, 2019.
+[28] J. Hamm, “Minimax filter: Learning to preserve privacy from inference attacks,” The Journal of Machine Learning Research, vol. 18, no. 1, pp. 4704–4734, 2017.
+[29] M. Coavoux, S. Narayan, and S. B. Cohen, “Privacy-preserving neural representations of text,” arXiv preprint arXiv:1808.09408, 2018.
+[30] T. Xiao, Y.-H. Tsai, K. Sohn, M. Chandraker, and M.-H. Yang, “Adversarial learning of privacy-preserving and task-oriented representations,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, pp. 12434–12441, 2020.
+[31] M. Dusmanu, J. L. Schönberger, S. N. Sinha, and M. Pollefeys, “Privacy-preserving visual feature descriptors through adversarial affine subspace embedding,” in IEEE Conference on Computer Vision and Pattern Recognition, 2021.
+[32] E. Adeli, Q. Zhao, A. Pfefferbaum, E. V. Sullivan, L. Fei-Fei, J. C. Niebles, and K. M. Pohl, “Representation learning with statistical independence to mitigate bias,” in IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 2513–2523, 2021.
+[33] V. Grari, O. E. Hajouji, S. Lamprier, and M. Detyniecki, “Learning unbiased representations via re’nyi minimization,” arXiv preprint arXiv:2009.03183, 2020.
+[34] B. K. Sriperumbudur, K. Fukumizu, and G. R. Lanckriet, “Universality, characteristic kernels and rkhs embedding of measures.,” Journal of Machine Learning Research, vol. 12, no. 7, 2011.
+[35] D. Greenfeld and U. Shalit, “Robust learning with the hilbert-schmidt independence criterion,” in International Conference on Machine Learning, pp. 3759–3768, PMLR, 2020.
+[36] M. I. Belghazi, A. Baratin, S. Rajeshwar, S. Ozair, Y. Bengio, A. Courville, and D. Hjelm, “Mutual information neural estimation,” in International Conference on Machine Learning, pp. 531–540, PMLR, 2018.
+[37] A. Gretton, O. Bousquet, A. Smola, and B. Schölkopf, “Measuring statistical dependence with hilbert-schmidt norms,” in International Conference on Algorithmic Learning Theory, pp. 63–77, Springer, 2005.
+[38] A. Gretton, R. Herbrich, A. Smola, O. Bousquet, and B. Schölkopf, “Kernel methods for measuring independence,” Journal of Machine Learning Research, vol. 6, no. Dec, pp. 2075– 2129, 2005.
+[39] F. R. Bach and M. I. Jordan, “Kernel independent component analysis,” Journal of Machine Learning Research, vol. 3, no. Jul, pp. 1–48, 2002.
+[40] K. Fukumizu, F. R. Bach, and A. Gretton, “Statistical consistency of kernel canonical correlation analysis.,” Journal of Machine Learning Research, vol. 8, no. 2, 2007.
+[41] E. Kreyszig, Introductory functional analysis with applications, vol. 1. wiley New York, 1978.
+[42] P. Diaconis and D. Freedman, “Asymptotics of graphical projection pursuit,” The Annals of Statistics, pp. 793–815, 1984.
+[43] P. Hall and K.-C. Li, “On almost linearity of low dimensional projections from high dimensional data,” The Annals of Statistics, pp. 867–889, 1993.
+[44] A. Rényi, “On measures of dependence,” Acta Mathematica Academiae Scientiarum Hungarica, vol. 10, no. 3-4, pp. 441–451, 1959.
+[45] J. Shawe-Taylor, N. Cristianini, et al., Kernel methods for pattern analysis. Cambridge university press, 2004.
+[46] Y. Bengio, A. Courville, and P. Vincent, “Representation learning: A review and new perspectives,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 8, pp. 1798–1828, 2013.
+[47] J. M. Mooij, J. Peters, D. Janzing, J. Zscheischler, and B. Schölkopf, “Distinguishing cause from effect using observational data: methods and benchmarks,” The Journal of Machine Learning Research, vol. 17, no. 1, pp. 1103–1204, 2016.
+# 440 Checklist
+1. For all authors...
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 3.2 and Section 3.3. Particularly, see Theorem 4 and Theorem 5.
+(b) Did you describe the limitations of your work? [Yes] See the discussion above equation (5) and below equation (8).
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] See Section 5.
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+2. If you are including theoretical results...
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3.1. Particularly, see Assumption 1 and Assumption 2 and discussion above equation (5) and below equation (8).
+(b) Did you include complete proofs of all theoretical results? [Yes] See supplementary material for the proofs of all Lemmas and Theorems.
+3. If you ran experiments...
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See supplementary material.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4 and supplementary material.
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Only one of the baseline methods, ARL, requires running multiple times with different random seeds. The error bar of ARL results is given in supplementary material.
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] This paper is not about computational complexity and/or execution time.
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+(a) If your work uses existing assets, did you cite the creators? [Yes] See supplementary material for the citation to the publicly available repository that we used.
+(b) Did you mention the license of the assets? [N/A]
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] We are not using any new assets.
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+5. If you used crowdsourcing or conducted research with human subjects...
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/YSzTMntO1KY/YSzTMntO1KY.md ADDED Viewed

	@@ -0,0 +1,232 @@

+# SIMONe: View-Invariant, Temporally-Abstracted Object Representations via Unsupervised Video Decomposition
+Rishabh Kabra1, Daniel Zoran1, Goker Erdogan1, Loic Matthey1 Antonia Creswell1, Matthew Botvinick1, Alexander Lerchner1, Christopher P. Burgess2∗
+1DeepMind, 2Wayve, ∗Work done at DeepMind {rkabra, danielzoran, gokererdogan, lmatthey, tonicreswell, botvinick, lerchner}@deepmind.com, chrisburgess@wayve.ai
+# Abstract
+To help agents reason about scenes in terms of their building blocks, we wish to extract the compositional structure of any given scene (in particular, the configuration and characteristics of objects comprising the scene). This problem is especially difficult when scene structure needs to be inferred while also estimating the agent’s location/viewpoint, as the two variables jointly give rise to the agent’s observations. We present an unsupervised variational approach to this problem. Leveraging the shared structure that exists across different scenes, our model learns to infer two sets of latent representations from RGB video input: a set of "object" latents, corresponding to the time-invariant, object-level contents of the scene, as well as a set of "frame" latents, corresponding to global time-varying elements such as viewpoint. This factorization of latents allows our model, SIMONe, to represent object attributes in an allocentric manner which does not depend on viewpoint. Moreover, it allows us to disentangle object dynamics and summarize their trajectories as time-abstracted, view-invariant, per-object properties. We demonstrate these capabilities, as well as the model’s performance in terms of view synthesis and instance segmentation, across three procedurally generated video datasets.
+# 1 Introduction
+The problem of unsupervised visual scene understanding has become an increasingly central topic in machine learning [1, 2]. The attention is merited by potential gains to reasoning, autonomous navigation, and myriad tasks. However, within the current literature, different studies frame the problem in different ways. One approach aims to decompose images into component objects and object features, supporting (among other things) generation of alternative data that permits insertion, deletion, or repositioning of individual objects [3–6]. Another approach aims at a very different form of decomposition—between allocentric scene structure and a variable viewpoint—supporting generation of views of a scene from new vantage points [7–9] and, if not supplied as input, estimation of camera pose [10]. Although there is work pursuing both of these approaches concurrently in the supervised setting [11–13], very few previous studies have approached the combined challenge in the unsupervised case. In this work, we introduce the Sequence-Integrating Multi-Object Net (SIMONe), a model which pursues that goal of object-level and viewpoint-level scene decomposition and synthesis without supervision. SIMONe is designed to handle these challenges without privileged information concerning camera pose, and in dynamic scenes.
+Given a video of a scene our model is able to decouple scene structure from viewpoint information (see Figure 1). To do so, it utilizes video-based cues, and a structured latent space which separates time-invariant per-object features from time-varying global features. These features are inferred using a transformer-based network which integrates information jointly across space and time.
+![](images/474ef165df2ffbd0478d6a7e3fa573cff337922fa05bcf1f109a4235f98414d0.jpg)
+Figure 1: Decomposition (A): SIMONe factorizes a scene sequence $\mathbf { X }$ into scene content (“object latents,” constant across the sequence) and view/global content (“frame latents,” one per frame) without supervision. Its spatio-temporal attention-based inference naturally allows stable object tracking (e.g. the green sphere is assigned the same segment across frames). Recomposition (B): Object latents of a given sequence X can be recomposed with the frame latents of a different (i.i.d.) sequence $\mathbf { X } ^ { \prime }$ to generate a consistent rendering of the same scene (i.e. objects and their properties, relative arrangements, and segmentation assignments) from entirely different viewpoints. Notice that both camera pose and lighting are transferred, as evidenced by the wall corners in the background and the shadows of the green sphere.
+Second, our method seeks to summarize objects’ dynamics. It learns to disentangle not only static object attributes (and their 2D spatial masks), but also object trajectories, without any prior notion of these objects, from videos alone. The learnt trajectory features are temporally abstract and per object; they are captured independently of the dynamics of camera pose, which being a global property, is captured in the model’s per-frame (time-varying) latents.1
+Our model thus advances the state of the art in unsupervised, object-centric scene understanding by satisfying the following desiderata: (1) decomposition of multi-object scenes from RGB videos alone; (2) handling of changing camera pose, and simultaneous inference of scene contents and viewpoint from correlated views (i.e. sequential observations of a moving agent); (3) learning of structure across diverse scene instances (i.e. procedurally sampled contents); (4) object representations which summarize static object attributes like color or shape, view-dissociated properties like position or size, as well as time-abstracted trajectory features like direction of motion; (5) no explicit assumptions of 3D geometry, no explicit dynamics model, no specialized renderer, and few a priori modeling assumptions about the objects being studied; and (6) simple, scalable modules (for inference and rendering) to enable large-scale use.
+# 2 Related Work
+Given the multifaceted problem it tackles, SIMONe connects across several areas of prior work. We describe its nearest neighbors from three scene understanding domains below:
+Scene decomposition models. (1) Our work builds on a recent surge of interest in unsupervised scene decomposition and understanding, especially using slot structure to capture the objects in a scene [14]. One line of work closely related to SIMONe includes methods like [3–6, 15], which all share SIMONe’s Gaussian mixture pixel likelihood model. While these prior methods handled only static scenes, more recent work [16–18, 12] has extended them to videos with promising results. Nevertheless, these approaches have no mechanism or inductive bias to separate view information from scene contents. Moreover, many of them are conditioned on extra inputs like the actions of an agent/camera to simplify inference. (2) Another family of decomposition models originated with Attend, Infer, Repeat (AIR) [19]. AIR’s recurrent attention mechanism does split images into components with separate appearance and pose latents each. Later work [6, 20–23] also extended the model to videos. Despite their structured latents, these models do not learn to distill object appearance into a time-invariant representation (as their appearance and pose latents are free to vary as a function of time). They also require separate object discovery and propagation modules to handle appearing/disappearing objects. In contrast, SIMONe processes a full sequence of images using spatio-temporal attention and produces a single time-invariant latent for each object, hence requiring no explicit transition model or discovery/propagation modules.
+Multi-view scene rendering models. Models which assume viewpoint information for each image like GQN [7], SRNs [8], and NeRF [9] have shown impressive success at learning implicit scene representations and generating novel views from different viewpoints. Recent work [24–27] has further extended these models to videos using deformation fields to model changes in scene geometry over time. In contrast to SIMONe, these models can achieve photorealistic reconstructions by assuming camera parameters (viewpoint information). To allow a direct comparison, we use a view-supervised version of SIMONe in Section 4.1. There is also recent work [28, 29] that relaxes the known viewpoint constraint, but they still model single scenes at a time, which prevents them from exploiting regularities over multiple scenes. A more recent line of work [30–32] explored amortizing inference by mapping from a given set of images to scene latents, but they cannot handle videos yet. Note that all of these models treat the whole scene as a single entity and avoid decomposing it into objects. One exception here is [33], which represents objects with separate pose and appearance latents. However, this model is purely generative and cannot infer object latents from a given scene. Another exception is [13], which can in fact infer object representations, but nevertheless depends on view supervision.
+Simultaneous localization and mapping. The problem of inferring scene representations in a novel environment by exploring it (rather than assuming given views and viewpoint information) is well studied in robotics and vision [10]. Classic SLAM techniques often rely on EM [34, 35] or particle filters [36] to infer viewpoint and scene contents jointly. While our problem is slightly simpler (we can leverage shared structure across scene instances; certain elements such as the shape of the room are held constant; and we use offline data rather than active exploration), our approach of using a factorized variational posterior provides a learning-based solution to the same computational problem. Our simplified setting is perhaps justified by our unsupervised take on the problem. On the other hand, we don’t assume simplifications which may be common in robotics practice (e.g. known camera properties like field of view; or the use of multiple cameras or depth sensors). Most popular SLAM benchmarks [37–39] are on unstructured 3D scenes and hence it was not straightforward for us to compare directly to classic methods. But an encouraging point of overlap is that object-centric SLAM formulations [11] as well as learning-based solutions [40, 41] are active topics of research. Our work could open new avenues in object-centric scene mapping without supervision.
+# 3 Model
+SIMONe is a variational auto-encoder [42] consisting of an inference network (encoder) which infers latent variables from a given input sequence, and a generative process (decoder) which decodes these latents back into pixels. Using the Evidence Lower Bound (ELBO), the model is trained to minimize a pixel reconstruction loss and latent compression KL loss. Crucially, SIMONe relies on a factorized latent space which enforces a separation of static object attributes from global, dynamic properties such as camera pose. We introduce our latent factorization and generative process in Section 3.1. Then in Section 3.2, we describe how the latents can be inferred using a transformer-based encoder, significantly simplifying the (recurrent or autoregressive) architectures used in prior work. Finally, we fully specify the training scheme in Section 3.3.
+![](images/ce156cadcf2724beb8d18887fd8697c8c846570c23660240b4d9540c10f68d0b.jpg)
+Figure 2: Architecture of the SIMONe inference network $\mathcal { E } _ { \phi }$ . The transformers integrate information jointly across space and time to infer (the posterior parameters of) the object and frame latents.
+# 3.1 Latent Structure and Generative Process
+Our model aims to capture the structure of a scene, observed as a sequence of images from multiple viewpoints (often along a smooth camera trajectory, though this is not a requirement). Like many recently proposed object-centric models we choose to represent the scene as a set of $K$ object latent variables $\mathbf { O } ^ { \mathsf { ^ { \prime } } } : = \{ \mathbf { o } _ { k } \} _ { k = 1 } ^ { K }$ . These are invariant by construction across all frames in the sequence (i.e. their distribution is constant through time, and expected to summarize information across the whole sequence). We also introduce $T$ frame latents $\mathbf { F } : = \{ \mathbf { f } _ { t } \} _ { t = 1 } ^ { T }$ , one for each frame in the sequence, that capture time-varying information. Note that by choosing this factorization we reduce the number of required latent variables from $K \cdot T$ to $K + T$ . The latent prior $\begin{array} { r } { p ( \mathbf { O } , \mathbf { F } ) = \prod _ { k } \mathcal { N } ( \mathbf { o } _ { k } \mid \mathbf { \Lambda } } \end{array}$ $\mathbf { 0 } , \mathbf { I } ) \prod _ { t } \mathcal { N } ( \mathbf { f } _ { t } \mid \bar { \mathbf { 0 } } , \mathbf { I } )$ is a unit spherical Gaussian, assuming and enforcing independence between object latents, frame latents, and their feature dimensions.
+Given the latent variables, we assume all pixels and all frames to be independent. Each pixel is modeled as a Gaussian mixture with $K$ components. The mixture weights for pixel $\mathbf { x } _ { t , i }$ capture which component $k$ “explains” that pixel $( 1 \leq i \leq H W$ ). The mixture logits $\hat { m } _ { k , t , i }$ and RGB (reconstruction) means $\mu _ { k , t , i }$ are computed for every component $k$ at a specific time-step $t$ and specific pixel location $\mathbf { l } _ { i }$ using a decoder $\mathcal { D } _ { \theta }$ :
+$$
+\begin{array} { c } { \hat { m } _ { k , t , i } , \pmb { \mu } _ { k , t , i } = \mathcal { D } _ { \theta } ( \mathbf { o } _ { k } , \mathbf { f } _ { t } ; \mathrm { \mathbf { l } } _ { i } , t ) } \\ { p ( \mathbf { x } _ { t , i } \mid \mathbf { o } _ { 1 } , . . . , \mathbf { o } _ { K } , \mathbf { f } _ { t } ; t , \mathrm { \mathbf { l } } _ { i } ) = \displaystyle \sum _ { k } m _ { k , t , i } \mathcal { N } ( \mathbf { x } _ { t , i } \mid \pmb { \mu } _ { k , t , i } ; \sigma _ { x } ) } \end{array}
+$$
+We decode each pixel independently, "querying" our pixel-wise decoder using the sampled latents, coordinates $\mathbf l _ { i } \in [ - 1 , 1 ] ^ { 2 }$ of the pixel, and time-step $t \in [ 0 , 1 )$ being decoded as inputs. The decoder’s architecture is an MLP or 1x1 CNN. (See Appendix A.3.1 for the exact parameterization as well as a diagram of the generative process). By constraining the decoder to work on individual pixels, we can use a subset of pixels as training targets (as opposed to full images; this is elaborated in Section 3.3). Once they are decoded, we obtain the mixture weights $m _ { k , t , i }$ by taking the softmax of the logits across the $K$ components: $m _ { k , t , i } = \tt s o f t m a x _ { k } ( \hat { m } _ { k , t , i } )$ . Equation 2 specifies the full pixel likelihood, where $\sigma _ { x }$ is a scalar hyperparameter.
+# 3.2 Inference
+Given a sequence of frames $\mathbf { X } : = \{ \mathbf { x } _ { t } \} _ { t = 1 } ^ { T }$ we now wish to infer the corresponding object latents $\mathbf { O }$ and frame latents $\mathbf { F }$ . The exact posterior distribution $p ( \mathbf { O } , \mathbf { F } \mid \mathbf { X } )$ is intractable so we resort to using a Gaussian approximate posterior $q ( \mathbf { O } , \mathbf { F } \mid \mathbf { X } )$ . The approximate posterior is parameterized as the output of an inference (encoder) network $\mathcal { E } _ { \phi } ( \mathbf { X } )$ which outputs the mean and (diagonal) log scale for all latent variables given the input sequence.
+SIMONe’s inference network is based on the principle that spatio-temporal data can be processed jointly across space and time using transformers. Beyond an initial step, we don’t need the translation invariance of a CNN, which forces spatial features to interact gradually via a widening receptive field. Nor do we need the temporal invariance of an RNN which forces sequential processing. Instead, we let feature maps interact simultaneously across the cross-product of space and time. See Figure 2 for an overview of our encoder architecture implementing this.
+Concretely, each frame $\mathbf { x } _ { t }$ in the sequence is passed through a CNN which outputs $I J$ spatial feature maps at each time-step (containing $C$ channels each). $I J$ can be larger than the number of object latents $K$ . (For all results in the paper, we set $I$ and $J$ to 8 each, and $K = 1 6$ .) The rest of the inference network consists of two transformers $\mathcal { T } _ { 1 }$ and $\mathcal { T } _ { 2 }$ . $\mathcal { T } _ { 1 }$ takes in all $T I J$ feature maps. Each feature map attends to all others as described. $\mathcal { T } _ { 1 }$ outputs $T I J$ transformed feature maps. When $I J > K$ , we apply a spatial pool to reduce the number of slots to $T K$ (see Appendix A.3.2 for details). These slots serve as the input to $\mathcal { T } _ { 2 }$ , which produces an equal number of output slots. Both transformers use absolute (rather than relative) positional embeddings, but these are 3D to denote the spatio-temporal position of each slot. We denote the output of $\mathcal { T } _ { 2 }$ as $\hat { \mathbf { e } } _ { k , t }$ . This intermediate output is aggregated along separate axes (and passed through MLPs) to obtain $T$ frame and $K$ object posterior parameters respectively. Specifically, $\lambda _ { \mathbf { o } _ { k } } = \mathrm { m l p } _ { o } ( 1 / T \sum _ { t } \hat { \mathbf { e } } _ { k , t } )$ while $\lambda _ { \mathbf { f } _ { t } } = \mathrm { m l p } _ { f } ( 1 / K \sum _ { k } \hat { \mathbf { e } } _ { k , t } )$ Using these posterior parameters we can sample the object latents $\mathbf { o } _ { k } \sim \mathcal { N } ( \lambda _ { \mathbf { o } _ { k } } ^ { \mu } , \mathrm { e x p } ( \lambda _ { \mathbf { o } _ { k } } ^ { \sigma } ) \mathbb { 1 } )$ , and the frame latents $\mathbf { f } _ { t } \sim \mathcal { N } ( \lambda _ { \mathbf { f } _ { t } } ^ { \mu } , \exp ( \lambda _ { \mathbf { f } _ { t } } ^ { \sigma } ) \mathbb { 1 } )$ .
+# 3.3 Loss and Training
+The model is trained end to end by minimizing the following negative-ELBO derivative:
+$$
+\begin{array} { l } { \displaystyle \frac { - \alpha } { T _ { d } H _ { d } W _ { d } } \sum _ { t = 1 } ^ { T _ { d } } \sum _ { i = 1 } ^ { H _ { d } W _ { d } } \log p ( \mathbf { x } _ { t , i } \mid \mathbf { o } _ { 1 } , . . . , \mathbf { o } _ { K } , \mathbf { f } _ { t } ; t , \mathbf { l } _ { i } ) + \displaystyle \frac { \beta _ { o } } { K } \sum _ { k } D _ { K L } ( q ( \mathbf { o } _ { k } \mid \mathbf { X } ) \parallel p ( \mathbf { o } _ { k } ) ) } \\ { + \displaystyle \frac { \beta _ { f } } { T } \sum _ { t } D _ { K L } ( q ( \mathbf { f } _ { t } \mid \mathbf { X } ) \parallel p ( \mathbf { f } _ { t } ) ) } \end{array}
+$$
+We normalize the data log-likelihood by the number of decoded pixels $\left( T _ { d } H _ { d } W _ { d } \right)$ to allow for decoding fewer than all input pixels $( T H W )$ . This helps scale the size of the decoder (without reducing the learning signal, due to the correlations prevalent between adjacent pixels). Normalizing by $1 / \bar { T _ { d } } \bar { H _ { d } } \bar { W _ { d } }$ ensures consistent learning dynamics regardless of the choice of how many pixels are decoded. $\alpha$ is generally set to 1, but available to tweak in case the scale of $\beta _ { o }$ and $\beta _ { f }$ is too small to be numerically stable. Unless explicitly mentioned, we set $\beta _ { o } = \beta _ { f }$ . See Appendix A.3 for details.
+# 4 Comparative Evaluation
+To evaluate the model we focus on two tasks: novel view synthesis and video instance segmentation. On the first task (Section 4.1), we highlight the benefit of view information when it is provided as ground truth to a simplified version of our model (denoted “SIMONe-VS” for view supervised), as well as baseline models like GQN [7] and NeRF-VAE [43]. On the second task (Section 4.2), we deploy the fully unsupervised version of our model; we showcase not only the possibility of inferring viewpoint from data, but also its benefit to extracting object-level structure in comparison to methods like MONet [3], Slot Attention [15], and Sequential IODINE [4].
+Our results are based on three procedurally generated video datasets of multi-object scenes. In increasing order of difficulty, they are: Objects Room 9 [44], CATER (moving camera) [45], and Playroom [46]. These were chosen to meet a number of criteria: we wanted at least 9-10 objects per scene (there can be fewer in view, or as many as 30 in the case of Playroom). We wanted a moving camera with a randomized initial position (the only exception is CATER, where the camera moves rapidly but is initialized at a fixed position to help localization). We wanted ground-truth object masks to evaluate our results quantitatively. We also wanted richness in terms of lighting, texture, object attributes, and other procedurally sampled elements. Finally, we wanted independently moving objects in one dataset (to evaluate trajectory disentangling and temporal abstraction), and unpredictable camera trajectories in another (the Playroom dataset is sampled using an arbitrary agent policy, so the agent is not always moving). Details on all datasets are in Appendix A.2.
+# 4.1 View synthesis (with viewpoint supervision)
+We first motivate view-invariant object representations by considering the case when ground-truth camera pose is provided to our model (a simplified variant we call “SIMONe-VS”). In this scenario, we don’t infer any frame latents. Rather, the encoder and decoder are conditioned on the viewpoint directly. This view-supervised setting is similar to models like GQN and NeRF which represent the contents of a scene implicitly and can be queried in different directions.
+![](images/04b14bb591866ade30a797d55eb4e50d260a7425b92b27c5488464ef25af4457.jpg)
+Figure 3: Comparison of scene representation and view synthesis capabilities between SIMONeVS, NeRF-VAE, and GQN. All models partially observe a procedurally generated Playroom from a given sequence of frames (we visualize 4 of the 16 input frames fed to the models). Then, we decode novel views on a circular trajectory around the room, with the yaw linearly spaced in $[ - \pi , \pi ]$ . NeRF-VAE retains very little object structure, while GQN hallucinates content. SIMONe-VS can produce fine reconstructions of objects that it observes even partially or at a distance (such as the bed or shelves in the scene). SIMONe-VS also segments the scene as a bonus. See Appendix A.5.1 for similar plots from different scenes/input sequences.
+We compare three such models on view synthesis in the Playroom. The models are provided a set of 16 consecutive frames as context, partially revealing a generated room. Having inferred a scene representation from the input context, the models are then tasked with generating unobserved views of the scene. This extrapolation task is performed without any retraining, and tests the coherence of the models’ inferred representations. The task is challenging given the compositional structure of each Playroom scene, as well as the variation across scenes (the color, position, size, and choice of all objects are procedurally sampled per scene; only the L-shaped layout of the room is shared across scenes in the dataset). Because each model is trained on and learns to represent many Playroom instances, NeRF itself is not directly suitable for the task. It needs to be retrained on each scene, whereas we want to infer the specifics of any given room at evaluation time. NeRF-VAE addresses this issue and makes it directly comparable to our model.
+To set up the comparison, we first trained SIMONe-VS and evaluated its log-likelihood on Playroom sequences. Then, we trained GQN and NeRF-VAE using constrained optimization (GECO [47]) to achieve roughly the same log likelihood per pixel. See Appendix A.5.1 for a comparison of the models in terms of the reconstruction-compression trade-off.
+Qualitatively, the models show vast differences in their perceived structure (see Figure 3). NeRF-VAE blurs out nearly all objects in the scene but understands the geometry of the room and is able to infer wall color. GQN produces more detailed reconstructions, but overfits to particular views and does not interpolate smoothly. SIMONe-VS on the other hand finely reproduces the object structure of the room. Even when it observes objects at a distance or up close, it places and sizes them correctly in totally novel views. This makes SIMONe-VS a powerful choice over NeRF-VAE and GQN-style models when the priority is to capture scene structure across diverse examples.
+# 4.2 Instance segmentation (fully unsupervised)
+Having shown the benefit of view information to inferring scene structure in the Section 4.1, we now turn to the added challenge of inferring viewpoint directly and simultaneously with scene contents (without any supervision).
+<table><tr><td></td><td colspan="3">Static ARI-F</td><td colspan="4">Video ARI-F</td></tr><tr><td></td><td>MONet</td><td>SA</td><td>S-IODINE</td><td>MONet</td><td>SA</td><td>S-IODINE</td><td>SIMONe</td></tr><tr><td>Objects Room 9</td><td>0.886 (±0.061)</td><td>0.784 (±0.138)</td><td>0.695 (±0.007)</td><td>0.865 (±0.007)</td><td>0.066 (±0.014)</td><td>0.673 (±0.0.002)</td><td>0.936</td></tr><tr><td>CATER</td><td>0.937</td><td>0.923</td><td>0.728</td><td>0.412</td><td>0.073</td><td>0.668</td><td>(±0.010) 0.918</td></tr><tr><td>Playroom</td><td>(±0.004) 0.647 (±0.012)</td><td>(±0.076) 0.653 (±0.024)</td><td>(±0.032) 0.439 (±0.009)</td><td>(±0.012) 0.442 (±0.010)</td><td>(±0.006) 0.059 (±0.002)</td><td>(±0.033) 0.356 (±0.006)</td><td>(±0.036) 0.800 (±0.043)</td></tr></table>
+Table 1: SIMONe segmentation performance (in terms of Adjusted Rand Index for foreground objects, ARI-F) compared to state-of-the-art unsupervised baselines: two static-frame models (MONet and Slot Attention, SA) and a video model (S-IODINE). We calculate static and video ARI-F scores separately. For static ARI-F, we evaluate the models per still image. For video ARI-F, we evaluate the models across space and time, taking an object’s full trajectory as a single class. The video ARI-F thus penalizes models (especially Slot Attention) which fail to track objects stably. We report the mean and standard deviation of scores across 5 random seeds in each case.
+![](images/0c7466d809c6bf3df48a2ccc0d1e8c7f5114c36d36595842e0cab66e7c99378d.jpg)
+Figure 4: Segmentations and reconstructions produced by SIMONe on CATER and Playroom. SIMONe copes well with clutter and different-sized objects. It learns to use object motion as a segmentation signal on CATER, evident from the fact that an object’s shadow is correctly assigned to that object’s segment as it moves. This is true even when there’s multiple shadows per object (due to multiple lights in the scene). SIMONe also overcomes color-based cues to segment two-toned objects such as beds in the Playroom as single objects. See Appendix A.5.2 to compare with baseline models.
+We compare SIMONe to a range of competitive but viewpoint-unaware scene decomposition approaches. First, we train two static-frame models: MONet and Slot Attention. MONet uses a similar generative process and training loss to our model, achieving segmentation by modeling the scene as a spatial mixture of components, and achieving disentangled representations using a $\beta$ -weighted KL information bottleneck. On the other hand, it uses a deterministic, recurrent attention network to infer object masks. Slot Attention is a transformer-based autoencoding model which focuses on segmentation performance rather than representation learning. Finally, we also compare against Sequential IODINE (“S-IODINE”), which applies a refinement network to amortize inference over time, separating objects by processing them in parallel. It also uses a $\beta$ -weighted KL loss to disentangle object representations. Note that S-IODINE is a simplified version of OP3 [17], which additionally attempts to model (pairwise) object dynamics using an agent’s actions as inputs. SIMONe and S-IODINE both avoid relying on this privileged information. Table 1 contains a quantitative comparison of segmentation performance across these models, while Figure 4 shows qualitative results.
+![](images/4bbe453caa2296f7c2e11f984cd4f4c05d0d48cd785a3a686e3cdb83baeb21db.jpg)
+Figure 5: Object attributes learnt by SIMONe. In each row, we manipulate a particular object latent attribute for an arbitrary target object (circled in red) in two scenes. This reveals the attributes’ relationship to interpretable object characteristics like color, size, position and identity.
+# 5 Analysis
+We take a closer look at the representations learnt by our model by decoding them in various ways. First, we manipulate latent attributes individually to assess the interpretability of object representations visually in Section 5.1. Next, we exploit SIMONe’s latent factorization to render views of a given scene using the camera trajectory of a different input sequence. These cross-over visualizations help identify how the model encodes object dynamics in Section 5.2. Finally, we measure the predictability of ground-truth camera dynamics and object dynamics from the two types of latents in Section 5.3. These analyses use a single, fully unsupervised model per dataset.
+# 5.1 Latent attribute traversals
+We visualize the object representations learnt by SIMONe on Playroom to highlight their disentanglement, across latent attributes and across object slots, in Figure 5. We seed all latents using a given input sequence, then manipulate one object latent attribute at a time by adding fixed offsets.
+Note that object position and size are well disentangled in each direction. Aided by the extraction of view-specific information in the frame latents, SIMONe also learns object features corresponding to identity. The decoder nevertheless obeys the biases in the dataset–for instance, shelves will slide along a wall when their position latent is traversed. The rubber duck does not morph into a chest of drawers because those are always located against a wall. This further suggests a well-structured latent representation, which the decoder can adapt to.
+# 5.2 Object and frame latent cross-overs
+We expect SIMONe to encode object trajectories and camera trajectories independently of each other. In fact, each object’s trajectory should be summarized in its own time-invariant latent code. To examine this, we recompose object latents from one sequence with frame latents from other sequences in the CATER dataset. The result, in Figure 6, is that we can observe object motion trajectories from multiple camera trajectories.
+Note the consistency of relative object positions (at any time-step) from all camera angles. In the single moving object case, its motion could in fact be interpreted as a time-varying global property of the scene. Despite this challenge, SIMONe is able to encode the object’s motion as desired in its specific time-invariant code. In Section 5.3, we further confirm that object trajectories are summarized in the object latents, which can be queried with time to recover allocentric object positions.
+![](images/898e0b3941c04bad1f27e5edb95e93f0dd4c2ceea8de314f4c150297097c6158.jpg)
+Figure 6: Separation of object trajectories from camera trajectories. Left: When encoding a sequence with consistent (i.i.d.) object dynamics, this information is extracted in the object latents and is unaffected by changing frame latents (see green cone). Right: Movement events are sequenced correctly; object relative positions also remain consistent (see pattern of shadows on the floor circled in yellow). See Appendix A.5.3 for cross-over plots showing more object trajectories.
+# 5.3 Camera pose and object trajectory prediction
+We assessed SIMONe’s frame latents by decoding the true camera position and orientation from them. We trained linear and MLP regressors to predict the camera pose at time $t$ from the corresponding frame latent $\mathbf { f } _ { t }$ on a subset of CATER sequences. We also trained an MLP on the time-invariant object latents $\mathbf { o } _ { 1 : K }$ for the same task. We evaluated these decoders on held-out data. Table 2 shows that frame latents describe the viewpoint almost perfectly.
+We also assessed if the object latents contain precise information about allocentric object positions (to be clear, position information is not provided in any form while training SIMONe). Table 3 shows that the correct object latent is predictive of the allocentric position of a dynamic object (when queried along with the timestep). Adding the frame latent does not provide more information, and using
+Table 2: Decoding camera pose. We show that ground-truth camera location or orientation is predictable from the corresponding frame latent, but cannot be predicted from all object latents put together. We report the test $\mathrm { \dot { ~ } R ^ { 2 } }$ score across 5 independently trained decoders per input type.
+<table><tr><td></td><td>Linear(ft)</td><td>MLP(ft)</td><td>MLP(0.1,..,.·0K)</td></tr><tr><td>Camera location</td><td>0.832 ± 0.0</td><td>0.949 ± 0.002</td><td>0.044 ± 0.026</td></tr><tr><td>Camera orientation (Rodrigues)</td><td>0.800 ± 0.0</td><td>0.946±0.002</td><td>0.292±0.025</td></tr></table>
+<table><tr><td></td><td>MLP(Ok)</td><td>MLP(Ok,t)</td><td>MLP(Ok,ft,t)</td><td>MLP({oj : j≠k})</td></tr><tr><td>Trained on all objects</td><td>0.710± 0.006</td><td>0.871 ± 0.006</td><td>0.876 ± 0.003</td><td>-0.062±0.006</td></tr><tr><td>Trained on moving objects</td><td>0.724± 0.007</td><td>0.894± 0.004</td><td>0.898± 0.005</td><td>一 -0.022 ± 0.025</td></tr></table>
+Table 3: Decoding object trajectories. We test MLP decoders on predicting allocentric object positions (of moving object in unseen scenes) based on the following inputs: (a) the corresponding object latent, (b) the timestep as well, and (c) the frame latent corresponding to that timestep as well, and (d) remaining object latents from the scene (not pertaining to the object of interest). The decoders were trained on arbitrary objects or a subset containing moving objects only. We report the test $R ^ { 2 }$ score across 5 independently trained decoders per input type.
+the “wrong” objects (from the same scene) is completely uninformative. To perform this analysis, we needed to align SIMONe’s inferred objects with the ground-truth set of objects. We used the Hungarian matching algorithm on the MSE of inferred object masks and ground-truth object masks to perform the alignment. Given SIMONe’s disentangling of object dynamics, its time-abstracted object representations could prove helpful for a variety of downstream tasks (e.g. “catch the flying ball!”).
+Taken together, Table 2 and Table 3 show the separation of information that is achieved between the object and frame latents, helping assert our two central aims of view-invariant and temporally abstracted object representations.
+# 6 Discussion and Future Work
+Scalability. The transformer-based inference network in SIMONe makes it amenable to processing arbitrarily large videos, just as transformer-based language models can process long text. SIMONe could be trained on windows of consecutive frames sampled from larger videos (aka "chunks"). For inference over a full video, one could add memory slots which carry information over time from one window to the next. Applying SIMONe on sliding windows of frames also presents the opportunity to amortize inference at any given time-step if the windows are partially overlapping (so the model could observe every given frame as part of two or more sequences). Our use of the standard transformer architecture also makes SIMONe amenable to performance improvements via alternative implementations.
+Limitations. (1) SIMONe cannot generate novel videos (e.g. sample a natural camera trajectory via consecutive frame latents) in its current version. This could be addressed in a similar fashion to the way GENESIS [5] built on MONet [3]–it should be possible (e.g. using recurrent networks) to learn conditional priors for objects in a scene and for successive frame latents, which would make SIMONe fully generative. (2) We see another possible limitation arising from our strict latent factorization. We have shown that temporally abstracted object features can predict object trajectories when queried by time. This can cover a lot of interesting cases (even multiple object-level "events" over time), but will start to break as object trajectories get more stochastic (i.e. objects transition considerably/chaotically through time). We leave it to future work to explore how temporal abstraction can be combined with explicit per-step dynamics modeling in those cases. For simpler settings, our approach to encoding object trajectories (distilling them across time) is surprisingly effective.
+# 7 Conclusion
+We’ve presented SIMONe, a latent variable model which separates the time-invariant, object-level properties of a scene video from the time-varying, global properties. Our choice of scalable modules such as transformers for inference, and a pixel-wise decoder, allow the model to extract this information effectively.
+SIMONe can learn the common structure across a variety of procedurally instantiated scenes. This enables it to recognize and generalize to novel scene instances from a handful of correlated views, as we showcased via 360-degree view traversal in the view-supervised setting. More significantly, SIMONe can learn to infer the two sets of latent variables jointly without supervision. Aided by crossframe spatio-temporal attention, it achieves state-of-the-art segmentation performance on complex 3D scenes. Our latent factorization (and information bottleneck pressures) further help with learning meaningful object representations. SIMONe can not only separate static object attributes (like size and position), but it can also separate the dynamics of different objects (as time-invariant localized properties) from global changes in view.
+We have discussed how the model can be applied to much longer videos in the future. It also has potential for applications in robotics (e.g. sim-to-real transfer) and reinforcement learning, where view-invariant object information (and summarizing their dynamics) could dramatically improve how agents reason about objects.
+# Acknowledgements
+We thank Michael Bloesch, Markus Wulfmeier, Arunkumar Byravan, Claudio Fantacci, and Yusuf Aytar for valuable discussions on the purview of our work. We are also grateful for David Ding’s support on the CATER dataset. The authors received no specific funding for this work.
+# References
+[1] Jitendra Malik. Scene understanding in the era of deep learning, Jun 2015. URL https://www.robots. ox.ac.uk/seminars/Extra/2015_06_22_JitendraMalik.pdf.
+[2] Farzad Husain, Babette Dellen, and Carme Torras. Chapter 20 - Scene Understanding Using Deep Learning, pages 373–382. Academic Press, 2017. ISBN 978-0-12-811318-9. doi: https://doi.org/10.1016/ B978-0-12-811318-9.00020-X.
+[3] Christopher P Burgess, Loic Matthey, Nicholas Watters, Rishabh Kabra, Irina Higgins, Matt Botvinick, and Alexander Lerchner. Monet: Unsupervised scene decomposition and representation. arXiv preprint arXiv:1901.11390, 2019.
+[4] Klaus Greff, Raphaël Lopez Kaufman, Rishabh Kabra, Nick Watters, Christopher Burgess, Daniel Zoran, Loic Matthey, Matthew Botvinick, and Alexander Lerchner. Multi-object representation learning with iterative variational inference. In International Conference on Machine Learning, pages 2424–2433. PMLR, 2019.
+[5] Martin Engelcke, Adam R. Kosiorek, Oiwi Parker Jones, and Ingmar Posner. Genesis: Generative scene inference and sampling with object-centric latent representations. In International Conference on Learning Representations, 2020.
+[6] Zhixuan Lin, Yi-Fu Wu, Skand Vishwanath Peri, Weihao Sun, Gautam Singh, Fei Deng, Jindong Jiang, and Sungjin Ahn. Space: Unsupervised object-oriented scene representation via spatial attention and decomposition. In International Conference on Learning Representations, 2020.
+[7] S M Ali Eslami, Danilo Jimenez Rezende, Frederic Besse, Fabio Viola, Ari S Morcos, Marta Garnelo, Avraham Ruderman, Andrei A Rusu, Ivo Danihelka, Karol Gregor, David P Reichert, Lars Buesing, Theophane Weber, Oriol Vinyals, Dan Rosenbaum, Neil Rabinowitz, Helen King, Chloe Hillier, Matt Botvinick, Daan Wierstra, Koray Kavukcuoglu, and Demis Hassabis. Neural scene representation and rendering. Science, 360(6394):1204–1210, June 2018.
+[8] Vincent Sitzmann, Michael Zollhoefer, and Gordon Wetzstein. Scene representation networks: Continuous 3D-Structure-Aware neural scene representations. In H Wallach, H Larochelle, A Beygelzimer, F d’Alché Buc, E Fox, and R Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.
+[9] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In European Conference on Computer Vision, pages 405–421. Springer, 2020.
+[10] C. Cadena, L. Carlone, H. Carrillo, Y. Latif, D. Scaramuzza, J. Neira, I. Reid, and J.J. Leonard. Past, present, and future of simultaneous localization and mapping: Towards the robust-perception age. IEEE Transactions on Robotics, 32(6):1309–1332, 2016.
+[11] Binbin Xu, Wenbin Li, Dimos Tzoumanikas, Michael Bloesch, Andrew Davison, and Stefan Leutenegger. Mid-fusion: Octree-based object-level multi-instance dynamic slam. In 2019 International Conference on Robotics and Automation (ICRA), pages 5231–5237. IEEE, 2019.
+[12] Li Nanbo, Cian Eastwood, and Robert B Fisher. Learning object-centric representations of multi-object scenes from multiple views. In Advances in Neural Information Processing Systems, 2020.
+[13] Chang Chen, Fei Deng, and Sungjin Ahn. Object-centric representation and rendering of 3d scenes. arXiv preprint arXiv:2006.06130, 2020.
+[14] Klaus Greff, Sjoerd van Steenkiste, and Jürgen Schmidhuber. On the binding problem in artificial neural networks. arXiv preprint arXiv:2012.05208, 2020.
+[15] Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold, Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf. Object-centric learning with slot attention. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 11525–11538. Curran Associates, Inc., 2020.
+[16] Antonia Creswell, Rishabh Kabra, Chris Burgess, and Murray Shanahan. Unsupervised object-based transition models for 3d partially observable environments. arXiv preprint arXiv:2103.04693, 2021.
+[17] Rishi Veerapaneni, John D Co-Reyes, Michael Chang, Michael Janner, Chelsea Finn, Jiajun Wu, Joshua Tenenbaum, and Sergey Levine. Entity abstraction in visual model-based reinforcement learning. In Conference on Robot Learning, pages 1439–1456. PMLR, 2020.
+[18] Polina Zablotskaia, Edoardo A Dominici, Leonid Sigal, and Andreas M Lehrmann. Unsupervised video decomposition using spatio-temporal iterative inference. arXiv preprint arXiv:2006.14727, 2020.
+[19] S M Ali Eslami, Nicolas Heess, Theophane Weber, Yuval Tassa, David Szepesvari, Koray Kavukcuoglu, and Geoffrey E Hinton. Attend, infer, repeat: fast scene understanding with generative models. In Proceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, pages 3233–3241, Red Hook, NY, USA, December 2016. Curran Associates Inc.
+[20] Adam Kosiorek, Hyunjik Kim, Yee Whye Teh, and Ingmar Posner. Sequential attend, infer, repeat: Generative modelling of moving objects. Adv. Neural Inf. Process. Syst., 31, 2018.
+[21] Eric Crawford and Joelle Pineau. Exploiting spatial invariance for scalable unsupervised object tracking. arXiv preprint arXiv:1911.09033, 2019.
+[22] Eric Crawford and Joelle Pineau. Spatially invariant unsupervised object detection with convolutional neural networks. AAAI, 33(01):3412–3420, July 2019.
+[23] Jindong Jiang, Sepehr Janghorbani, Gerard de Melo, and Sungjin Ahn. SCALOR: Generative world models with scalable object representations. arXiv preprint arXiv:1910.02384, October 2019.
+[24] Albert Pumarola, Enric Corona, Gerard Pons-Moll, and Francesc Moreno-Noguer. D-NeRF: Neural radiance fields for dynamic scenes. arXiv preprint arXiv:2011.13961, November 2020.
+[25] Keunhong Park, Utkarsh Sinha, Jonathan T Barron, Sofien Bouaziz, Dan B Goldman, Steven M Seitz, and Ricardo Martin-Brualla. Nerfies: Deformable neural radiance fields. arXiv preprint arXiv:2011.12948, 2020.
+[26] Zhengqi Li, Simon Niklaus, Noah Snavely, and Oliver Wang. Neural scene flow fields for Space-Time view synthesis of dynamic scenes. arXiv preprint arXiv:2011.13084, November 2020.
+[27] Yilun Du, Yinan Zhang, Hong-Xing Yu, Joshua B Tenenbaum, and Jiajun Wu. Neural radiance flow for 4D view synthesis and video processing. arXiv preprint arXiv:2012.09790, December 2020.
+[28] Chen-Hsuan Lin, Wei-Chiu Ma, Antonio Torralba, and Simon Lucey. BARF: Bundle-Adjusting neural radiance fields. arXiv preprint arXiv:2104.06405, April 2021.
+[29] Zirui Wang, Shangzhe Wu, Weidi Xie, Min Chen, and Victor Adrian Prisacariu. NeRF–: Neural radiance fields without known camera parameters. arXiv preprint arXiv:2102.07064, February 2021.
+[30] Adam Kosiorek, Heiko Strathmann, Daniel Zoran, Pol Moreno, Rosalia Schneider, Sona Mokrá, and ˇ Danilo Jimenez Rezende. NeRF-VAE: A geometry aware 3D scene generative model. In International Conference on Machine Learning. PMLR, 2021.
+[31] Alex Trevithick and Bo Yang. GRF: Learning a general radiance field for 3D scene representation and rendering. arXiv preprint arXiv:2010.04595, October 2020.
+[32] Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa. pixelNeRF: Neural radiance fields from one or few images. arXiv preprint arXiv:2012.02190, December 2020.
+[33] Michael Niemeyer and Andreas Geiger. GIRAFFE: Representing scenes as compositional generative neural feature fields. arXiv preprint arXiv:2011.12100, November 2020.
+[34] H. Durrant-Whyte and T. Bailey. Simultaneous localization and mapping: part i. IEEE Robotics Automation Magazine, 13(2):99–110, 2006. doi: 10.1109/MRA.2006.1638022.
+[35] T. Bailey and H. Durrant-Whyte. Simultaneous localization and mapping (slam): part ii. IEEE Robotics Automation Magazine, 13(3):108–117, 2006. doi: 10.1109/MRA.2006.1678144.
+[36] Josep Aulinas, Yvan Petillot, Joaquim Salvi, and Xavier Lladó. The slam problem: A survey. In Proceedings of the 2008 Conference on Artificial Intelligence Research and Development: Proceedings of the 11th International Conference of the Catalan Association for Artificial Intelligence, page 363–371, NLD, 2008. IOS Press. ISBN 9781586039257.
+[37] Michael Burri, Janosch Nikolic, Pascal Gohl, Thomas Schneider, Joern Rehder, Sammy Omari, Markus W Achtelik, and Roland Siegwart. The euroc micro aerial vehicle datasets. The International Journal of Robotics Research, 2016. doi: 10.1177/0278364915620033.
+[38] Andreas Geiger, Philip Lenz, and Raquel Urtasun. Are we ready for autonomous driving? the kitti vision benchmark suite. In Conference on Computer Vision and Pattern Recognition (CVPR), 2012.
+[39] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers. A benchmark for the evaluation of rgb-d slam systems. In Proc. of the International Conference on Intelligent Robot Systems (IROS), Oct. 2012.
+[40] Ruihao Li, Sen Wang, and Dongbing Gu. Deepslam: A robust monocular slam system with unsupervised deep learning. IEEE Transactions on Industrial Electronics, 68(4):3577–3587, 2021. doi: 10.1109/TIE. 2020.2982096.
+[41] Mingyang Geng, Suning Shang, Bo Ding, Huaimin Wang, and Pengfei Zhang. Unsupervised learningbased depth estimation-aided visual slam approach. Circuits, Systems, and Signal Processing, 39(2): 543–570, 2020.
+[42] Diederik P Kingma and Max Welling. An introduction to variational autoencoders. arXiv preprint arXiv:1906.02691, 2019.
+[43] Adam R Kosiorek, Heiko Strathmann, Daniel Zoran, Pol Moreno, Rosalia Schneider, Sona Mokrá, ˇ and Danilo J Rezende. Nerf-vae: A geometry aware 3d scene generative model. arXiv preprint arXiv:2104.00587, 2021.
+[44] Rishabh Kabra, Chris Burgess, Loic Matthey, Raphael Lopez Kaufman, Klaus Greff, Malcolm Reynolds, and Alexander Lerchner. Multi-object datasets. https://github.com/deepmind/multi-object-datasets/, 2019.
+[45] Rohit Girdhar and Deva Ramanan. CATER: A diagnostic dataset for compositional actions & temporal reasoning. In International Conference on Learning Representations, 2020.
+[46] Josh Abramson, Arun Ahuja, Arthur Brussee, Federico Carnevale, Mary Cassin, Stephen Clark, Andrew Dudzik, Petko Georgiev, Aurelia Guy, Tim Harley, et al. Imitating interactive intelligence. arXiv preprint arXiv:2012.05672, 2020.
+[47] Danilo Jimenez Rezende and Fabio Viola. Taming vaes. arXiv preprint arXiv:1810.00597, 2018.
+[48] Vasu Parameswaran and Rama Chellappa. View invariance for human action recognition. International Journal of Computer Vision, 66(1):83–101, 2006.
+[49] Yang Liu, Zhaoyang Lu, Jing Li, and Tao Yang. Hierarchically learned view-invariant representations for cross-view action recognition. IEEE Transactions on Circuits and Systems for Video Technology, 29(8): 2416–2430, 2018.
+[50] Ting Liu, Jennifer J Sun, Long Zhao, Jiaping Zhao, Liangzhe Yuan, Yuxiao Wang, Liang-Chieh Chen, Florian Schroff, and Hartwig Adam. View-invariant, occlusion-robust probabilistic embedding for human pose. arXiv preprint arXiv:2010.13321, 2020.
+[51] Jennifer J Sun, Jiaping Zhao, Liang-Chieh Chen, Florian Schroff, Hartwig Adam, and Ting Liu. Viewinvariant probabilistic embedding for human pose. In European Conference on Computer Vision, pages 53–70. Springer, 2020.
+[52] Yana Hasson, Bugra Tekin, Federica Bogo, Ivan Laptev, Marc Pollefeys, and Cordelia Schmid. Leveraging photometric consistency over time for sparsely supervised hand-object reconstruction. In CVPR, 2020.
+[53] Emily L Denton and Vighnesh Birodkar. Unsupervised learning of disentangled representations from video. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
+[54] Yizhe Zhu, Martin Renqiang Min, Asim Kadav, and Hans Peter Graf. S3VAE: Self-supervised sequential VAE for representation disentanglement and data generation. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, June 2020.
+[55] Emily Denton and Rob Fergus. Stochastic video generation with a learned prior. arXiv preprint arXiv:1802.07687, February 2018.
+[56] Ruben Villegas, Jimei Yang, Seunghoon Hong, Xunyu Lin, and Honglak Lee. Decomposing motion and content for natural video sequence prediction. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
+[57] Li Yingzhen and Stephan Mandt. Disentangled sequential autoencoder. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 5670–5679. PMLR, 2018.
+[58] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
+[59] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021.
+[60] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-End object detection with transformers. In Computer Vision – ECCV 2020, pages 213–229. Springer International Publishing, 2020.
+[61] Huiyu Wang, Yukun Zhu, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen. MaX-DeepLab: End-to-End panoptic segmentation with mask transformers. arXiv preprint arXiv:2012.00759, December 2020.
+[62] Sixiao Zheng, Jiachen Lu, Hengshuang Zhao, Xiatian Zhu, Zekun Luo, Yabiao Wang, Yanwei Fu, Jianfeng Feng, Tao Xiang, Philip H S Torr, and Li Zhang. Rethinking semantic segmentation from a Sequence-toSequence perspective with transformers. arXiv preprint arXiv:2012.15840, December 2020.
+[63] Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. arXiv preprint arXiv:2104.14294, 2021.
+[64] Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is Space-Time attention all you need for video understanding? arXiv preprint arXiv:2102.05095, February 2021.
+[65] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan Asselmann. Video transformer network. arXiv preprint arXiv:2102.00719, February 2021.
+[66] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia Schmid. ViViT: ´ A video vision transformer. arXiv preprint arXiv:2103.15691, March 2021.
+[67] Yuqing Wang, Zhaoliang Xu, Xinlong Wang, Chunhua Shen, Baoshan Cheng, Hao Shen, and Huaxia Xia. End-to-End video instance segmentation with transformers. arXiv preprint arXiv:2011.14503, November 2020.
+[68] Fabio Viola, Louise Deason, and Marcel Büsching. Datasets used to train generative query networks (gqns) in the ‘neural scene representation and rendering’ paper. https://github.com/deepmind/gqn-datasets, 2018.
+[69] Felix Hill, Olivier Tieleman, Tamara von Glehn, Nathaniel Wong, Hamza Merzic, and Stephen Clark. Grounded language learning fast and slow. arXiv preprint arXiv:2009.01719, 2020.
+[70] Nicholas Watters, Loic Matthey, Christopher P Burgess, and Alexander Lerchner. Spatial broadcast decoder: A simple architecture for learning disentangled representations in vaes. arXiv preprint arXiv:1901.07017, 2019.

parse/train/YSzTMntO1KY/YSzTMntO1KY_content_list.json ADDED Viewed

	@@ -0,0 +1,973 @@

+[
+    {
+        "type": "text",
+        "text": "SIMONe: View-Invariant, Temporally-Abstracted Object Representations via Unsupervised Video Decomposition ",
+        "text_level": 1,
+        "bbox": [
+            196,
+            122,
+            802,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Rishabh Kabra1, Daniel Zoran1, Goker Erdogan1, Loic Matthey1 Antonia Creswell1, Matthew Botvinick1, Alexander Lerchner1, Christopher P. Burgess2∗ ",
+        "bbox": [
+            189,
+            250,
+            812,
+            280
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1DeepMind, 2Wayve, ∗Work done at DeepMind {rkabra, danielzoran, gokererdogan, lmatthey, tonicreswell, botvinick, lerchner}@deepmind.com, chrisburgess@wayve.ai ",
+        "bbox": [
+            196,
+            280,
+            794,
+            321
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            357,
+            535,
+            375
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To help agents reason about scenes in terms of their building blocks, we wish to extract the compositional structure of any given scene (in particular, the configuration and characteristics of objects comprising the scene). This problem is especially difficult when scene structure needs to be inferred while also estimating the agent’s location/viewpoint, as the two variables jointly give rise to the agent’s observations. We present an unsupervised variational approach to this problem. Leveraging the shared structure that exists across different scenes, our model learns to infer two sets of latent representations from RGB video input: a set of \"object\" latents, corresponding to the time-invariant, object-level contents of the scene, as well as a set of \"frame\" latents, corresponding to global time-varying elements such as viewpoint. This factorization of latents allows our model, SIMONe, to represent object attributes in an allocentric manner which does not depend on viewpoint. Moreover, it allows us to disentangle object dynamics and summarize their trajectories as time-abstracted, view-invariant, per-object properties. We demonstrate these capabilities, as well as the model’s performance in terms of view synthesis and instance segmentation, across three procedurally generated video datasets. ",
+        "bbox": [
+            233,
+            390,
+            766,
+            611
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            638,
+            310,
+            656
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The problem of unsupervised visual scene understanding has become an increasingly central topic in machine learning [1, 2]. The attention is merited by potential gains to reasoning, autonomous navigation, and myriad tasks. However, within the current literature, different studies frame the problem in different ways. One approach aims to decompose images into component objects and object features, supporting (among other things) generation of alternative data that permits insertion, deletion, or repositioning of individual objects [3–6]. Another approach aims at a very different form of decomposition—between allocentric scene structure and a variable viewpoint—supporting generation of views of a scene from new vantage points [7–9] and, if not supplied as input, estimation of camera pose [10]. Although there is work pursuing both of these approaches concurrently in the supervised setting [11–13], very few previous studies have approached the combined challenge in the unsupervised case. In this work, we introduce the Sequence-Integrating Multi-Object Net (SIMONe), a model which pursues that goal of object-level and viewpoint-level scene decomposition and synthesis without supervision. SIMONe is designed to handle these challenges without privileged information concerning camera pose, and in dynamic scenes. ",
+        "bbox": [
+            174,
+            671,
+            825,
+            864
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given a video of a scene our model is able to decouple scene structure from viewpoint information (see Figure 1). To do so, it utilizes video-based cues, and a structured latent space which separates time-invariant per-object features from time-varying global features. These features are inferred using a transformer-based network which integrates information jointly across space and time. ",
+        "bbox": [
+            176,
+            871,
+            823,
+            898
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/474ef165df2ffbd0478d6a7e3fa573cff337922fa05bcf1f109a4235f98414d0.jpg",
+        "image_caption": [
+            "Figure 1: Decomposition (A): SIMONe factorizes a scene sequence $\\mathbf { X }$ into scene content (“object latents,” constant across the sequence) and view/global content (“frame latents,” one per frame) without supervision. Its spatio-temporal attention-based inference naturally allows stable object tracking (e.g. the green sphere is assigned the same segment across frames). Recomposition (B): Object latents of a given sequence X can be recomposed with the frame latents of a different (i.i.d.) sequence $\\mathbf { X } ^ { \\prime }$ to generate a consistent rendering of the same scene (i.e. objects and their properties, relative arrangements, and segmentation assignments) from entirely different viewpoints. Notice that both camera pose and lighting are transferred, as evidenced by the wall corners in the background and the shadows of the green sphere. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            85,
+            772,
+            372
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            540,
+            821,
+            569
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Second, our method seeks to summarize objects’ dynamics. It learns to disentangle not only static object attributes (and their 2D spatial masks), but also object trajectories, without any prior notion of these objects, from videos alone. The learnt trajectory features are temporally abstract and per object; they are captured independently of the dynamics of camera pose, which being a global property, is captured in the model’s per-frame (time-varying) latents.1 ",
+        "bbox": [
+            174,
+            574,
+            825,
+            643
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our model thus advances the state of the art in unsupervised, object-centric scene understanding by satisfying the following desiderata: (1) decomposition of multi-object scenes from RGB videos alone; (2) handling of changing camera pose, and simultaneous inference of scene contents and viewpoint from correlated views (i.e. sequential observations of a moving agent); (3) learning of structure across diverse scene instances (i.e. procedurally sampled contents); (4) object representations which summarize static object attributes like color or shape, view-dissociated properties like position or size, as well as time-abstracted trajectory features like direction of motion; (5) no explicit assumptions of 3D geometry, no explicit dynamics model, no specialized renderer, and few a priori modeling assumptions about the objects being studied; and (6) simple, scalable modules (for inference and rendering) to enable large-scale use. ",
+        "bbox": [
+            174,
+            650,
+            825,
+            789
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
+            321,
+            830
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given the multifaceted problem it tackles, SIMONe connects across several areas of prior work. We describe its nearest neighbors from three scene understanding domains below: ",
+        "bbox": [
+            176,
+            849,
+            823,
+            877
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Scene decomposition models. (1) Our work builds on a recent surge of interest in unsupervised scene decomposition and understanding, especially using slot structure to capture the objects in a scene [14]. One line of work closely related to SIMONe includes methods like [3–6, 15], which all share SIMONe’s Gaussian mixture pixel likelihood model. While these prior methods handled only static scenes, more recent work [16–18, 12] has extended them to videos with promising results. Nevertheless, these approaches have no mechanism or inductive bias to separate view information from scene contents. Moreover, many of them are conditioned on extra inputs like the actions of an agent/camera to simplify inference. (2) Another family of decomposition models originated with Attend, Infer, Repeat (AIR) [19]. AIR’s recurrent attention mechanism does split images into components with separate appearance and pose latents each. Later work [6, 20–23] also extended the model to videos. Despite their structured latents, these models do not learn to distill object appearance into a time-invariant representation (as their appearance and pose latents are free to vary as a function of time). They also require separate object discovery and propagation modules to handle appearing/disappearing objects. In contrast, SIMONe processes a full sequence of images using spatio-temporal attention and produces a single time-invariant latent for each object, hence requiring no explicit transition model or discovery/propagation modules. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            313
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-view scene rendering models. Models which assume viewpoint information for each image like GQN [7], SRNs [8], and NeRF [9] have shown impressive success at learning implicit scene representations and generating novel views from different viewpoints. Recent work [24–27] has further extended these models to videos using deformation fields to model changes in scene geometry over time. In contrast to SIMONe, these models can achieve photorealistic reconstructions by assuming camera parameters (viewpoint information). To allow a direct comparison, we use a view-supervised version of SIMONe in Section 4.1. There is also recent work [28, 29] that relaxes the known viewpoint constraint, but they still model single scenes at a time, which prevents them from exploiting regularities over multiple scenes. A more recent line of work [30–32] explored amortizing inference by mapping from a given set of images to scene latents, but they cannot handle videos yet. Note that all of these models treat the whole scene as a single entity and avoid decomposing it into objects. One exception here is [33], which represents objects with separate pose and appearance latents. However, this model is purely generative and cannot infer object latents from a given scene. Another exception is [13], which can in fact infer object representations, but nevertheless depends on view supervision. ",
+        "bbox": [
+            174,
+            319,
+            825,
+            525
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Simultaneous localization and mapping. The problem of inferring scene representations in a novel environment by exploring it (rather than assuming given views and viewpoint information) is well studied in robotics and vision [10]. Classic SLAM techniques often rely on EM [34, 35] or particle filters [36] to infer viewpoint and scene contents jointly. While our problem is slightly simpler (we can leverage shared structure across scene instances; certain elements such as the shape of the room are held constant; and we use offline data rather than active exploration), our approach of using a factorized variational posterior provides a learning-based solution to the same computational problem. Our simplified setting is perhaps justified by our unsupervised take on the problem. On the other hand, we don’t assume simplifications which may be common in robotics practice (e.g. known camera properties like field of view; or the use of multiple cameras or depth sensors). Most popular SLAM benchmarks [37–39] are on unstructured 3D scenes and hence it was not straightforward for us to compare directly to classic methods. But an encouraging point of overlap is that object-centric SLAM formulations [11] as well as learning-based solutions [40, 41] are active topics of research. Our work could open new avenues in object-centric scene mapping without supervision. ",
+        "bbox": [
+            174,
+            532,
+            825,
+            726
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Model ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            751,
+            258,
+            768
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "SIMONe is a variational auto-encoder [42] consisting of an inference network (encoder) which infers latent variables from a given input sequence, and a generative process (decoder) which decodes these latents back into pixels. Using the Evidence Lower Bound (ELBO), the model is trained to minimize a pixel reconstruction loss and latent compression KL loss. Crucially, SIMONe relies on a factorized latent space which enforces a separation of static object attributes from global, dynamic properties such as camera pose. We introduce our latent factorization and generative process in Section 3.1. Then in Section 3.2, we describe how the latents can be inferred using a transformer-based encoder, significantly simplifying the (recurrent or autoregressive) architectures used in prior work. Finally, we fully specify the training scheme in Section 3.3. ",
+        "bbox": [
+            174,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce156cadcf2724beb8d18887fd8697c8c846570c23660240b4d9540c10f68d0b.jpg",
+        "image_caption": [
+            "Figure 2: Architecture of the SIMONe inference network $\\mathcal { E } _ { \\phi }$ . The transformers integrate information jointly across space and time to infer (the posterior parameters of) the object and frame latents. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            90,
+            789,
+            256
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Latent Structure and Generative Process ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            320,
+            496,
+            335
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our model aims to capture the structure of a scene, observed as a sequence of images from multiple viewpoints (often along a smooth camera trajectory, though this is not a requirement). Like many recently proposed object-centric models we choose to represent the scene as a set of $K$ object latent variables $\\mathbf { O } ^ { \\mathsf { ^ { \\prime } } } : = \\{ \\mathbf { o } _ { k } \\} _ { k = 1 } ^ { K }$ . These are invariant by construction across all frames in the sequence (i.e. their distribution is constant through time, and expected to summarize information across the whole sequence). We also introduce $T$ frame latents $\\mathbf { F } : = \\{ \\mathbf { f } _ { t } \\} _ { t = 1 } ^ { T }$ , one for each frame in the sequence, that capture time-varying information. Note that by choosing this factorization we reduce the number of required latent variables from $K \\cdot T$ to $K + T$ . The latent prior $\\begin{array} { r } { p ( \\mathbf { O } , \\mathbf { F } ) = \\prod _ { k } \\mathcal { N } ( \\mathbf { o } _ { k } \\mid \\mathbf { \\Lambda } } \\end{array}$ $\\mathbf { 0 } , \\mathbf { I } ) \\prod _ { t } \\mathcal { N } ( \\mathbf { f } _ { t } \\mid \\bar { \\mathbf { 0 } } , \\mathbf { I } )$ is a unit spherical Gaussian, assuming and enforcing independence between object latents, frame latents, and their feature dimensions. ",
+        "bbox": [
+            173,
+            347,
+            825,
+            486
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given the latent variables, we assume all pixels and all frames to be independent. Each pixel is modeled as a Gaussian mixture with $K$ components. The mixture weights for pixel $\\mathbf { x } _ { t , i }$ capture which component $k$ “explains” that pixel $( 1 \\leq i \\leq H W$ ). The mixture logits $\\hat { m } _ { k , t , i }$ and RGB (reconstruction) means $\\mu _ { k , t , i }$ are computed for every component $k$ at a specific time-step $t$ and specific pixel location $\\mathbf { l } _ { i }$ using a decoder $\\mathcal { D } _ { \\theta }$ : ",
+        "bbox": [
+            176,
+            491,
+            823,
+            560
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0ae86e079d36b4f6693a38606ccf3c83b38c25a8c68b5d2bfc9abcafc776083c.jpg",
+        "text": "$$\n\\begin{array} { c } { \\hat { m } _ { k , t , i } , \\pmb { \\mu } _ { k , t , i } = \\mathcal { D } _ { \\theta } ( \\mathbf { o } _ { k } , \\mathbf { f } _ { t } ; \\mathrm { \\mathbf { l } } _ { i } , t ) } \\\\ { p ( \\mathbf { x } _ { t , i } \\mid \\mathbf { o } _ { 1 } , . . . , \\mathbf { o } _ { K } , \\mathbf { f } _ { t } ; t , \\mathrm { \\mathbf { l } } _ { i } ) = \\displaystyle \\sum _ { k } m _ { k , t , i } \\mathcal { N } ( \\mathbf { x } _ { t , i } \\mid \\pmb { \\mu } _ { k , t , i } ; \\sigma _ { x } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            566,
+            696,
+            621
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We decode each pixel independently, \"querying\" our pixel-wise decoder using the sampled latents, coordinates $\\mathbf l _ { i } \\in [ - 1 , 1 ] ^ { 2 }$ of the pixel, and time-step $t \\in [ 0 , 1 )$ being decoded as inputs. The decoder’s architecture is an MLP or 1x1 CNN. (See Appendix A.3.1 for the exact parameterization as well as a diagram of the generative process). By constraining the decoder to work on individual pixels, we can use a subset of pixels as training targets (as opposed to full images; this is elaborated in Section 3.3). Once they are decoded, we obtain the mixture weights $m _ { k , t , i }$ by taking the softmax of the logits across the $K$ components: $m _ { k , t , i } = \\tt s o f t m a x _ { k } ( \\hat { m } _ { k , t , i } )$ . Equation 2 specifies the full pixel likelihood, where $\\sigma _ { x }$ is a scalar hyperparameter. ",
+        "bbox": [
+            173,
+            626,
+            826,
+            738
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Inference ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            753,
+            279,
+            768
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a sequence of frames $\\mathbf { X } : = \\{ \\mathbf { x } _ { t } \\} _ { t = 1 } ^ { T }$ we now wish to infer the corresponding object latents $\\mathbf { O }$ and frame latents $\\mathbf { F }$ . The exact posterior distribution $p ( \\mathbf { O } , \\mathbf { F } \\mid \\mathbf { X } )$ is intractable so we resort to using a Gaussian approximate posterior $q ( \\mathbf { O } , \\mathbf { F } \\mid \\mathbf { X } )$ . The approximate posterior is parameterized as the output of an inference (encoder) network $\\mathcal { E } _ { \\phi } ( \\mathbf { X } )$ which outputs the mean and (diagonal) log scale for all latent variables given the input sequence. ",
+        "bbox": [
+            174,
+            779,
+            825,
+            849
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "SIMONe’s inference network is based on the principle that spatio-temporal data can be processed jointly across space and time using transformers. Beyond an initial step, we don’t need the translation invariance of a CNN, which forces spatial features to interact gradually via a widening receptive field. Nor do we need the temporal invariance of an RNN which forces sequential processing. Instead, we let feature maps interact simultaneously across the cross-product of space and time. See Figure 2 for an overview of our encoder architecture implementing this. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Concretely, each frame $\\mathbf { x } _ { t }$ in the sequence is passed through a CNN which outputs $I J$ spatial feature maps at each time-step (containing $C$ channels each). $I J$ can be larger than the number of object latents $K$ . (For all results in the paper, we set $I$ and $J$ to 8 each, and $K = 1 6$ .) The rest of the inference network consists of two transformers $\\mathcal { T } _ { 1 }$ and $\\mathcal { T } _ { 2 }$ . $\\mathcal { T } _ { 1 }$ takes in all $T I J$ feature maps. Each feature map attends to all others as described. $\\mathcal { T } _ { 1 }$ outputs $T I J$ transformed feature maps. When $I J > K$ , we apply a spatial pool to reduce the number of slots to $T K$ (see Appendix A.3.2 for details). These slots serve as the input to $\\mathcal { T } _ { 2 }$ , which produces an equal number of output slots. Both transformers use absolute (rather than relative) positional embeddings, but these are 3D to denote the spatio-temporal position of each slot. We denote the output of $\\mathcal { T } _ { 2 }$ as $\\hat { \\mathbf { e } } _ { k , t }$ . This intermediate output is aggregated along separate axes (and passed through MLPs) to obtain $T$ frame and $K$ object posterior parameters respectively. Specifically, $\\lambda _ { \\mathbf { o } _ { k } } = \\mathrm { m l p } _ { o } ( 1 / T \\sum _ { t } \\hat { \\mathbf { e } } _ { k , t } )$ while $\\lambda _ { \\mathbf { f } _ { t } } = \\mathrm { m l p } _ { f } ( 1 / K \\sum _ { k } \\hat { \\mathbf { e } } _ { k , t } )$ Using these posterior parameters we can sample the object latents $\\mathbf { o } _ { k } \\sim \\mathcal { N } ( \\lambda _ { \\mathbf { o } _ { k } } ^ { \\mu } , \\mathrm { e x p } ( \\lambda _ { \\mathbf { o } _ { k } } ^ { \\sigma } ) \\mathbb { 1 } )$ , and the frame latents $\\mathbf { f } _ { t } \\sim \\mathcal { N } ( \\lambda _ { \\mathbf { f } _ { t } } ^ { \\mu } , \\exp ( \\lambda _ { \\mathbf { f } _ { t } } ^ { \\sigma } ) \\mathbb { 1 } )$ . ",
+        "bbox": [
+            173,
+            126,
+            825,
+            311
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Loss and Training ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            324,
+            339,
+            340
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The model is trained end to end by minimizing the following negative-ELBO derivative: ",
+        "bbox": [
+            171,
+            349,
+            751,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b14b286357304ad1becbebf7230a051b938ceb7e3afb1fe58d987b580d3ef783.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { - \\alpha } { T _ { d } H _ { d } W _ { d } } \\sum _ { t = 1 } ^ { T _ { d } } \\sum _ { i = 1 } ^ { H _ { d } W _ { d } } \\log p ( \\mathbf { x } _ { t , i } \\mid \\mathbf { o } _ { 1 } , . . . , \\mathbf { o } _ { K } , \\mathbf { f } _ { t } ; t , \\mathbf { l } _ { i } ) + \\displaystyle \\frac { \\beta _ { o } } { K } \\sum _ { k } D _ { K L } ( q ( \\mathbf { o } _ { k } \\mid \\mathbf { X } ) \\parallel p ( \\mathbf { o } _ { k } ) ) } \\\\ { + \\displaystyle \\frac { \\beta _ { f } } { T } \\sum _ { t } D _ { K L } ( q ( \\mathbf { f } _ { t } \\mid \\mathbf { X } ) \\parallel p ( \\mathbf { f } _ { t } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            369,
+            785,
+            452
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We normalize the data log-likelihood by the number of decoded pixels $\\left( T _ { d } H _ { d } W _ { d } \\right)$ to allow for decoding fewer than all input pixels $( T H W )$ . This helps scale the size of the decoder (without reducing the learning signal, due to the correlations prevalent between adjacent pixels). Normalizing by $1 / \\bar { T _ { d } } \\bar { H _ { d } } \\bar { W _ { d } }$ ensures consistent learning dynamics regardless of the choice of how many pixels are decoded. $\\alpha$ is generally set to 1, but available to tweak in case the scale of $\\beta _ { o }$ and $\\beta _ { f }$ is too small to be numerically stable. Unless explicitly mentioned, we set $\\beta _ { o } = \\beta _ { f }$ . See Appendix A.3 for details. ",
+        "bbox": [
+            174,
+            453,
+            825,
+            537
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Comparative Evaluation ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            555,
+            410,
+            573
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the model we focus on two tasks: novel view synthesis and video instance segmentation. On the first task (Section 4.1), we highlight the benefit of view information when it is provided as ground truth to a simplified version of our model (denoted “SIMONe-VS” for view supervised), as well as baseline models like GQN [7] and NeRF-VAE [43]. On the second task (Section 4.2), we deploy the fully unsupervised version of our model; we showcase not only the possibility of inferring viewpoint from data, but also its benefit to extracting object-level structure in comparison to methods like MONet [3], Slot Attention [15], and Sequential IODINE [4]. ",
+        "bbox": [
+            173,
+            585,
+            825,
+            684
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our results are based on three procedurally generated video datasets of multi-object scenes. In increasing order of difficulty, they are: Objects Room 9 [44], CATER (moving camera) [45], and Playroom [46]. These were chosen to meet a number of criteria: we wanted at least 9-10 objects per scene (there can be fewer in view, or as many as 30 in the case of Playroom). We wanted a moving camera with a randomized initial position (the only exception is CATER, where the camera moves rapidly but is initialized at a fixed position to help localization). We wanted ground-truth object masks to evaluate our results quantitatively. We also wanted richness in terms of lighting, texture, object attributes, and other procedurally sampled elements. Finally, we wanted independently moving objects in one dataset (to evaluate trajectory disentangling and temporal abstraction), and unpredictable camera trajectories in another (the Playroom dataset is sampled using an arbitrary agent policy, so the agent is not always moving). Details on all datasets are in Appendix A.2. ",
+        "bbox": [
+            174,
+            690,
+            825,
+            842
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 View synthesis (with viewpoint supervision) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
+            516,
+            872
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We first motivate view-invariant object representations by considering the case when ground-truth camera pose is provided to our model (a simplified variant we call “SIMONe-VS”). In this scenario, we don’t infer any frame latents. Rather, the encoder and decoder are conditioned on the viewpoint directly. This view-supervised setting is similar to models like GQN and NeRF which represent the contents of a scene implicitly and can be queried in different directions. ",
+        "bbox": [
+            176,
+            883,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/04b14bb591866ade30a797d55eb4e50d260a7425b92b27c5488464ef25af4457.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of scene representation and view synthesis capabilities between SIMONeVS, NeRF-VAE, and GQN. All models partially observe a procedurally generated Playroom from a given sequence of frames (we visualize 4 of the 16 input frames fed to the models). Then, we decode novel views on a circular trajectory around the room, with the yaw linearly spaced in $[ - \\pi , \\pi ]$ . NeRF-VAE retains very little object structure, while GQN hallucinates content. SIMONe-VS can produce fine reconstructions of objects that it observes even partially or at a distance (such as the bed or shelves in the scene). SIMONe-VS also segments the scene as a bonus. See Appendix A.5.1 for similar plots from different scenes/input sequences. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            89,
+            821,
+            296
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            450,
+            825,
+            492
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare three such models on view synthesis in the Playroom. The models are provided a set of 16 consecutive frames as context, partially revealing a generated room. Having inferred a scene representation from the input context, the models are then tasked with generating unobserved views of the scene. This extrapolation task is performed without any retraining, and tests the coherence of the models’ inferred representations. The task is challenging given the compositional structure of each Playroom scene, as well as the variation across scenes (the color, position, size, and choice of all objects are procedurally sampled per scene; only the L-shaped layout of the room is shared across scenes in the dataset). Because each model is trained on and learns to represent many Playroom instances, NeRF itself is not directly suitable for the task. It needs to be retrained on each scene, whereas we want to infer the specifics of any given room at evaluation time. NeRF-VAE addresses this issue and makes it directly comparable to our model. ",
+        "bbox": [
+            174,
+            498,
+            825,
+            651
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To set up the comparison, we first trained SIMONe-VS and evaluated its log-likelihood on Playroom sequences. Then, we trained GQN and NeRF-VAE using constrained optimization (GECO [47]) to achieve roughly the same log likelihood per pixel. See Appendix A.5.1 for a comparison of the models in terms of the reconstruction-compression trade-off. ",
+        "bbox": [
+            174,
+            657,
+            825,
+            713
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Qualitatively, the models show vast differences in their perceived structure (see Figure 3). NeRF-VAE blurs out nearly all objects in the scene but understands the geometry of the room and is able to infer wall color. GQN produces more detailed reconstructions, but overfits to particular views and does not interpolate smoothly. SIMONe-VS on the other hand finely reproduces the object structure of the room. Even when it observes objects at a distance or up close, it places and sizes them correctly in totally novel views. This makes SIMONe-VS a powerful choice over NeRF-VAE and GQN-style models when the priority is to capture scene structure across diverse examples. ",
+        "bbox": [
+            174,
+            719,
+            825,
+            816
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Instance segmentation (fully unsupervised) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            840,
+            509,
+            856
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Having shown the benefit of view information to inferring scene structure in the Section 4.1, we now turn to the added challenge of inferring viewpoint directly and simultaneously with scene contents (without any supervision). ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/397bd35552a08851b9820a1ef07bac710214de5eec71d1141befc6b19c74d860.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Static ARI-F</td><td colspan=\"4\">Video ARI-F</td></tr><tr><td></td><td>MONet</td><td>SA</td><td>S-IODINE</td><td>MONet</td><td>SA</td><td>S-IODINE</td><td>SIMONe</td></tr><tr><td>Objects Room 9</td><td>0.886 (±0.061)</td><td>0.784 (±0.138)</td><td>0.695 (±0.007)</td><td>0.865 (±0.007)</td><td>0.066 (±0.014)</td><td>0.673 (±0.0.002)</td><td>0.936</td></tr><tr><td>CATER</td><td>0.937</td><td>0.923</td><td>0.728</td><td>0.412</td><td>0.073</td><td>0.668</td><td>(±0.010) 0.918</td></tr><tr><td>Playroom</td><td>(±0.004) 0.647 (±0.012)</td><td>(±0.076) 0.653 (±0.024)</td><td>(±0.032) 0.439 (±0.009)</td><td>(±0.012) 0.442 (±0.010)</td><td>(±0.006) 0.059 (±0.002)</td><td>(±0.033) 0.356 (±0.006)</td><td>(±0.036) 0.800 (±0.043)</td></tr></table>",
+        "bbox": [
+            187,
+            88,
+            803,
+            205
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: SIMONe segmentation performance (in terms of Adjusted Rand Index for foreground objects, ARI-F) compared to state-of-the-art unsupervised baselines: two static-frame models (MONet and Slot Attention, SA) and a video model (S-IODINE). We calculate static and video ARI-F scores separately. For static ARI-F, we evaluate the models per still image. For video ARI-F, we evaluate the models across space and time, taking an object’s full trajectory as a single class. The video ARI-F thus penalizes models (especially Slot Attention) which fail to track objects stably. We report the mean and standard deviation of scores across 5 random seeds in each case. ",
+        "bbox": [
+            173,
+            209,
+            826,
+            306
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c7466d809c6bf3df48a2ccc0d1e8c7f5114c36d36595842e0cab66e7c99378d.jpg",
+        "image_caption": [
+            "Figure 4: Segmentations and reconstructions produced by SIMONe on CATER and Playroom. SIMONe copes well with clutter and different-sized objects. It learns to use object motion as a segmentation signal on CATER, evident from the fact that an object’s shadow is correctly assigned to that object’s segment as it moves. This is true even when there’s multiple shadows per object (due to multiple lights in the scene). SIMONe also overcomes color-based cues to segment two-toned objects such as beds in the Playroom as single objects. See Appendix A.5.2 to compare with baseline models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            335,
+            764,
+            608
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare SIMONe to a range of competitive but viewpoint-unaware scene decomposition approaches. First, we train two static-frame models: MONet and Slot Attention. MONet uses a similar generative process and training loss to our model, achieving segmentation by modeling the scene as a spatial mixture of components, and achieving disentangled representations using a $\\beta$ -weighted KL information bottleneck. On the other hand, it uses a deterministic, recurrent attention network to infer object masks. Slot Attention is a transformer-based autoencoding model which focuses on segmentation performance rather than representation learning. Finally, we also compare against Sequential IODINE (“S-IODINE”), which applies a refinement network to amortize inference over time, separating objects by processing them in parallel. It also uses a $\\beta$ -weighted KL loss to disentangle object representations. Note that S-IODINE is a simplified version of OP3 [17], which additionally attempts to model (pairwise) object dynamics using an agent’s actions as inputs. SIMONe and S-IODINE both avoid relying on this privileged information. Table 1 contains a quantitative comparison of segmentation performance across these models, while Figure 4 shows qualitative results. ",
+        "bbox": [
+            174,
+            734,
+            825,
+            915
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/4bbe453caa2296f7c2e11f984cd4f4c05d0d48cd785a3a686e3cdb83baeb21db.jpg",
+        "image_caption": [
+            "Figure 5: Object attributes learnt by SIMONe. In each row, we manipulate a particular object latent attribute for an arbitrary target object (circled in red) in two scenes. This reveals the attributes’ relationship to interpretable object characteristics like color, size, position and identity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            89,
+            740,
+            385
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            454,
+            276,
+            472
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We take a closer look at the representations learnt by our model by decoding them in various ways. First, we manipulate latent attributes individually to assess the interpretability of object representations visually in Section 5.1. Next, we exploit SIMONe’s latent factorization to render views of a given scene using the camera trajectory of a different input sequence. These cross-over visualizations help identify how the model encodes object dynamics in Section 5.2. Finally, we measure the predictability of ground-truth camera dynamics and object dynamics from the two types of latents in Section 5.3. These analyses use a single, fully unsupervised model per dataset. ",
+        "bbox": [
+            174,
+            487,
+            825,
+            584
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 Latent attribute traversals ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            597,
+            397,
+            611
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We visualize the object representations learnt by SIMONe on Playroom to highlight their disentanglement, across latent attributes and across object slots, in Figure 5. We seed all latents using a given input sequence, then manipulate one object latent attribute at a time by adding fixed offsets. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            665
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that object position and size are well disentangled in each direction. Aided by the extraction of view-specific information in the frame latents, SIMONe also learns object features corresponding to identity. The decoder nevertheless obeys the biases in the dataset–for instance, shelves will slide along a wall when their position latent is traversed. The rubber duck does not morph into a chest of drawers because those are always located against a wall. This further suggests a well-structured latent representation, which the decoder can adapt to. ",
+        "bbox": [
+            174,
+            671,
+            825,
+            755
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Object and frame latent cross-overs ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            767,
+            460,
+            781
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We expect SIMONe to encode object trajectories and camera trajectories independently of each other. In fact, each object’s trajectory should be summarized in its own time-invariant latent code. To examine this, we recompose object latents from one sequence with frame latents from other sequences in the CATER dataset. The result, in Figure 6, is that we can observe object motion trajectories from multiple camera trajectories. ",
+        "bbox": [
+            174,
+            794,
+            825,
+            863
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note the consistency of relative object positions (at any time-step) from all camera angles. In the single moving object case, its motion could in fact be interpreted as a time-varying global property of the scene. Despite this challenge, SIMONe is able to encode the object’s motion as desired in its specific time-invariant code. In Section 5.3, we further confirm that object trajectories are summarized in the object latents, which can be queried with time to recover allocentric object positions. ",
+        "bbox": [
+            176,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/898e0b3941c04bad1f27e5edb95e93f0dd4c2ceea8de314f4c150297097c6158.jpg",
+        "image_caption": [
+            "Figure 6: Separation of object trajectories from camera trajectories. Left: When encoding a sequence with consistent (i.i.d.) object dynamics, this information is extracted in the object latents and is unaffected by changing frame latents (see green cone). Right: Movement events are sequenced correctly; object relative positions also remain consistent (see pattern of shadows on the floor circled in yellow). See Appendix A.5.3 for cross-over plots showing more object trajectories. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            88,
+            820,
+            295
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            393,
+            823,
+            422
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3 Camera pose and object trajectory prediction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            433,
+            531,
+            448
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We assessed SIMONe’s frame latents by decoding the true camera position and orientation from them. We trained linear and MLP regressors to predict the camera pose at time $t$ from the corresponding frame latent $\\mathbf { f } _ { t }$ on a subset of CATER sequences. We also trained an MLP on the time-invariant object latents $\\mathbf { o } _ { 1 : K }$ for the same task. We evaluated these decoders on held-out data. Table 2 shows that frame latents describe the viewpoint almost perfectly. ",
+        "bbox": [
+            174,
+            458,
+            825,
+            527
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also assessed if the object latents contain precise information about allocentric object positions (to be clear, position information is not provided in any form while training SIMONe). Table 3 shows that the correct object latent is predictive of the allocentric position of a dynamic object (when queried along with the timestep). Adding the frame latent does not provide more information, and using ",
+        "bbox": [
+            174,
+            534,
+            825,
+            590
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/24ddfe55612c61698c7954a9f049082d5b3aaef76538a78057dd50a375cb37df.jpg",
+        "table_caption": [
+            "Table 2: Decoding camera pose. We show that ground-truth camera location or orientation is predictable from the corresponding frame latent, but cannot be predicted from all object latents put together. We report the test $\\mathrm { \\dot { ~ } R ^ { 2 } }$ score across 5 independently trained decoders per input type. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Linear(ft)</td><td>MLP(ft)</td><td>MLP(0.1,..,.·0K)</td></tr><tr><td>Camera location</td><td>0.832 ± 0.0</td><td>0.949 ± 0.002</td><td>0.044 ± 0.026</td></tr><tr><td>Camera orientation (Rodrigues)</td><td>0.800 ± 0.0</td><td>0.946±0.002</td><td>0.292±0.025</td></tr></table>",
+        "bbox": [
+            220,
+            613,
+            774,
+            671
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/d84b0506ff637e94cce4b5d372f9ab41e13ae7a2409030c2b7808d6958136d58.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MLP(Ok)</td><td>MLP(Ok,t)</td><td>MLP(Ok,ft,t)</td><td>MLP({oj : j≠k})</td></tr><tr><td>Trained on all objects</td><td>0.710± 0.006</td><td>0.871 ± 0.006</td><td>0.876 ± 0.003</td><td>-0.062±0.006</td></tr><tr><td>Trained on moving objects</td><td>0.724± 0.007</td><td>0.894± 0.004</td><td>0.898± 0.005</td><td>一 -0.022 ± 0.025</td></tr></table>",
+        "bbox": [
+            173,
+            747,
+            825,
+            804
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Decoding object trajectories. We test MLP decoders on predicting allocentric object positions (of moving object in unseen scenes) based on the following inputs: (a) the corresponding object latent, (b) the timestep as well, and (c) the frame latent corresponding to that timestep as well, and (d) remaining object latents from the scene (not pertaining to the object of interest). The decoders were trained on arbitrary objects or a subset containing moving objects only. We report the test $R ^ { 2 }$ score across 5 independently trained decoders per input type. ",
+        "bbox": [
+            173,
+            814,
+            825,
+            900
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "the “wrong” objects (from the same scene) is completely uninformative. To perform this analysis, we needed to align SIMONe’s inferred objects with the ground-truth set of objects. We used the Hungarian matching algorithm on the MSE of inferred object masks and ground-truth object masks to perform the alignment. Given SIMONe’s disentangling of object dynamics, its time-abstracted object representations could prove helpful for a variety of downstream tasks (e.g. “catch the flying ball!”). ",
+        "bbox": [
+            174,
+            92,
+            825,
+            161
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Taken together, Table 2 and Table 3 show the separation of information that is achieved between the object and frame latents, helping assert our two central aims of view-invariant and temporally abstracted object representations. ",
+        "bbox": [
+            176,
+            166,
+            823,
+            209
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Discussion and Future Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            250,
+            442,
+            267
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Scalability. The transformer-based inference network in SIMONe makes it amenable to processing arbitrarily large videos, just as transformer-based language models can process long text. SIMONe could be trained on windows of consecutive frames sampled from larger videos (aka \"chunks\"). For inference over a full video, one could add memory slots which carry information over time from one window to the next. Applying SIMONe on sliding windows of frames also presents the opportunity to amortize inference at any given time-step if the windows are partially overlapping (so the model could observe every given frame as part of two or more sequences). Our use of the standard transformer architecture also makes SIMONe amenable to performance improvements via alternative implementations. ",
+        "bbox": [
+            173,
+            294,
+            825,
+            419
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Limitations. (1) SIMONe cannot generate novel videos (e.g. sample a natural camera trajectory via consecutive frame latents) in its current version. This could be addressed in a similar fashion to the way GENESIS [5] built on MONet [3]–it should be possible (e.g. using recurrent networks) to learn conditional priors for objects in a scene and for successive frame latents, which would make SIMONe fully generative. (2) We see another possible limitation arising from our strict latent factorization. We have shown that temporally abstracted object features can predict object trajectories when queried by time. This can cover a lot of interesting cases (even multiple object-level \"events\" over time), but will start to break as object trajectories get more stochastic (i.e. objects transition considerably/chaotically through time). We leave it to future work to explore how temporal abstraction can be combined with explicit per-step dynamics modeling in those cases. For simpler settings, our approach to encoding object trajectories (distilling them across time) is surprisingly effective. ",
+        "bbox": [
+            173,
+            424,
+            825,
+            577
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            617,
+            299,
+            636
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We’ve presented SIMONe, a latent variable model which separates the time-invariant, object-level properties of a scene video from the time-varying, global properties. Our choice of scalable modules such as transformers for inference, and a pixel-wise decoder, allow the model to extract this information effectively. ",
+        "bbox": [
+            174,
+            662,
+            825,
+            718
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "SIMONe can learn the common structure across a variety of procedurally instantiated scenes. This enables it to recognize and generalize to novel scene instances from a handful of correlated views, as we showcased via 360-degree view traversal in the view-supervised setting. More significantly, SIMONe can learn to infer the two sets of latent variables jointly without supervision. Aided by crossframe spatio-temporal attention, it achieves state-of-the-art segmentation performance on complex 3D scenes. Our latent factorization (and information bottleneck pressures) further help with learning meaningful object representations. SIMONe can not only separate static object attributes (like size and position), but it can also separate the dynamics of different objects (as time-invariant localized properties) from global changes in view. ",
+        "bbox": [
+            174,
+            724,
+            825,
+            849
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have discussed how the model can be applied to much longer videos in the future. It also has potential for applications in robotics (e.g. sim-to-real transfer) and reinforcement learning, where view-invariant object information (and summarizing their dynamics) could dramatically improve how agents reason about objects. ",
+        "bbox": [
+            176,
+            856,
+            823,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            338,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We thank Michael Bloesch, Markus Wulfmeier, Arunkumar Byravan, Claudio Fantacci, and Yusuf Aytar for valuable discussions on the purview of our work. We are also grateful for David Ding’s support on the CATER dataset. The authors received no specific funding for this work. ",
+        "bbox": [
+            174,
+            119,
+            825,
+            161
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            186,
+            266,
+            202
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[1] Jitendra Malik. Scene understanding in the era of deep learning, Jun 2015. URL https://www.robots. ox.ac.uk/seminars/Extra/2015_06_22_JitendraMalik.pdf.   \n[2] Farzad Husain, Babette Dellen, and Carme Torras. Chapter 20 - Scene Understanding Using Deep Learning, pages 373–382. Academic Press, 2017. ISBN 978-0-12-811318-9. doi: https://doi.org/10.1016/ B978-0-12-811318-9.00020-X.   \n[3] Christopher P Burgess, Loic Matthey, Nicholas Watters, Rishabh Kabra, Irina Higgins, Matt Botvinick, and Alexander Lerchner. Monet: Unsupervised scene decomposition and representation. arXiv preprint arXiv:1901.11390, 2019.   \n[4] Klaus Greff, Raphaël Lopez Kaufman, Rishabh Kabra, Nick Watters, Christopher Burgess, Daniel Zoran, Loic Matthey, Matthew Botvinick, and Alexander Lerchner. Multi-object representation learning with iterative variational inference. In International Conference on Machine Learning, pages 2424–2433. PMLR, 2019.   \n[5] Martin Engelcke, Adam R. Kosiorek, Oiwi Parker Jones, and Ingmar Posner. Genesis: Generative scene inference and sampling with object-centric latent representations. In International Conference on Learning Representations, 2020.   \n[6] Zhixuan Lin, Yi-Fu Wu, Skand Vishwanath Peri, Weihao Sun, Gautam Singh, Fei Deng, Jindong Jiang, and Sungjin Ahn. Space: Unsupervised object-oriented scene representation via spatial attention and decomposition. In International Conference on Learning Representations, 2020.   \n[7] S M Ali Eslami, Danilo Jimenez Rezende, Frederic Besse, Fabio Viola, Ari S Morcos, Marta Garnelo, Avraham Ruderman, Andrei A Rusu, Ivo Danihelka, Karol Gregor, David P Reichert, Lars Buesing, Theophane Weber, Oriol Vinyals, Dan Rosenbaum, Neil Rabinowitz, Helen King, Chloe Hillier, Matt Botvinick, Daan Wierstra, Koray Kavukcuoglu, and Demis Hassabis. Neural scene representation and rendering. Science, 360(6394):1204–1210, June 2018.   \n[8] Vincent Sitzmann, Michael Zollhoefer, and Gordon Wetzstein. Scene representation networks: Continuous 3D-Structure-Aware neural scene representations. In H Wallach, H Larochelle, A Beygelzimer, F d’Alché Buc, E Fox, and R Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.   \n[9] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In European Conference on Computer Vision, pages 405–421. Springer, 2020.   \n[10] C. Cadena, L. Carlone, H. Carrillo, Y. Latif, D. Scaramuzza, J. Neira, I. Reid, and J.J. Leonard. Past, present, and future of simultaneous localization and mapping: Towards the robust-perception age. IEEE Transactions on Robotics, 32(6):1309–1332, 2016.   \n[11] Binbin Xu, Wenbin Li, Dimos Tzoumanikas, Michael Bloesch, Andrew Davison, and Stefan Leutenegger. Mid-fusion: Octree-based object-level multi-instance dynamic slam. In 2019 International Conference on Robotics and Automation (ICRA), pages 5231–5237. IEEE, 2019.   \n[12] Li Nanbo, Cian Eastwood, and Robert B Fisher. Learning object-centric representations of multi-object scenes from multiple views. In Advances in Neural Information Processing Systems, 2020.   \n[13] Chang Chen, Fei Deng, and Sungjin Ahn. Object-centric representation and rendering of 3d scenes. arXiv preprint arXiv:2006.06130, 2020.   \n[14] Klaus Greff, Sjoerd van Steenkiste, and Jürgen Schmidhuber. On the binding problem in artificial neural networks. arXiv preprint arXiv:2012.05208, 2020.   \n[15] Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold, Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf. Object-centric learning with slot attention. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 11525–11538. Curran Associates, Inc., 2020.   \n[16] Antonia Creswell, Rishabh Kabra, Chris Burgess, and Murray Shanahan. Unsupervised object-based transition models for 3d partially observable environments. arXiv preprint arXiv:2103.04693, 2021.   \n[17] Rishi Veerapaneni, John D Co-Reyes, Michael Chang, Michael Janner, Chelsea Finn, Jiajun Wu, Joshua Tenenbaum, and Sergey Levine. Entity abstraction in visual model-based reinforcement learning. In Conference on Robot Learning, pages 1439–1456. PMLR, 2020.   \n[18] Polina Zablotskaia, Edoardo A Dominici, Leonid Sigal, and Andreas M Lehrmann. Unsupervised video decomposition using spatio-temporal iterative inference. arXiv preprint arXiv:2006.14727, 2020.   \n[19] S M Ali Eslami, Nicolas Heess, Theophane Weber, Yuval Tassa, David Szepesvari, Koray Kavukcuoglu, and Geoffrey E Hinton. Attend, infer, repeat: fast scene understanding with generative models. In Proceedings of the 30th International Conference on Neural Information Processing Systems, NIPS’16, pages 3233–3241, Red Hook, NY, USA, December 2016. Curran Associates Inc.   \n[20] Adam Kosiorek, Hyunjik Kim, Yee Whye Teh, and Ingmar Posner. Sequential attend, infer, repeat: Generative modelling of moving objects. Adv. Neural Inf. Process. Syst., 31, 2018.   \n[21] Eric Crawford and Joelle Pineau. Exploiting spatial invariance for scalable unsupervised object tracking. arXiv preprint arXiv:1911.09033, 2019.   \n[22] Eric Crawford and Joelle Pineau. Spatially invariant unsupervised object detection with convolutional neural networks. AAAI, 33(01):3412–3420, July 2019.   \n[23] Jindong Jiang, Sepehr Janghorbani, Gerard de Melo, and Sungjin Ahn. SCALOR: Generative world models with scalable object representations. arXiv preprint arXiv:1910.02384, October 2019.   \n[24] Albert Pumarola, Enric Corona, Gerard Pons-Moll, and Francesc Moreno-Noguer. D-NeRF: Neural radiance fields for dynamic scenes. arXiv preprint arXiv:2011.13961, November 2020.   \n[25] Keunhong Park, Utkarsh Sinha, Jonathan T Barron, Sofien Bouaziz, Dan B Goldman, Steven M Seitz, and Ricardo Martin-Brualla. Nerfies: Deformable neural radiance fields. arXiv preprint arXiv:2011.12948, 2020.   \n[26] Zhengqi Li, Simon Niklaus, Noah Snavely, and Oliver Wang. Neural scene flow fields for Space-Time view synthesis of dynamic scenes. arXiv preprint arXiv:2011.13084, November 2020.   \n[27] Yilun Du, Yinan Zhang, Hong-Xing Yu, Joshua B Tenenbaum, and Jiajun Wu. Neural radiance flow for 4D view synthesis and video processing. arXiv preprint arXiv:2012.09790, December 2020.   \n[28] Chen-Hsuan Lin, Wei-Chiu Ma, Antonio Torralba, and Simon Lucey. BARF: Bundle-Adjusting neural radiance fields. arXiv preprint arXiv:2104.06405, April 2021.   \n[29] Zirui Wang, Shangzhe Wu, Weidi Xie, Min Chen, and Victor Adrian Prisacariu. NeRF–: Neural radiance fields without known camera parameters. arXiv preprint arXiv:2102.07064, February 2021.   \n[30] Adam Kosiorek, Heiko Strathmann, Daniel Zoran, Pol Moreno, Rosalia Schneider, Sona Mokrá, and ˇ Danilo Jimenez Rezende. NeRF-VAE: A geometry aware 3D scene generative model. In International Conference on Machine Learning. PMLR, 2021.   \n[31] Alex Trevithick and Bo Yang. GRF: Learning a general radiance field for 3D scene representation and rendering. arXiv preprint arXiv:2010.04595, October 2020.   \n[32] Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa. pixelNeRF: Neural radiance fields from one or few images. arXiv preprint arXiv:2012.02190, December 2020.   \n[33] Michael Niemeyer and Andreas Geiger. GIRAFFE: Representing scenes as compositional generative neural feature fields. arXiv preprint arXiv:2011.12100, November 2020.   \n[34] H. Durrant-Whyte and T. Bailey. Simultaneous localization and mapping: part i. IEEE Robotics Automation Magazine, 13(2):99–110, 2006. doi: 10.1109/MRA.2006.1638022.   \n[35] T. Bailey and H. Durrant-Whyte. Simultaneous localization and mapping (slam): part ii. IEEE Robotics Automation Magazine, 13(3):108–117, 2006. doi: 10.1109/MRA.2006.1678144.   \n[36] Josep Aulinas, Yvan Petillot, Joaquim Salvi, and Xavier Lladó. The slam problem: A survey. In Proceedings of the 2008 Conference on Artificial Intelligence Research and Development: Proceedings of the 11th International Conference of the Catalan Association for Artificial Intelligence, page 363–371, NLD, 2008. IOS Press. ISBN 9781586039257.   \n[37] Michael Burri, Janosch Nikolic, Pascal Gohl, Thomas Schneider, Joern Rehder, Sammy Omari, Markus W Achtelik, and Roland Siegwart. The euroc micro aerial vehicle datasets. The International Journal of Robotics Research, 2016. doi: 10.1177/0278364915620033.   \n[38] Andreas Geiger, Philip Lenz, and Raquel Urtasun. Are we ready for autonomous driving? the kitti vision benchmark suite. In Conference on Computer Vision and Pattern Recognition (CVPR), 2012.   \n[39] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers. A benchmark for the evaluation of rgb-d slam systems. In Proc. of the International Conference on Intelligent Robot Systems (IROS), Oct. 2012.   \n[40] Ruihao Li, Sen Wang, and Dongbing Gu. Deepslam: A robust monocular slam system with unsupervised deep learning. IEEE Transactions on Industrial Electronics, 68(4):3577–3587, 2021. doi: 10.1109/TIE. 2020.2982096.   \n[41] Mingyang Geng, Suning Shang, Bo Ding, Huaimin Wang, and Pengfei Zhang. Unsupervised learningbased depth estimation-aided visual slam approach. Circuits, Systems, and Signal Processing, 39(2): 543–570, 2020.   \n[42] Diederik P Kingma and Max Welling. An introduction to variational autoencoders. arXiv preprint arXiv:1906.02691, 2019.   \n[43] Adam R Kosiorek, Heiko Strathmann, Daniel Zoran, Pol Moreno, Rosalia Schneider, Sona Mokrá, ˇ and Danilo J Rezende. Nerf-vae: A geometry aware 3d scene generative model. arXiv preprint arXiv:2104.00587, 2021.   \n[44] Rishabh Kabra, Chris Burgess, Loic Matthey, Raphael Lopez Kaufman, Klaus Greff, Malcolm Reynolds, and Alexander Lerchner. Multi-object datasets. https://github.com/deepmind/multi-object-datasets/, 2019.   \n[45] Rohit Girdhar and Deva Ramanan. CATER: A diagnostic dataset for compositional actions & temporal reasoning. In International Conference on Learning Representations, 2020.   \n[46] Josh Abramson, Arun Ahuja, Arthur Brussee, Federico Carnevale, Mary Cassin, Stephen Clark, Andrew Dudzik, Petko Georgiev, Aurelia Guy, Tim Harley, et al. Imitating interactive intelligence. arXiv preprint arXiv:2012.05672, 2020.   \n[47] Danilo Jimenez Rezende and Fabio Viola. Taming vaes. arXiv preprint arXiv:1810.00597, 2018.   \n[48] Vasu Parameswaran and Rama Chellappa. View invariance for human action recognition. International Journal of Computer Vision, 66(1):83–101, 2006.   \n[49] Yang Liu, Zhaoyang Lu, Jing Li, and Tao Yang. Hierarchically learned view-invariant representations for cross-view action recognition. IEEE Transactions on Circuits and Systems for Video Technology, 29(8): 2416–2430, 2018.   \n[50] Ting Liu, Jennifer J Sun, Long Zhao, Jiaping Zhao, Liangzhe Yuan, Yuxiao Wang, Liang-Chieh Chen, Florian Schroff, and Hartwig Adam. View-invariant, occlusion-robust probabilistic embedding for human pose. arXiv preprint arXiv:2010.13321, 2020.   \n[51] Jennifer J Sun, Jiaping Zhao, Liang-Chieh Chen, Florian Schroff, Hartwig Adam, and Ting Liu. Viewinvariant probabilistic embedding for human pose. In European Conference on Computer Vision, pages 53–70. Springer, 2020.   \n[52] Yana Hasson, Bugra Tekin, Federica Bogo, Ivan Laptev, Marc Pollefeys, and Cordelia Schmid. Leveraging photometric consistency over time for sparsely supervised hand-object reconstruction. In CVPR, 2020.   \n[53] Emily L Denton and Vighnesh Birodkar. Unsupervised learning of disentangled representations from video. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.   \n[54] Yizhe Zhu, Martin Renqiang Min, Asim Kadav, and Hans Peter Graf. S3VAE: Self-supervised sequential VAE for representation disentanglement and data generation. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, June 2020.   \n[55] Emily Denton and Rob Fergus. Stochastic video generation with a learned prior. arXiv preprint arXiv:1802.07687, February 2018.   \n[56] Ruben Villegas, Jimei Yang, Seunghoon Hong, Xunyu Lin, and Honglak Lee. Decomposing motion and content for natural video sequence prediction. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.   \n[57] Li Yingzhen and Stephan Mandt. Disentangled sequential autoencoder. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 5670–5679. PMLR, 2018.   \n[58] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.   \n[59] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021.   \n[60] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-End object detection with transformers. In Computer Vision – ECCV 2020, pages 213–229. Springer International Publishing, 2020.   \n[61] Huiyu Wang, Yukun Zhu, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen. MaX-DeepLab: End-to-End panoptic segmentation with mask transformers. arXiv preprint arXiv:2012.00759, December 2020.   \n[62] Sixiao Zheng, Jiachen Lu, Hengshuang Zhao, Xiatian Zhu, Zekun Luo, Yabiao Wang, Yanwei Fu, Jianfeng Feng, Tao Xiang, Philip H S Torr, and Li Zhang. Rethinking semantic segmentation from a Sequence-toSequence perspective with transformers. arXiv preprint arXiv:2012.15840, December 2020.   \n[63] Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. arXiv preprint arXiv:2104.14294, 2021.   \n[64] Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is Space-Time attention all you need for video understanding? arXiv preprint arXiv:2102.05095, February 2021.   \n[65] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan Asselmann. Video transformer network. arXiv preprint arXiv:2102.00719, February 2021.   \n[66] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luciˇ c, and Cordelia Schmid. ViViT: ´ A video vision transformer. arXiv preprint arXiv:2103.15691, March 2021.   \n[67] Yuqing Wang, Zhaoliang Xu, Xinlong Wang, Chunhua Shen, Baoshan Cheng, Hao Shen, and Huaxia Xia. End-to-End video instance segmentation with transformers. arXiv preprint arXiv:2011.14503, November 2020.   \n[68] Fabio Viola, Louise Deason, and Marcel Büsching. Datasets used to train generative query networks (gqns) in the ‘neural scene representation and rendering’ paper. https://github.com/deepmind/gqn-datasets, 2018.   \n[69] Felix Hill, Olivier Tieleman, Tamara von Glehn, Nathaniel Wong, Hamza Merzic, and Stephen Clark. Grounded language learning fast and slow. arXiv preprint arXiv:2009.01719, 2020.   \n[70] Nicholas Watters, Loic Matthey, Christopher P Burgess, and Alexander Lerchner. Spatial broadcast decoder: A simple architecture for learning disentangled representations in vaes. arXiv preprint arXiv:1901.07017, 2019. ",
+        "bbox": [
+            173,
+            194,
+            826,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            49,
+            828,
+            915
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            73,
+            828,
+            915
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            89,
+            828,
+            703
+        ],
+        "page_idx": 13
+    }
+]

parse/train/YSzTMntO1KY/YSzTMntO1KY_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff