Properties from mechanisms: an equivariance perspective on identifiable representation learning

1 INTRODUCTION . Modern unsupervised representation learning techniques can generate images of our world with intricate detail ( e.g . Karras et al. , 2019 ; Song et al. , 2020 ; Razavi et al. , 2019 ) , and yet , the latent representations from which these images are generated remain entangled and challenging to interpret ( Schölkopf et al. , 2021 ; Locatello et al. , 2019 ) . At the same time , the success of pre-trained transformers ( Devlin et al. , 2018 ; Brown et al. , 2020 ) shows that advances in our ability to disentangle the underlying generative factors can lead to dramatic improvements in the sample complexity of downstream supervised tasks ( Bengio & LeCun , 2007 ; Bengio et al. , 2013 ) . In order to consistently replicate this success , we need methods that can reliably invert the data generating process . This is a challenging task because unsupervised representation learning with independent and identically distributed ( IID ) data is hopelessly unidentified : even in the nice case in which observations are generated via some p ( x|z ) , there exists an infinitely large class of possible latent distributions p̃ ( z|x ) that are consistent with the observed marginal distribution , p ( x ) , and only one of them corresponds to the true latent distribution p ( z|x ) ( Locatello et al. , 2019 ; Khemakhem et al. , 2020a ) . If we want to build systems which are able to provably identify1 the true latent variables z that generated our observed data x = g ( z ) for some observation model g , then we need to exploit structural assumptions that constrain this set of possible distributions . Most of the prior work that can provide such guarantees was developed in the independent component analysis ( ICA ) literature . The key ICA assumption is that the latent factors are ( conditionally ) independent and non-Gaussian . Then , if the observation model , g : Z → X is linear and invertible , z = g−1 ( x ) is identified ( Comon , 1994 ) , and for nonlinear g there are a number of recent approaches that leverage non-stationarity in the distributions over z to identify g−1 ( Hyvarinen & Morioka , 2016 ; 2017 ; Hyvarinen et al. , 2019 ; 1A problem is identified if there exists a unique solution in the infinite data limit and no constraints on model capacity . Khemakhem et al. , 2020a ) . These results give a tantalizing demonstration that representation learning with identification guarantees is possible , but the requirement that the latent factors are statistically ( conditionally ) independent is limiting2 ( Higgins et al. , 2018 ; Schölkopf et al. , 2021 ) . In this paper we take a very different approach . We study how the mechanisms that govern an environment ’ s evolution constrain the set of possible latent representations that are consistent with the data . As a simple concrete example , consider the bouncing ball environment shown in Figure 1 . The latent state can be completely described by a vector , z , containing the position , velocity and acceleration of the balls3 , and given this latent state , the images , x , shown in Figure 1 , are produced via some rendering engine g : Z → X . Our task is to leverage sequences of observations x1 , . . . , xT and knowledge of m to recover z1 , . . . , zT . If we can show that this task has a unique solution that is consistent with the observations and mechanisms , then the problem is identified . Our main result is that when we know the true mechanism , m , the system is identified up to any equivariances of the mechanism ; or equivalently , in Section 2.2 we prove that we can identify z up to any invertible transformation a : Z → Z that commutes with m , such that the composition m◦a = a◦m . For example , in the bouncing balls environment shown in Figure 1 , the laws of physics are equivariant with respect to your choice of units of measurement—changing from representing z in meters to inches leaves the output of the mechanism unchanged up to a corresponding unit change— and hence we can only hope to identify z up to some scaling function a ( z ) which corresponds to an arbitrary choice of units of measurement . Interestingly , when the environment evolves according to multiple known mechanisms , the sources of non-identifiability are even further constrained : such a system is identified up to equivariances that are shared by all n mechanisms . Perfectly knowing the true mechanisms and when they are applied is unlikely , but in Section 2.3 we show that we can relax that assumption to a setting where we instead know a hypothesis class of possible mechanisms that could have been applied . This weaker assumption leads to an additional 2As a simple example of dependence between latent variables , assume the bouncing balls shown in Figure 1 have different masses indicated by their colors . If the initial conditions were such that all the balls have the same momentum , then mass and velocity will be inversely correlated . 3A complete generative model of these images would also need to track the colors and shapes of the elements ; we will return to this issue in the discussion of the results . source of non-identifiability : the mechanisms in our hypothesis class can imitate each other if there exists an invertible transformation a such that for any two mechanisms m1 and m2 in our hypothesis class , m2 = a−1 ◦m1 ◦ a . For example , if we are in a setting where our hypothesis class includes both a product mechanism , m1 ( z ) = ∏ i zi , and a sum mechanism , m2 ( z ) = ∑ i zi , then m1 can imitate m2 if a ( z ) = exp ( z ) , since ∑ i zi = log ( ∏ i exp ( zi ) ) . As before , this result is complete , in the sense that these two sources of non-identifiability—equivariance and imitation—are the only sources of non-identifiability in such systems . This gives us a natural way of thinking about the way knowledge of deterministic mechanisms constrains a representation learning task : with complete knowledge , the only source of non-identifiability is any equivariances inherent in the mechanism , but by allowing a hypothesis class of possible mechanisms , we introduce potential non-identifiability via imitation . That said , the relationship between the size of the hypothesis class and the size of the set of a ’ s that commute via imitation is not necessarily monotonic : for environments governed by multiple mechanisms , a larger hypothesis class can lead to fewer imitations . Section 3 , shows that we can derive analogous results for stochastic mechanisms , m ( z , U ) , where the mechanism defines a conditional distribution p ( zt+1|zt ) . This generalization gives us a way of comparing our mechanism-based perspective with existing identifiability results . We demonstrate this in Section 4 by showing that it is possible to view the distributional assumptions made in Klindt et al . ( 2020 ) as a particular choice of mechanism , and by doing so , we can leverage our theory to give alternative proofs of these results . This strategy required weaker distributional assumptions , thereby generalizing their result . Finally , we give a mechanism-based perspective on the related work in Section 5 and Section 6 concludes with a discussion of the open problems that need to be addressed in order to reliably leverage this approach in practice . 2 MECHANISM BASED IDENTIFICATION . 2.1 DATA GENERATION PROCESS . The state of the system at time t ∈ { 1 , · · · , T } is given by zt ∈ Z ⊆ Rd . At each time t , we observe g ( zt ) = xt ∈ X ⊆ Rn , which is some transformation of the latent zt . We can think of g : Z → X as a function that transforms the ( typically low dimensional ) state variables to the ( typically high dimensional ) observed variables ; for example , in the bouncing ball environment described in the introduction , g is the rendering engine that produces the images shown in Figure 1 . We assume g is injective 4 with respect to Rn—i.e . g ( z1 ) = g ( z2 ) implies z1 = z2 ; or equivalently , any change to the underlying state , z , is reflected in some pixel-level change to the observation x—and we make g bijective by restricting its inverse g−1 to any x on the data manifold which we denote X ( i.e . X is the image of g ) . The state transition from time t to t+ 1 is governed by a mechanism mt : Z → Z . There may be multiple mechanisms in a given environment . For example in Figure 1 the transition from z1 to z2 does not involve any collisions so the state evolves according to Newton ’ s first law of motion ( Newton , 1687 ) ; the transition from z2 to z3 involves a collision between two of the balls that is described by Newton ’ s third law . Together mt and g describe the data-generation from time t to t+ 1 as follows , xt ← g ( zt ) , zt+1 ← mt ( zt ) . ( 1 ) If we fix the initial conditions , z0 , this data generation process is deterministic . In Section 2.2 , we provide results when the underlying mechanism is known and in Section 2.3 , we we extend those results to the case when the mechanisms are not known . In Section 3 , we extend our results to stochastic mechanisms , m ( Zt , U ) that take samples from some distribution U ∼ Uniform ( 0 , 1 ) as input . These stochastic mechanisms can represent any conditional distribution P ( Zt+1|Zt ) over states.5 2.2 IDENTIFYING ENCODERS WHEN THE UNDERLYING MECHANISM IS KNOWN . We begin in the simplest version of the system described by equation ( 1 ) : assume that at each time t the same mechanism m : Z → Z is used , and we know this mechanism . From these assumptions we 4for ball collision example , if Z equals position and velocity , we can achieve injectivity by frame stacking . 5For example , if Zt+1|Zt ∼ N ( Zt , σ ) then m ( z , U ) = z + σΦ−1 ( U ) where Φ is the cumulative density of a standard Gaussian . Strictly , representing a joint ( Zt+1 , Zt ) as ( m ( Zt , U ) , Zt ) requires a restriction to nice spaces . See Austin ( 2015 , Lemma 3.1 ) . can derive an identity that describes how the observations xt and xt+1 are related , zt+1 = m ( zt ) , g −1 ( xt+1 ) = m ◦ g−1 ( xt ) , xt+1 = g ◦m ◦ g−1 ( xt ) . ( 2 ) It may be helpful to think of this identity as describing an autoencoder where we require that the encoder g−1 ( xt ) inverts the data generating process to produce some latent zt from xt , and that the decoder has to reproduce xt+1 from m ( zt ) ; i.e . the representation transformed by the mechanism . Importantly , this identity describes the true data generating process , rather than some model of it . Our hypothesis class over the possible encoder / decoder functions is the set , G , of all bijective functions from Z → X . By assuming that bijectivity we are essentially assuming that the reconstuction task is solved:6 G is the set of all autoencoders that perfectly reproduce the data , such that for any x on the data manifold X , and any encoder / decoder pair ( g̃−1 , g̃ ) with g̃ ∈ G , their composition is the identity function , x = g̃ ◦ g̃−1 ◦ x . Because we assumed that the true g is bijective , we know that it is in our search space , G. We can constrain this set using our knowledge of the mechanism by only considering solutions that also satisfy the identity given in equation 2 , such that for every pair of observations ( xt , xt+1 ) , an analogous identity holds , xt+1 = g̃ ◦m ◦ g̃−1 ( xt ) ( 3 ) Now suppose that we have access to observations xt from the entire set X , then the above identities have to hold for all xt , ∈ X with corresponding xt+1 from equation 2 , so we can conclude that the following functions are equal , g ◦m ◦ g−1 = g̃ ◦m ◦ g̃−1 ( 4 ) This relationship will hold for any of the possible decoders g̃ ( and corresponding encoders g̃−1 ) that are observationally equivalent given our assumptions . We denote the set of all such decoders , Gid = { g̃ | g̃ ∈ G , g ◦m ◦ g−1 = g̃ ◦m ◦ g̃−1 } . If Gid = { g } then we have shown that the problem is exactly identified , which means that if we manage to find a g̃ that satisfies equation 2 , then g̃ = g ; but if Gid , also includes other functions g̃ 6= g , then these functions are the sources of non-identifiability . Equivariant mechanisms To see an example of a setting where the problem is not exactly identified , consider a mechanism which is equivariant with respect to some bijective transformation a : Z → Z . A mechanism m is said to be equivariant w.r.t a if a ◦ m = m ◦ a . For example , a mechanism derived from Newtonian mechanics may be equivariant with respect to your choice of units of measurement , such that m ( cz ) = cm ( z ) for some scaling constant c. Similarly , if a mechanism transforms sets of items , any permutation of z would lead to a corresponding permutation of the mechanism ’ s output . If we have a mechanism that is equivariant with respect to some transformation a ( where a is not identity map ) , that implies that there exists a function g̃ 6= g in Gid , so the problem is not exactly identified . We can see this as follows , g ◦m ◦ g−1 = g ◦ a−1 ◦ a ◦m ◦ g−1 = g ◦ a−1 ◦m ◦ a ◦ g−1 = g̃ ◦m ◦ g̃−1 where the first equality uses the fact that a−1 ◦ a is the identity function , the second applies the definition of equivariance , and the final equality defines g̃ : = g ◦ a−1 and g̃−1 : = a ◦ g−1 . This shows that if the mechanism is equivariant , an encoder , g̃−1 , can output a transformed z̃ = a ( z ) , and the decoder , g̃ , inverts this transformation before producing its output , thereby leaving the observed variables , x , unchanged . If we denote the set of all equivariances of the mechanism E = { a | a is a bijection , a ◦ m = m ◦ a } , then we can define the set of all such sources of non-identification as , Geq = { g̃ | g̃ = g ◦ a−1 , a ∈ E } . This is a natural source of non-identification : given that we are relying on the mechanism to constrain the encoder g̃−1 , it is unsurprising we can not prevent transformations that are not affected by the mechanism . The more interesting observation , which we will show below , is that this set is the only source of non-identification when the mechanism is known , and hence we recover the true z up to equivariances in the mechanism . To state this theorem , we first define the notion of identifiability up to an equivalence class defined by a family of bijections A , where a ∈ A is a map a : Z → Z . Definition 1 . Identifiability up to A . If the learned encoder g̃−1 and the true encoder g−1 are related by some bijection a ∈ A , such that g̃−1 = a ◦ g−1 ( or equivalently g̃ = g ◦ a−1 ) , then g̃−1 is said to learn g−1 up to bijections in A . We denote this g̃−1 ∼A g−1 . 6This is obviously a strong assumption—learning autoencoders that perfectly reconstructed the data is not at all easy—but it focuses the discussion on the identification issues that remain after reconstruction is solved . Suppose , for example , A is a family of a permutations . Identifiability up to A implies that the true latent variables will be recovered , but that they would not necessarily be ordered in the same way as they were in the original data generation process . In this setting where the mechanism , m , is known , the following theorem shows that A is just E , the set of equivariances of m. Theorem 1 . If the data generation process follows equation 2 , the encoders that solve the observation identity in equation 4 identify true encoder g−1 up to equivariances of m ( g̃−1 ∼E g−1 ) . To prove the above theorem , we need to establish that the set of solutions to the identity in equation 4 and the set of maps derived from equivariances are equal Gid = Geq . The proof is given in Section A.1 of the appendix . From Theorem 1 , we can derive a number of observations . First , notice that if we have a standard autoencoder7 then the mechanism is the identity map , m ( z ) = z , and its set of equivariances E is any invertible function a , and hence Theorem 1 shows that the encoder is essentially unconstrained . However , if the mechanism is any non-trivial function , m ( z′ ) 6= z′ for some z′ , then the space of possible encoders is significantly reduced to just those invertible transformations that commute with m. If the system involves multiple known mechanisms , { m1 , . . . , mT } , where at each time zt+1 = mt ( zt ) , then the encoder is even further constrained . Define the set of all the mechanisms that are used at least once in the evolution of the system asM∗ = ∪Tt=1 { mt } . Suppose E i denotes the equivariances of mi ∈ M∗ . Define the equivariances that are shared across all the mechanisms as E∗ = ∩iE i ; Corollary 1 . If the data generation process follows equation 1 , then the encoders that satisfy observation identity in equation 4 for all m ∈M∗ identify true encoder g−1 up to the equivariances shared across all the mechanisms inM∗ , E∗ ( g̃−1 ∼E∗ g−1 ) . The proof of the above claim is in Section A.3 . This corollary implies a blessing that comes with more complex environments : if an object is transformed by multiple mechanisms which are diverse ( in the sense that they share few equivariances ) , then it becomes easier to identify . Given access to inputs and outputs of a mechanism , if we can not tell apart whether some transformation was applied to the input or the output , then the mechanism is equivariant with respect to the transformation . Together Theorem 1 and Corollary 1 , state that we can learn to invert the data generation process but we can not distinguish latents that were transformed by equivariances shared across the mechanisms .

Summary: This work gives several identifiability results for representation learning from sequences of observed variables when the mechanism governing the underlying latent variables is known, and the observation renderer is bijective. The identifiability of an inverse renderer is shown to depend only upon the equivariances of the mechanism(s) governing the evolution of the underlying latent variables. Extensions are given to the case of a known mechanism class and to stochastic mechanisms.