Weakly Supervised Representation Learning with Sparse Perturbations
Authors: Kartik Ahuja
Jason Hartford
Yoshua Bengio
Venue: NeurIPS 2022 Accept (2022)
Keywords: representation learning
identifiability

Abstract: The theory of representation learning aims to build methods that provably invert the data generating process with minimal domain knowledge or any source of supervision. Most prior approaches require strong distributional assumptions on the latent variables and weak supervision (auxiliary information such as timestamps) to provide provable identification guarantees. In this work, we show that if one has weak supervision from observations generated by sparse perturbations of the latent variables--e.g. images in a reinforcement learning environment where actions move individual sprites--identification is achievable under unknown continuous latent distributions. We show that if the perturbations are applied only on mutually exclusive blocks of latents, we identify the latents up to those blocks. We also show that if these perturbation blocks overlap, we identify latents up to the smallest blocks shared across perturbations. Consequently, if there are blocks that intersect in one latent variable only, then such latents are identified up to permutation and scaling. We propose a natural estimation procedure based on this theory and illustrate it on low-dimensional synthetic and image-based experiments. 

Weakly Supervised Representation Learning with Sparse Perturbations
Kartik Ahuja"
Jason Hartford'
Yoshua Bengio
Abstract
The theory of representation learning aims to build methods that provably invert the data generating process with minimal domain knowledge any source of supervision. Most prior approaches require strong distributional assumptions on the latent variables and weak supervision (auxiliary information such as timestamps) to provide provable identification guarantees. In this work, we show that if one has weak supervision from observations generated by sparse perturbations of the latent variables-e.g. images in a reinforcement learning environment where actions move individual sprites-identificatinn is achievable under unknown continuous latent distributions. We show that the perturbations are applied only on mutually exclusive blocks of latents, we identify the latents up to those blocks. We also show that if these perturbation blocks overlap, we identify latents to the smallest blocks shared across perturbations. Consequently, if there blocks that intersect in one latent variable only, then such latents are identified up to permutation and scaling. We propose a natural estimation procedure based on this theory and illustrate it on low-dimensional synthetic and mmage-based experiments.
1 Introduction
If you are reading this paper computer, press one of the arrow keys... all the text you are reading jumps as the screen refreshes in response to your action. Now imagine you were playing video game like Atari's Space Invaders- the same keystroke would cause small sprite at the bottom of your screen to move in response. These actions induce changes in pixels that are very different, but in both cases, the visual feedback in response to our actions indicates the presence of some object on the screen-a virtual paper and a virtual spacecraft, respectively- with properties that can manipulate. Our keystrokes induce sparse changes program's 's state, and these changes are reflected on the screen, albeit not necessarily in a correspondingly sparse way (e.g.. most pixels change when scrolling). Similarly, many of our interactions with the real world induce sparse changes to the underlying causal factors of our environment: lift coffee cup and the cup moves, but not the rest of the objects your desk; the room shift, but only in the horizontal direction. These examples hint at the main question we aim answer in this paper: if know that actions have sparse effects on the latent factors of our system, can we use that knowledge as weak supervision to help disentangle these latent factors from pixel-level data?
your head laterally, and the coordinates of all the objects in
learning approaches have made phenomenal progress in the last few years, with large-scale systems like GPT-3 (Brown al., 2020) offering large improvements all natural language benchmarks, and CLIP (Radford et al., 2021) outperforming state-of-the-an supervised models from six years ago (Szegedy et al., 2016) the ImageNet challenge (Deng et al., 2009) without using any of the labels.
Self- and weakly-superee
Mila Quebec AI Institute, Université de Montréal. Correspondence to: kartik aakjaaaaajaaaaaa www..immiiaaacc.....mmmm
CIFAR Fellow.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Figure 1: Ball agent interaction environment. Different frames show the effect of sparse perturbations.
Yet, despite these advances, these systems are still far from human reasoning abilities and often fail on out-of oof-distribution examples (Geirhos et al., 2020). To robustly generalize out of distribution, we need models that can infer the causal mechanisms that relate latent variables (Schölkopf et al., 2021; Schölkopf and von Kügelgen, 2022) because these mechanisms are invariant under distribution shift. The field of causal inference has developed theory and methods to infer causal mechanisms from data (Pearl, 2009; Peters et al., 2017), but these methods assume access to high-level abstract features, instead of low-level signal data such as video, text and images. We need representation learning methods that reliably recover these abstract features if are bridge the gap between causal inference and deep learning.
This is a challenging task because the problem of inferring latent variables is not identified with independent and identically distributed (IID) data (Hyvärinen and Pajunen, 1999; Locatello et al., 2019), even in the limit of an infinite number of such IID examples. However, there has been signifi- cant recent progress factors Z (e.g.. object positions, object colors, etc.) underlying complex data X (e.g. image), where X + (Z), by going beyond the IID setting and using observations of along with minimal domain knowledge and supervision (Hyvarinen and Morioka, 2016, 2017; Locatello et al., 2020; Khemakhem et al., 2020a). These works establish provable identification latents by leveraging strong structural assumptions such independence conditional auxiliary information (c.g.. timestamps). In this work, we aim to relax these distributional assumptions on the latent variables to achieve identification for arbitrary continuous latent distributions. Instead of distributional assumptions, to data generated under sparse perturbations that change only a few latent variables at a time as a source of weak supervision. Figure I illustrates working example of this assumption: simple environment where an agent's actions perturb the coordinates of a few balls at a time. Our main contributions are summarized follows.
developing representation learning approaches that provably recover latent
assume access
We show that sparse perturbations that impact one latent at time are sufficient to learn the latents (up to permutation and scaling) that follow any unknown continuous distribution.
Next, we consider more general settings, where perturbations impact one block of latent variables a time. the setting where blocks do not overlap, we recover the latents up to affine transformation of these blocks.
Further, we show that when perturbation blocks overlap, we get stronger identification. In this setting, we prove identification up to affine transformation of the smallest intersecting block. Consequently, if there are blocks that intersect one latent variable only, then such latents are identified up to permutation and scaling.
We leverage these results to propose natural estimation procedure and experimentally illustrate the theoretical claims on low-dimensional synthetic and high-dimensional image- based data.

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2
Related works
Many of the works on provable identification of representations trace their roots to non-linewt ICA (Hyvärinen and Pajunen, 1999). Hyvarinen and Morioka (2016, 2017) were the first to use auxiliary information in the form of timestamps and additional structure on the latent evolution to achieve provable identification. Since then, these works have been generalized in many exciting ways. Khemakhem et al. (2020a) assume independence of latents conditional on auxiliary information, and several of these assumptions were further relaxed by Khemakhem et al. (2020b).
Our work builds on the machinery developed by Ahuja et al. (2022). Ahuja et al. show that if we know the mechanisms that drive the evolution of latents, then the latents are identified up to equivariances of these mechanisms. However, the authors leave the question of achieving identification without such knowledge open. Here we consider a class of mechanisms where an agent's 's actions impact the latents through unknown perturbations. We show how to achieve identification by exploiting the sparsity in the perturbations. This class of perturbations was first leveraged to prove identification by Locatello et al. (2020). However, Locatello et al. assume that the latents are independent, whereas we make no assumptions on the distribution other than continuity. Lachapelle et al. (2022) also use sparse interventions on the latents to strengthen the identification guarantees in Khemakhem et al. (2020a) for conditional exponential distributions. However, the form of sparsity that is leveraged in their work is different from ours. In our work, we assume that the vector of changes in the latents is sparse, i.e., some components change and the rest of the components do not change. In Lachapelle et al. (2022), all the components of the latents change post interventions but the graphical model capturing the interaction of interventions (described random variables) and the latents is sparse. Klindt et al. (2021) also use sparsity in time-series settings to attain identification. Klindt et al. (2021) enforce soft f norm driven sparsity in the vector of changes in latents by assuming that latents evolve independently under a Laplace distribution but do not require access to data under interventions.
Yao et al. (2021) and Lippe et al. (2022) model the latent evolution as a structural causal model unrolled in time. Yao et al. exploit non-stationarity and sufficient variability dictated by the auxiliary information to provide identification guarantees. Lippe et al. exploit causal interventions the latents to provide identification guarantees but require the knowledge of intervention targets and assume an invariant causal model describing the relations between any adjacent time frames. In concurrent work, Brehmer et al. (2022) leverage data generated under causal interventions as a source of weak supervision and prove identification for structural causal models that are diffeomorphic transforms of exogenous noise. Our work also connects to an insightful line of work on multi-view ICA (Gresele al., 2020), which proves identification under independent latents, in the following sense. We can interpret the data under different perturbations as different views of the same underlying latent. In addition, some recent papers explain the success of self-supervised contrastive learning Zimmermann et al. (2021); Von Kügelgen et al. (2021) through the lens of identification of representations.
Above, we focused provable representation identification, which is central to this work. We now give a brief overview of empirical works on disentanglement which have shown success on some benchmark tasks, but do not theoretically characterize conditions for successful disentanglement. Many variations of variational autoencoders (VAE) were developed over the years to achieve disen- tanglement. -AV (Higgins et al., 2016) uses hyperparameter in front of the KI regularizer to make the learned latent independent. Factor VAE (Kim and Mnih, 2018) proposes an adversarial training-bar based approach, where the discriminator encourages the learned representation to have independent components. Annealed 3- VAE (Burgess et al., 2018) proposes to progressively increase the capacity of bottleneck B enforce independence one component disentanglement have been also used in reinforcement learning; Higgins et al. (2017), Dittadi et al. (2020), and Miladinović et al. (2019) are some of the representative works in the area. Locatello et al. (2019) showed that most of the above methods could often fail to disentangle in the absence of supervision or inductive biases. As a result, there has been a surge in the interest in building approaches that achieve provable representation identification. Lastly, there is a line of work, which does not focus on disentanglement or representation identification, but has shown the benefits of sparsity based inductive biases sparse changes in latents over time (Goyal et al., 2019) or sparse interactions between latents (Goyal et al., 2021) under distribution shifts.
time. Ideas based on
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3 Latent identification under sparse perturbations
Data Generation Process work. There are two classes of variables we consider - a) unobserved latent variables Z E Z C R
We start by describing the data generation process used for the rest of the
and b) observed variables X E X C I The latent variables Z are sampled from a distribution Pz and then transformed by map g IR + R", where is injective and analytic", to generate X. We
write this as follows
2 ~ Pz
x + g(2)
(1)
where z and x are realizations of the random variables Z and X respectively. is impossible to invert g just from the realizations of X (Hyvärinen and Pajunen, 1999; Locatello et al., 2019). Most work has gone into understanding how structure of latents Z and auxiliary information in the form of timestamps labels play role in solving the above problem. In this work, we depart from these assumptions and instead investigate the role of data generated under perturbations of latents to achieve identification. Define the set of perturbations I = {1.... ..... m] and the corresponding perturbation vectors as A {ō), b, where 5; is the 8th perturbation. Each latent is sampled from an arbitrary and unknown distribution Ipz. The same set of unknown perturbations in are applied to each to generate m perturbed latents {²x per sampled x and the corresponding observed vectors (5xx im R= Each of these latents are transformed by the map and we observe (x,, T1,- :.m)- Our goal is to these observations and estimate the underlying latents. We summarize this data generation process (DGP) in the following assumption.
Assumption 1. The DGP follows
2 ~~ PZ,I + g(2)
E + 6x, EI
+ 9(k), Ak EI
(2)
where g is injective and analytic, and Z is a continuous random vector with full support over Rdd
The above DGP is very close to the DGP in Locatello et al. (2020) except we do not require latent dimensions to be mutually independent. To better understand the above DGP, let us turn to some examples. Consider a setting where an agent is interacting with an environment containing several balls (See Figure 1). The latent captures the properties of the objects; for example, in Figure 1, just captures the positions of each ball, but in general it could include more properties such as velocity, shape, color, etc.. The agent perturbs the objects in the scene by dg which can modify a single property associated with one object or multiple properties from one or more objects depending on how the agent acts. Note that when the agent perturbs latent, it can lead to downstream effects. For instance, if the agent moves a ball to the edge of the table, the ball falls in subsequent frames. For this work, we only consider the observations just before and after the perturbation and not the downstream effects. In the Appendix (Section A.2.5), we explain these downstream effects using structural causal models. We also explain the connection between the perturbations in equation (24) and causal interventions leveraged in Brehmer et al. (2022); Lachapelle et al. (2022). The above example is typical of a reinforcement learning environment, other examples include natural videos with sparse changes (e.g., MPI3D data (Gondal al., 2019)).
In the above DGP in equation (24), we assumed that for each scene x there are multiple perturbations. It is possible to extend our results to settings where we perturb each scene only once, given sufficiently diverse set of perturbations, i.c., for a small neighborhood of a scene around T, each scene in the neighbourhood receives different perturbation. We compare these two approaches experimentally.
Learning objective The learner's objective is to use the observed samples (1r, (.... .... gen- crated by the DGP in Assumption and Iearn an encoder f R" + Rd that inverts the function and recovers the true latents. For each observed sample (z, ..m)). the learner compares all the pairs (x., 3t)) pre- and post-perturbation. For every unknown perturbation òū used in the DGP in equation (24), the learner guesses the perturbation 6; and enforces that the latents predicted by the encoder for x and Z are consistent with the guess. We write this as V (I, T1, ... Tmm) generated by DGP in (24)
f(xx) f(x) t
2 analytic function, g. is an infinitely differentiable function such that for all z' in its domain, the Taylor series evaluated converges pointwise to g(z')
'The assumption on the support of Z can be relaxed.

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We denote the set of guessed perturbations as A {6i. 5m) J. where 6, is the guess for perturbation oi. We can turn the above identity into a mean square error loss given as
min E (f(ox)- f(x) -6) -()))))) A
(4)
where the expectation is taken over observed samples generated by the DGP in (24) and the mini- mization is over all the possible maps f and perturbation guesses in the set A' Note that a trivial solution to the above problem is an encoder that maps everything zero, and all guesses equal zero In the next section, we get rid of these trivial solutions by imposing an additional requirement that the span of the set A is Rd It is worth pointing out that we do not restrict the set of f's to injective maps in theory and experiments. We denote the latent estimated by the encoder for a point x as 2 f(x). It is related to the true latent as follows 2 = f g(z) = a(z), where a is some function that relates true z to estimated E. the next section, we show that perturbations are diverse, then a an affine transform. Further, we show that if perturbations are sparse, then takes an even simpler form.
3.1 Sparse perturbations
We first show that it is possible to identify the true latents up to affine transformation without any sparsity assumptions. Later, we leverage sparsity to strengthen identification guarantees.
Assumption 2. The dimension of the span of the perturbations in (24) is d, i.e., dim (im(span(4)) d.
The above assumption implies that the perturbations are diverse. We now state a regularity condition on the function a.
Assumption 3. is analytic function. For each component i {1.... dd} of a(2) and each component j E {1, (1, ,d] of z, define the set 8 = {0 [ + b) = vai(()) + (0);
Rd), where b is a fixed vector in Rdd Each set S has non-zero Lebesgue measure in Rd.
If we restrict the encoder f to be analytic, then a is analytic since g is also analytic, thus satisfying the first part of the above assumption. The second part of the above assumption can understood as follows: suppose we have scalar valued function h IR - IR that is differentiable. If we expand h(u + v) around (u), by the mean value theorem we get h(u + v) h(u) + 1'(c)v, where E [u. u u+ v). If we vary u to take all the values in IR, then also varies. The above assumption states that the set of e's has a non-zero Lebesgue measure. Under the above assumptions, we show that an encoder that solves equation (3) identifies true latents up to affine transform, i.e., = Az + where A E Rddd is a matrix and c E Id is an offset.
Proposition If Assumptions 2. and hold, then the encoder that solves equation (3) (with A s.t. dim (ppann")) (A') d) identifies true latents up an invertible affine transform, i.e. Az + c, where Rddd an invertible matrix and e Rdd is an offset.
The proof of above proposition follows the proof technique from Ahuja et al. (2022), for further details refer to the Appendix (Section A. We interpret the above result in the context of the agent interacting with balls (as shown in Figure 1), where the latent vector z captures the x and y coordinates of the Tiballs- Under each perturbation, the balls move along the vector dictated by the perturbation. If there are least 2nblls perturbations, then the latents estimated by the learned encoder are guaranteed to be an affine transformation of the actual positions of the balls.
3.1.1 Non-overlapping perturbations
In Proposition 1, we showed affine identification guarantees for the DGP from Assumption 1. We now explore identification when perturbations are one-sparse, i.c., one latent changes at a time.
Assumption 4. The perturbations in A are one-sparse, iee., each 6, E A has one non-zero component.
Next, we show that under one-sparse perturbations, the latents estimated identify true latents up to permutation and scaling.
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Theorem 1. If Assumptions 1-4 hold and the mumber of perturbations per example equals the latent dimension, m. d, then the encoder that solves equation (3) (with A as one-sparse and dim (span( (A)) = d) identifies true latents up to permutation and scaling, i.e. = IIAa + c, where A E Raxd is an invertible diagonal matrix, IT E Rdxd is a permutation matrix and e. is an offset.
For the proof of above theorem. refer to Section A.I in the Appendix. The theorem does not require that learner knows either the identity or amount each component changed. However, the learner has to use one-sparse perturbations as guesses. Suppose the learner does not know that the actual perturbations are one-sparse and instead uses guesses that are p-sparse, i.e., P latents change at one time. In such a case, the and true 2 are related to each other through a permutation and block diagonal matrix, i.e., we can replace A in the above result to be a block diagonal matrix instead of a diagonal matrix (see the Appendix for details). In the context of the ball agent interaction environment from Figure 1, the above result states that provided the agent interacts with one coordinate of each ball at a time, possible to learn the position of each ball up to scaling errors.
We now consider natural extension of the setting above, where the perturbations simultancously operate on blocks of latents. In the ball agent interaction environment, this can lead to multiple scenarios- - i) the agent interacts with one ball at a time but perturbs both coordinates simultancously, ii) the agent interacts with several balls simultaneously.
Consider a perturbation 8; E A (from equation (24)). We define the block of latents that is impacted under perturbation 6; E A as {j 6. * 0.j E (], where of is the jth component of 6, We group the perturbations in I based on the block they act upon, i.e. perturbations in the same group act on the same block of latents. Define the set of the groups corresponding to perturbations in I as Gz. Define the set of corresponding blocks as By- = {B,.... Bg}, where B& is the block impacted by perturbations in group k. If By partitions the set of latent components indexed {I,..-,d then it implies all the distinct blocks are non non- conovvrlapping. We formally define this below.
Definition 1. Blockwise and non-overlapping perturbations. If the the set of blocks Bz corre- sponding to perturbations I form a partition of {1.... ... ...... then I is said to be blockwise and non-overlapping. Formally stated, any two distinct B,, Bj E By do not intersect, i.e., B, n B, 0. and U.B. = (1, dd..
From the above definition it follows that two perturbations either act on the same block or completely different blocks with no overlapping variables.
The perturbations I (used in equation (24)) are blockwise and non-overlapping (see Definition 1). Each perturbation in I is p-sparse, i.e., it impucts blocks of length p (p S d) at a time. Assumption 6. The learner knows the group label for each perturbation i E I. Therefore, any two perturbations in A associated with same group in G7 impact the same block of latents.
Assumption 5.
We illustrate the above Assumptions 5, 6 in the following example. Consider the ball agent interaction environment (Figure 1). = [21z, B1y: where //e/y is the x/y coordinate of ball i. If the agent randomly perturbs ball i, then it changes the block (Ziz: =sg). We would call such a system 2-sparse. All the perturbations on ball i are in one group. Since the agent knows the group of the perturbation, it does not know the ball index but it knows whenever we interact with the same ball.
Amaas Znwsyl is the vector of positions of all balls,
Definition 2. If the latent variables recovered HÂ + c, where II is a permutation matrix and Å is a bluck-diagonal mutrix, then the latent variables are said to be recovered up to permulations and block diagonll transforms.
In the theorem that follows, show that under the assumptions made in this section, we achieve identification up to permutations and diagonal transforms with invertible p X p blocks.
Theorem 2. If Assumptions 1-3, 6 hold, then the encoder that solves equation (3) (where A is p-sparse, dim (span(A) (A)) = d) identifies true latents up to permutation and block- diagonal transforms, i.e. f(x) 2 ILÁ= + where Å pdxdd is an invertible block-diagond matrix with blocks of size p x p. I ddd is permutation matrix and e E IRd is an offset.
"We can relax this condition to 2 d, refer to the Appendix for details.

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For the proof of the above theorem, refer to Section A.1 in the Appendix. From the above theorem, we gather that the learner can separate the perturbed blocks. However, the latent dimensions within the block are linearly entangled. In the ball agent interaction with 2-sparse perturbations, the above theorem implies that the agent can separate each ball out but not their respective J and y coordinates. In the above theorem, we require the learner to know the group of each intervention (Assumption 6). In the Appendix Section A.2.2, we relax Assumption 6 and show that we can continue to achieve identification up to permutation and block diagonal transforms. However, we need computationally expensive procedure that searches over subsets of latent dimensions to identify the dimensions impacted under the current intervention.
We briefly compare with Von Kügelgen et al. (2021), where the authors also establish block identifica- tion guarantees. In Von Kügelgen et al. (2021), the latent vector is divided into two parts the content block and the style block. Across augmentations, style varied and content is fixed. Von Kügelgen et al. leverage this invariance of the content across augmentations to learn the content block and not the style block. To summarize, invariance of content across different views is the key signal that is used to achieve identification. In our case, the perturbations act on different blocks of latents. In contrast to Von Kügelgen et al., we leverage sparsity of changes, i.e., we exploit both the varying part and the invariant part to identify all the distinct blocks and not just the content block.
3.1.2 Overlapping perturbations
In the previous section, we assumed that the blocks across different perturbations are non-overlapping. This section relaxes this assumption and allows the perturbation blocks to overlap. We start with a motivating example to show how overlapping perturbations can lead to stronger identification.
Consider the agent interacting with two balls, where [zim: Ziys Z2z, ø2y] describes the coordinates of the two balls. The agent perturbs the first ball and then perturbs the second ball. For the purpose of this example, assume that these perturbations satisfy the assumptions Theorem 2. We obtain that the estimated position of each ball /sz/v is linearly entangled w.r.t the true I and y coordinates. For the first ball we get &12- 21 + 0221y + 43- We also have the agent perturb the coordinates of the first and second ball together and then it does the same with the coordinates. We apply Theorem 2 and obtain that the estimated r coordinates of each ball are linearly entangled. We write this as f1x = by #1x + b222n
by. We take a difference of the two relations for Ê1 to get
(a1 - b1 21z + 2221y by 222 + 03 by =
(5)
Since the above has to hold for all øir, Ziy, 22n, we get 01 = b,, 02 = O, b2 = 0 and 03 = by. Thus
21 01 Ziz + ag. Similarly, we can disentangle the rest of the balls.
We take the insights from the above example and generalize them below. Let us suppose that from the set of perturbations I we can construct at least two distinct subsets I₃ and Iz such that both subsets form a blockwise non-overlapping perturbation (see Definition 1). Perturbations in I₁ (I2) partition {1. d) into blocks Bz, (Bz,) respectively. It follows that there exists at least two blocks B' E Bz, and B E Br, such that B¹ n B² * O. From Theorem 2, we know that we can identify latents in block B and 1 up to affine transforms. In the next theorem, show that we can identify BB up to affine transforms.
latents in each of the blocks Ba n 152, B3 \ 132, ,22 \
Assumption 7. Each perturbation in is p-sparse. The perturbations in each group span a p- dimensional space, i.e. Vqq E Gz, dim span (di))ee)
p. There exist at least two distinct subsets of perturbations In I and I2 C I that are both blockwise and non-overlapping.
Theorem 3. Suppose Assumptions 1, 3, 6 and 7 hold. Consider the subsets In and Lz that satisfy Assumption For every pair of blocks, B' E Bz, and B² BB,, the encoder that solves equation (3) (where A' is p-sparse, dim (span(
d) identifies latents in each of the blocks B¹ n BB B1 8 B² B² up to invertible affine transforms.
For the proof of the above theorem, refer to Section A. in the Appendix. From the above theorem, it follows that blocks overlap at one latent only, then all such latents are identified up to permutation and scaling. We now construct an example to show the identification of all the latents under overlapping perturbations. Suppose we have 4 dimensional latent. The set of all contiguous blocks of length 2 is given as follows ([1,2), (2,3), {3, 4}, 1)). Different 2 2-sparse perturbations impact
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these blocks. Observe that every component between 1 to 4 gets to be the first element of block exactly once and the last element of the block exactly once. As result, each latent gets to be the only element at the intersection of two blocks. We apply Theorem 3 to this case and get that all the latents are identified up to permutation and scaling. We generalize this example below.
Assumption 8. BI is a set of all the contiguous blocks of length p, where p < d. The perturbations in each block span a dimensional space. Further, also assume that d mod 0.
In the above assumption, we construct contiguous blocks of length p. The construction ensures that each index in (1, d) forms the first element of exactly one block and last element of exactly one block. In Theorem 1 (Locatello et al., 2019) and in Theorem 5 (Lachapelle assumption is made that requires exactly one latent is at the intersection of multiple blocks. In the next theorem, we show that under the above assumption, we achieve identification up to permutation and scaling.
al., 2022) a similar
Theorem 4. Suppose Assumptions 1, 3, 6 and 8 hold, then the encoder that solves the identity in =d) d) identifies true latents up to permutations and scaling, L.e. 2 = ITA2 + G, where II E Rddd matrix and A E Rddd is a diagonal matrix.
equation (3) (where A is p-sparse, dim ssaan (span(>')) =
For the proof of the above theorem, refer to Section A. in the Appendix. The total number of perturbations required in the above theorem is p x d. If we plug p = 1, we recover Theorem 1 as a special case. The above result highlights that if the block lengths are larger. then we need scale the number of perturbations accordingly by the same factor to achieve identification up to permutation and scaling. We assumed a special class of perturbations operating on contiguous blocks. In general, the total number of distinct blocks can be up to (g). Suppose s distinct random blocks of length p are selected for perturbations. As s grows, we reach a point where each latent component is the intersection of two blocks from different sets of blockwise non-overlapping perturbations. At that point, we identify all latents up to permutation and scaling.
Extensions In the discussion so far, we made some assumptions for ease of exposition. In the appendix, we describe how to relax them. In the DGP in Assumption 1, the perturbations used are deterministic. In Section A.2.3, we extend the DGP in Assumption 1 to incorporate stochastic perturbations. Specifically, instead of f 2 + 5 we consider DGP where ă + + 8 + n, where is the noise vector added to the perturbation 5. We show that the key results presented in the paper extend provided the noise vector n follow the same sparsity pattern as We also present experiments for the same model in Section A.3. In the DGP in Assumption 1, the perturbations used are independent of the value of z. Instead of +- z + we consider a DGP given as f- z + m(z). where m(-) is general non-linear perturbation map. In Section A.2.4, we show that the key results presented in the paper extend to this setting with non-linear perturbation mechanisms
Experiments
Data generation processes We conducted two sets of experiments low-dimensional synthetic and image-based inputs - that follow the DGP in equation (24). In the low-dimensional synthetic experiments we experimented with two choices for Pz a) uniform distribution with independent latents, b) normal distribution with latents that are blockwise independent (with block length d/2). We used invertible multi-layer perceptron (MLP) (with hidden layers) from Zimmermann et al. (2021) for g. We evaluated for latent dimensions d E {6. 10, 20}. The training and test data size was 10000 and 5000 respectively. For the image- masee-based experiments we used Py (Shinners, 2011)'s rendering engine for and generated 64 x 64 pixel images that look like those shown in Figure 1. The coordinates of each ball, 2,, were drawn independently from a uniform distribution, 2; (0.1,, 1,.01,..9, 0.9). We varied the number of balls from 2 (d 4) to 4 (d = 8). For these experiments, there was no fixed-size training set; instead the images are generated online and we trained to convergence. Because these problems are high dimensional, we only sampled a single perturbation for each image.
Loss function, architecture, evaluation metrics In all the experiments we optimized equation (4) with square error loss. The encoder ʃ was an MLP with two hidden layers of size 100 for the low-dimensional synthetic experiments and a ResNet-18 (He et al., 2015) for the image-based experiments. Further training details such as the optimizers used, hyperparameters etc. are in the
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Table 1:
Comparing MCC and BMCC for non-overlapping perturbations.
The number of perturbations applied for each example is given in parenthesis
d
Pz
MCC
MCC
BMCC
BMCC
C-wise (d)
C-wise (1)
B-wise (d)
B-wise (1)
6
Normal
0.99 + 0.00
0.99 + 0.00
0.99 + 0.00
0.99 + 0.01
10
Normal
0.99 + 0.00
0.99 = 0.01
0.99 = 0.00
0.91 = 0.02
20
Normal
0.99 + 0.00
0.88 + 0.03
0.99 + 0.00
0.90 + 0.01
Uniform
0.99 + 0.00
0.99 + 0.00
0.99 + 0.00
0.96 0.04
10
Uniform
0.99 + 0.00
0.99 0.01
0.99 = 0.00
0.81 = 0.05
20
Uniform
0.99 + 0.00
0.82 + 0.02
0.85 + 0.08
0.51 + 0.04
Predicted Z1
Prodicted 22
10
Producted Z3
Table 2: MCC for B-wise (overlap).
d
Distribution
MCC
Troe 21
Trae
True
6
Normal
0.95 = 0.01
Producted 211
Produced 22
10
10
Normal
0.96 = 0.01
Prodictal
20
Normal
0.99 = 0.01
6
Uniform
0.86 = 0.03
10
Uniform
0.88 + 0.03
Trae Z2
True 22
True 22
20
Uniform
0.81 + 0.03
Figure 2: Illustrating blockwise dependence (d = 10).
Appendix (Section A.3). We used the mean correlation coefficient (MCC) (Hyvarinen and Morioka, 2016) to verify the claims in Theorems I and 4. If MCC equals one, then the estimated latents identify true latents up to permutation and scaling. We extend MCC to blockwise MCC (BMCC) to verify the claims in Theorem 2. If BMCC equals one, then the estimated latents identify true latents up to permutation and block-diagonal transforms. Further details are in the Appendix (Section A.3).
Non-overlapping perturbations We first conducted experiments with one-sparse perturbations, the set A consists of m d one-sparse perturbations that span d dimensional space. In the context of the image experiments, these perturbations correspond to moving each ball individually along single axis. The learner solves the identity in equation (3) using set of random one-sparse perturbations A that span d dimensional space. In Table 1, we used the -ow-dimensional synthetic data generating process to compare the effect of (i) applying all m = d perturbations to each instance z (following the DGP in (24)), against more practical setting (ii) where perturbation is selected uniform at random from A and applied to each instance z. The results for (i) are shown in black and the results for (ii) are shown in gray font in the C-wise (componentwise) column in Table 1. We observed high MCCs in both settings. The results were similar in the more challenging image-based experiments (see Table 3, C-wise column) with MCC scores 0.97 for all the settings that we tested, as expected given the results presented in Theorem 1.
In our next experiments, the set of perturbations A comprised of d 2-sparse non-overlapping pertur- bations that span d dimensional space. We repeated the same synthetic experiments as above with one and d perturbations per instance. Under these assumptions we should expect to see that pairs of latents are separated blockwise but linearly entangled within the blocks (c.f. Theorem 2). We found this to be the case. The high BMCC numbers in Table I displayed under B-wise (blockwise) column (except for d = 20 and one perturbation per sample) show disentanglement between the blocks of latents. In Figure 2, the first two rows and columns show how the predicted latents corresponding to a block are correlated with their true counterpart (see Predicted Z; vs True Z,) and the other latent in the block (Predicted 7, vs True Z₂ and vice versa). The plots the last column show that the predicted latents did not bear correlation with randomly selected latent from outside the block.

---

Table 3: Image experiments
1]
MCC C-wise
MCC
MCC
2(n
B-wise
B-wise
d
())
2tm 13
4
0.994
0.710
0.864
f(a)
((a)
6
0.981
0.817
0.912
0.975
0.866
0.934
-1.
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Figure 3: (Left) Results for the image-base experiments. (Centre) Example images in which the bottom left ball is shifted to the right. (Right) A trained encoder's predictions for the two images shown in (centre). The green ball prediction shifts right by and the other balls left by s 2 1j For further illustrations, refer to the animations in the supplement.
Overlapping perturbations In this section, we experimented with blocks of size two that overlap in order to conform with the setting described in Theorem 4. We used the same distributions as before and only changed the type of perturbations. The low-dimensional synthetic results are summarized in Table 2. The results were largely as expected, with a strong correspondence between the predicted and true latents reflected by high MCC values.
On the image datasets (see Table 3), we found that the MCC scores depended on both the number of balls and how the blocks were selected. We compared two strategies for selecting blocks of latents to perturb: either select uniformly from all adjacent pairs I = {(i mod d, i + mod d)) (d blocks), or uniformly from all combinations of latent indices, I = {ii,j) : i {1, f1.....dd d}. > i} ((5) blocks). The latter lead to higher MCC scores (ranging from 0.86 to 0.93) as it placed more constraints on the solution space. The dependence on the number of balls is more surprising. To investigate the implied entanglement from the lower MCC scores, we evaluated trained encoders on images where we kept Aballs 1 balls in fixed location and moved one of the balls (see Section A.3 in the Appendix for example images). If the coordinates were perfectly disentangled, the encoder should predict no movement for static balls. When the moving ball shifted by units, the predicted location of the static balls shifted by II and the moving ball shifted R units. We further verified this claim and ran blockwise experiments with rbllls = 10 balls (d = 20) and got MCC scores of 0.930 and 0.969 for d and (o) blocks respectively. In the Appendix (Section A.3), we show that this solution stationary point, and we achieve a perfect MCC of one when Ahalls xo. Finally, the code to reproduce the experiments presented above can be found at https: : //
thub. wto.bccoo///......ooo/mo...oo.o. com/ahuj ak/WSRL. WSRL.
5 Discussion and limitations
Our work presents the first systematic analysis of the role of sparsity in achieving latent identification under unknown arbitrary latent distributions. We assume that every sample (or at least every neighbor- hood of a sample) experiences the same set of perturbations. results to settings where this assumption may not hold. Data augmentation provides rich source of perturbations; our results cover translations, but extending them to other forms of augmentation an important future direction. We followed the literature made two assumptions - the map that mixes latents is injective, and ii) the dimension of the latent d is known. We believe future works should aim to relax these assumptions. In reinforcement learning (RL) environments, the effects of actions can often be sparse. Therefore, we believe illustrating the efficacy of the proposed approach in RL environments (Ahmed al., 2020) is an important direction to further the case of the proposed theory and methods in real-world applications.
natural question how to extend our
non-linear ICA (Hyvarinen et al., 2019) and
6 Acknowledgements
We thank Sébastien Lachapelle and Anirudh Goyal for insightful discussions. Kartik Ahuja acknowl- edges the support from the IVADO postdoctoral fellowship. Jason Hartford acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Recursion Pharmaceuticals. Yoshua Bengio acknowledges the support from CIFAR, Samsung and IBM.
10

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Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] Each of the main contributions listed in the introduction correspond a theorem in Section and the final claim corresponds the experiments described in Section 4.
(b) Did you describe the limitations of your work? [Yes] See Section 5.
(c) Did you discuss any potential negative societal impacts of your work? [No] We do not foresee any potential negative societal impacts specific to this work.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3.
(b) Did you include complete proofs of all theoretical results? [Yes] See the Appendix (Section A.1) in the Supplementary Material.
3. If you ran experiments...
(a) Did you include the code. data, and instructions needed reproduce the main perimental results (either in the supplemental material or as a URL)? [Yes] We have provided the codes in the Supplementary Material.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See the Appendix (Section A.3).
(c) Did you report error bars (c.g., with respect the random seed after rumning experi- ments multiple times)? [Yes] For the low-dimensional synthetic experiments, which were computationally less demanding we ran five seeds. For image-based experiments, the run time for each case was on average 12 hours. Since we had several such cases, running several seeds not feasible given the compute
had at our disposal.
(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)? [Yes] See the Appendix (Section A.3).
4. If you are using existing assets (c.g., code, data, models) or curating/relcusing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] See the Appendix (Section A.3).
(b) Did you mention the license of the assets? [Yes] See the Appendix (Section A.3).
(c) Did you include any new assets either in the supplemental material
URL? [Yes] The codes can be found at https: //gi thub. com/ jujjaaujjaaa a /WSL..
(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 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, applicable? [N/A]
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13


Reviews:
[{"review_id": "GWk7zogR2T8", "reviewer": "Reviewer_EVcJ", "summary": "The authors study how well they can recover causal structure in inferred latent variables given sparsity in perturbations.\nRoughly speaking, assuming a generative process x = g(z), the authors study the effects of applying perturbations z'=z + delta leading to x'=g(z') and whether the delta can be recovered from those under what kind of conditions/assumptions.\n\nIn this work, the authors propose a hierarchy of more and more specific assumptions leading them to formalize how more specific structured can be inferred with different types of sparsity, i.e. identification up to affine transformations, blockwise identification, etc.\n\nThe paper then has 2 simple simulation examples in which they demonstrate that the theoretical properties hold.\n\nEdit:\nIncreased score post rebuttal to 6.", "questions": "-Can the authors make statements about robustness to uncertainty/noise even in their simplified setting with exactly known number of latents? Even if the theory can't keep up now, I would be curious to see how this shakes out empirically.\n- Are there any more 'appealing' experiments the authors could share so we see a less artificial example where this may work?", "limitations": "no concerns", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n-I really enjoyed the hierarchy of assumptions and how these permit making stronger statements quite precisely about recovery of latents under unknown arbitrary latent structure. This core part of the paper seems lucid, well explained, and interesting.\nIt is also surprisingly simple due to the simplified setup here.\n-Another strength of the paper that stands out is that it is well positioned among the current literature, with precise statements about links to contemporary papers and the detailed conditions and assumptions they share or how they differ. \n-Overall the paper is also easy to read and intuitive, which is a positive result.\n\nWeaknesses:\n-There is some emphasis on literature from recent years where weakly supervised learning and disentanglement have been around a few more years in the ML community, here there is an implicit belief that after ICA the main new result is Locatello et al. What about older/other work on disentangled representations in both VAEs and RL? Is it not related just because the term causality is used less?\n- The empirical setup is almost artificially toy. I have little doubt that the techniques described in this work would be applicable in various domains fitting the assumptions, but find that the experiments chosen are almost insultingly toy and hard to compare with other work.\nWe have a nice RL simulator from Stefan Bauer's recent work, that might be a useful testbed here in somewhat more ambitious environments. My key issue here is that the experiments are useful, but not really on the level of utility the cited work typically tackles.\n- Given the simplicity of the experiments we also see that the encoder structure is simple and suffices here. I would be curious how work like this behaves under noise and uncertainty, the idealized scenario here is noiseless and assumes exact numbers of known latents. \nIt would be necessary to start to go beyond that very narrow scenario to study such mechanisms for real world scenarios.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "F6__lzw91AF", "reviewer": "Reviewer_8xUA", "summary": "This paper extends the literature on using sparse latent perturbations for identifiable representation learning. It assumes that there is a fixed set of perturbations, each of which only affects a single latent or a block of latents, in which case each block or the smallest overlap between two blocks can be identified up to affine transformations. In contrast to previous work, no independence assumptions on the latent distributions are necessary and the assumptions are a bit closer to reinforcement learning settings.", "questions": "* In one experiment you are selecting the perturbations randomly. What happens if you also perturb the scale of each action, so if each delta_k is scaled with a gamma that is drawn from a narrow Gaussian distribution around 1?\n* At least intuitively the assumptions seem quite strong. Do you have an intuition for what assumptions could potentially be relaxed in future work?\n* In the image-experiment, why do the latent factors change for all objects upon translation of one? This sounds counter-intuitive and not really in line with the training objective. Also, if this effect is robustly present, it seems to contradict the theory.", "limitations": "I don't see negative societal implications.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper adds an interesting angle that is amenable to situations in which an agent controls its environment through discrete and finite actions. The manuscript is very well written, the definitions and proofs seem solid and the experiments are mostly sufficient to verify the claims (with one exception, see below). At the same time, some of the assumptions seem quite unrealistic, in particular that all perturbations need to be present for each sample, and that the set of perturbations are completely fixed. Nonetheless, the direction is interesting, in particular because it relaxes the strong assumptions on the marginal that so far had to be in place.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "457u4v0LyCB", "reviewer": "Reviewer_zmwx", "summary": "This paper shows that if given data generated with block-wise (i.e. sparse) perturbations on the latent, then the latents are identifiable up to affine transformation on the blocks.\n- In particular, in the special case of block size = 1, i.e. the perturbations change 1 coordinate at a time, then the latent is identifiable up to coordinate-wise permutation and scaling.\n- For general blocks, the latent is identifiable up to affine transformations on the set of intersections and differences of the blocks (for non-overlapping blocks, this set contains the blocks themselves).\n\nThe paper also presents empirical results on synthetic data to corroborate the theory.\n", "questions": "Numbers in Table 1-3 need more contexts: how to interpret an imperfect MCC value, e.g. 0.99 vs 0.95 vs 0.85?\n\nSide note: Perhaps considering having the same numbering for theorems in the main paper and the appendix for easier reference.\n\n=== Update === The authors have addressed my concerns in their response.", "limitations": "The paper discusses limitations and future directions.\n\nThere is no direct societal impact of the work.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- The paper discusses the related work well.\n- Assumption 6 (i.e. knowing the group id) is unrealistic but is relaxed in the appendix.\n- The paper is well structured and easy to follow.\n\nI have no major complaints about the paper, but I'm concerned about the novelty and significance and hope to hear the authors' comment on this.\n- Novelty: It's true that this perturbation setup is different from prior work which rely on auxiliary variables, mechanisms, or assumptions on the structure of the latent. However, I find the idea of coordinate/block-wise perturbation conceptually very similar to the idea of block-wise augmentation (i.e. only act on \"style\") in Von Kügelgen et. al. 21.\n- Significance / applicability of the results: The identifiability results requires the learner to know the type of perturbation, which is often unrealistic.\n- I find using the word \"sparsity\" misleading, since the identifiability results are more about being block-wise, regardless of the size of the block (i.e. sparsity). Perhaps rephrasing it could make the main idea clearer.\n\n=== Update === The authors have addressed my concerns in their response.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "3fV2eRkE57S", "reviewer": "Reviewer_ZbtV", "summary": "In this paper, the authors propose a theoretical framework on representation learning. Different from prior approaches, the authors show a different assumption and theoretically prove that sparse perturbation on latent variables is sufficient to learn the latents that follow any continuous distribution under weak supervision. The authors further discuss more general cases, and propose an estimation procedure based on the theory. Experimental results on two datasets are shown to support the claims in theory.", "questions": "Given the current assumption, how would your method be applied in real-world applications?\n\nFor the experiment of image-base experiments, could you consider other baseline representation learning methods as comparison?\n", "limitations": "The authors adequately addressed the limitations and potential negative social impact of their work. ", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "$\\textbf{originality}$ \n\nThis manuscript provides a novel theoretical framework for representation learning which is first used in this area to the best of my knowledge. The related works are adequately cited.\n\n$\\textbf{quality}$ \n\nThe authors seem to provide sound theoretical justifcation for the proposed framework. And shows experimental results to support the claim.  \n\n$\\textbf{clarity}$ \n\nThe paper is overall clear in presenting intuition and experimental results. The central thoery part is difficult to follow. The authors also adequately discuss the limitation of the proposed methods, and point out future direction for improvement.\n\n$\\textbf{significance}$\n\nLack of applications in real-world. The significance of the proposed method requires further illustration as it might be difficult to find a real-world application given the current assumptions. The applications meet the assumptions might be very specific which undermines the significance of this manuscript.\n\nOverall, I think the paper in the current form might be interesting to the NeurIPS community.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept