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Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)

Peter Sorrenson, Carsten Rother, Ullrich Köthe

arXiv:2001.04872v1cs.LGstat.ML

TL;DR

The paper addresses the known-dimension assumption in nonlinear ICA, which the authors describe as unrealistic beyond artificial datasets. It extends the theory to unknown intrinsic dimension and introduces GIN, reporting dimension discovery and 22 meaningful variables from EMNIST; scalability and extensions to less-supervised settings remain open.

  • Problem

    Existing nonlinear ICA theory assumes the intrinsic problem dimension is known, which the paper describes as unrealistic for datasets beyond artificially generated ones.

    Source: §1 · p. 1

  • Method

    The authors extend nonlinear ICA theory to unknown intrinsic dimension and propose GIN, an invertible neural network modified from RealNVP, as an implementation.

    Source: §1 · pp. 1–2

  • Results

    In the Gaussian latent-space case, intrinsic dimension can be discovered and informative variables separated from noise; the EMNIST experiments extract 22 meaningful variables.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    The EMNIST experiments demonstrate the viability of theory-inspired nonlinear ICA on real-world data, even when not all theoretical conditions are met.

    Source: §5 · p. 8

    Whether the method scales to larger problems or extends to semisupervised or unsupervised learning remains unclear.

    Source: §5 · p. 8

Abstract

from arXiv · show

A central question of representation learning asks under which conditions it is possible to reconstruct the true latent variables of an arbitrarily complex generative process. Recent breakthrough work by Khemakhem et al. (2019) on nonlinear ICA has answered this question for a broad class of conditional generative processes. We extend this important result in a direction relevant for application to real-world data. First, we generalize the theory to the case of unknown intrinsic problem dimension and prove that in some special (but not very restrictive) cases, informative latent variables will be automatically separated from noise by an estimating model. Furthermore, the recovered informative latent variables will be in one-to-one correspondence with the true latent variables of the generating process, up to a trivial component-wise transformation. Second, we introduce a modification of the RealNVP invertible neural network architecture (Dinh et al. (2016)) which is particularly suitable for this type of problem: the General Incompressible-flow Network (GIN). Experiments on artificial data and EMNIST demonstrate that theoretical predictions are indeed verified in practice. In particular, we provide a detailed set of exactly 22 informative latent variables extracted from EMNIST.

1 INTRODUCTION

The paper extends conditional nonlinear ICA to unknown intrinsic dimension, proposes GIN as an invertible implementation, and demonstrates the approach on artificial data and EMNIST, extracting 22 meaningful variables. Its theory identifies when informative variables can be separated from noise and related to the generating latents.

1 INTRODUCTION

  • Nonlinear ICA provides a framework for recovering latent structure when data are generated conditionally on an observed variable, such as a class label or time index.The cited prior theory permits recovery of the joint data and latent-space distribution up to a simple latent-space transformation.Source: §1 · p. 1
  • For Gaussian latent distributions, the intrinsic dimension can be discovered and informative variables separated from noise, with recovered variables corresponding to the true latents up to component-wise translation and scaling.Similar results are noted for two-parameter exponential-family distributions such as beta and gamma.Source: §1 · pp. 1–2
  • The authors introduce GIN, a RealNVP modification, as an invertible neural-network implementation of their nonlinear ICA approach.GIN preserves volume and is described as more expressive than earlier volume-preserving flows such as NICE.Source: §1 · pp. 1–2
  • Invertible architectures allow direct likelihood optimization and do not require separately specifying latent-space dimension; their forward and backward models share parameters.An invertible neural network has a latent space with the same dimension as the data.Source: §1 · p. 2
  • On EMNIST, the model extracts 22 meaningful variables encoding both global and local features, alongside demonstrations on artificial data.Source: §1 · p. 2

2 RELATED WORK

Related work traces nonlinear ICA from linear identifiability results to conditional nonlinear methods, while contrasting supervised, volume-preserving approaches with unsupervised and alternative flow designs.

2 RELATED WORK

  • Linear ICA identifiability results specify conditions for discovering generating latents under linear mixing, but this approach does not extend to general nonlinear functions.Source: §2 · p. 2
  • Nonlinear ICA results progressed from time-series methods to broader formulations and generative-model theory, including Khemakhem et al. (2019).Source: §2 · p. 2
  • Unsupervised disentanglement approaches include β-VAE and InfoGAN, but unsupervised models without latent-space conditioning are generally unidentifiable according to Locatello et al. (2018).Source: §2 · p. 2
  • Compared with Gaussian-mixture VAE approaches, this work restricts itself to supervised tasks and enforces volume preservation, which the authors say VAEs cannot do.Source: §2 · p. 2
  • Unlike Ardizzone et al. (2019), which applies conditioning as a network parameter and reports rotated latent variables, this work conditions latent distributions and reports non-rotated meaningful variables.Source: §2 · p. 2

3 THEORY

The section extends nonlinear ICA theory to settings where the intrinsic dimension is unknown, then shows how a Gaussian latent space enables dimension discovery and variable-wise recovery. It also motivates volume preservation as a way to retain a link between latent-variable standard deviation and importance.

3.1 EXISTING THEORY OF NONLINEAR ICA

  • Matching conditional likelihoods in the infinite-data, perfect-convergence limit implies an affine relation between generating and estimated sufficient statistics.This generalizes the earlier same-dimension setting, where the relation between latent spaces is square and invertible.Source: §3.1 · pp. 3–4

3.2 NONLINEAR ICA WITH A GAUSSIAN LATENT SPACE

  • With Gaussian generating and estimated latent spaces, each generating variable corresponds to one estimated variable up to scaling and translation, while unmatched estimated variables encode noise.The informative estimated variables therefore reveal the unknown generating dimension; similar results are stated for common continuous two-parameter exponential-family distributions, including gamma and beta.Source: §3.2 · p. 4

3.3 VOLUME PRESERVATION VIA INCOMPRESSIBLE FLOW

  • Volume preservation is motivated by retaining the correspondence between a latent variable’s standard deviation and its importance in explaining the data.The section connects this rationale to invertible networks whose tractable Jacobian determinant can be constrained to unity.Source: §3.3 · p. 4

4 EXPERIMENTS

The experiments test GIN on artificial data and EMNIST, assessing whether it identifies informative latent dimensions and yields interpretable variables. Results include conditions affecting recovery and limits on how closely the EMNIST setup meets the theory.

4.1 MODEL DESCRIPTION

  • GIN is a RealNVP-like invertible network that preserves volume by enforcing a unit Jacobian determinant.The method’s coupling-layer scaling contributions sum to zero, while allowing some output dimensions to grow as others shrink proportionally.Source: §4.1 · pp. 4–5

4.2 OPTIMIZATION

  • The optimization fits a conditional Gaussian mixture with diagonal class-conditional variances by minimizing latent-space negative log-likelihood.Volume preservation removes the Jacobian term from the likelihood calculation.Source: §4.2 · p. 5

4.3.1 EXPERIMENT 1: CONDITIONS OF THEORY FULFILLED

  • 2 estimated dimensions receive importance when mixture components overlap sufficiently; insufficient overlap can split a true dimension, and a too-low initial learning rate can also split variables.The authors attribute overlap-related failures to finite data and report that more samples can help when component gaps are not too large.Source: §4.3.1 · p. 6

4.3.2 EXPERIMENT 2: CONDITIONS OF THEORY NOT FULFILLED

  • Three mixture components still yield essentially the same results, suggesting the theory’s identifiability conditions are sufficient but not necessary.The experiment uses fewer than the five conditions predicted by the theory for two dimensions with two sufficient statistics per dimension.Source: §4.3.2 · p. 6

4.4.1 EXPERIMENT

  • EMNIST Digits provides 240,000 labeled training images, but its ten label conditions fall outside theoretical guarantees if there are more than four informative factors.The authors also note that handwriting may not meet the method’s assumptions and that rare or subtle variations may be underrepresented.Source: §4.4.1 · p. 7

4.4.2 RESULTS

  • 22 broadly interpretable EMNIST variables encode digit shape; the remaining variables are treated as noise.The first eight encode global information and the following 14 more local information; some global variables are not fully disentangled.Source: Figure 3 · p. 6, §4.4.2 · p. 7
  • GIN’s global variables include separate upper- and lower-half width factors, as well as digit height and bending.The authors note that the upper and lower halves can vary independently; width is somewhat entangled with slant.Source: Figure 4 · p. 7, §4.4.2 · p. 8
  • Local variables can alter specific strokes or regions while leaving digits without those features unaffected.Examples include changing the top-left stroke of 2, 3, and 7, or the lower-right stroke of a 2.Source: Figure 5 · p. 7, §4.4.2 · p. 8

5 CONCLUSION AND OUTLOOK

The paper extends nonlinear ICA theory to unknown intrinsic dimension and demonstrates GIN on EMNIST, where the discovered variables are interpretable. The authors note that scaling to larger problems and extension to semisupervised or unsupervised learning remain unresolved.

5 CONCLUSION AND OUTLOOK

  • EMNIST experiments demonstrate the viability of applying models inspired by the theory to real-world data, even when not all theoretical conditions are met.The variables discovered by GIN are described as interpretable and detailed enough to suggest that the assumptions about the generating process are approximately true.Source: §5 · p. 8
  • It remains unclear whether the methods will scale to larger problems or extend to semisupervised or unsupervised learning.The authors describe scaling as a plausible prospect given advances in related flow-based density-estimation models, but do not establish it here.Source: §5 · p. 8

A PROOF OF IDENTIFIABILITY

Under the stated regularity and condition-variation assumptions, the proof relates generating and estimated sufficient statistics through a constant full-rank matrix. It also explains when estimated latent variables encode noise and when condition (ii) holds almost surely.

A PROOF OF IDENTIFIABILITY

  • Theorem 1 relates generating and estimated sufficient statistics through a constant, full-rank matrix, assuming differentiable sufficient statistics with nonzero derivatives and an invertible condition matrix.The theorem assumes nk + 1 distinct conditions; its sufficient-statistic relationship uses a constant full-rank nk × dk matrix and a constant vector.Source: A PROOF OF IDENTIFIABILITY · p. 11
  • The proof equates conditional densities, subtracts the expression for a baseline condition to cancel condition-independent terms, and uses the resulting matrix equations to establish full rank.The proof is adapted from Khemakhem et al. (2019) for invertible neural networks (normalizing flows).Source: A PROOF OF IDENTIFIABILITY · pp. 11–12

A.1 REMARK REGARDING CONDITION (II)

  • When coefficient vectors are generated randomly and independently, condition (ii) holds almost surely because the sampled vectors are in general position.Source: §A.1 · p. 12

A.2 REMARK REGARDING NOISE VARIABLES IN THE ESTIMATED LATENT SPACE

  • An estimated latent variable whose distribution does not depend on the conditioning variable encodes only noise and has no effect on the generating latent space; the converse also holds.Source: §A.2 · p. 12

B SPARSITY IN UNMIXING MATRIX: GAUSSIAN DISTRIBUTION

For conditional Gaussian latent spaces, the proof constrains the unmixing matrix so each generating variable is linked to one estimated variable. Unlinked estimated variables encode only noise.

B SPARSITY IN UNMIXING MATRIX: GAUSSIAN DISTRIBUTION

  • The argument assumes both generating and estimated latent samples follow conditional Gaussian distributions and that the estimating model reproduces the observed conditional density.Under these conditions, the latent spaces can be related using equation (4).Source: B SPARSITY IN UNMIXING MATRIX:… · p. 12
  • Cross terms in the squared expression must vanish, forcing each row of the first block of the unmixing matrix to have exactly one nonzero entry in a distinct column.Source: B SPARSITY IN UNMIXING MATRIX:… · p. 13
  • Each generating Gaussian latent variable is related to exactly one estimated latent variable by an affine transformation, while unassociated estimated variables encode only noise.The proof also establishes that the associated estimated variables are distinct, so the generating variables are recovered up to affine transformation and permutation as a subset.Source: B SPARSITY IN UNMIXING MATRIX:… · p. 13

C SPARSITY IN UNMIXING MATRIX: TWO-PARAMETER EXPONENTIAL

For two-parameter exponential-family members, the proof uses higher-order terms to constrain the relation between generating and estimated latent variables. It concludes that each generating variable is linked to exactly one estimated variable.

C SPARSITY IN UNMIXING MATRIX: TWO-PARAMETER EXPONENTIAL

  • The appendix extends the Gaussian-case results to other two-parameter exponential-family members and examines their specific transformation forms for further constraints.Source: C SPARSITY IN UNMIXING MATRIX:… · pp. 13–14
  • When the transformation has a convergent Taylor expansion, its higher-order terms create cross terms that cannot be reconciled with the other side of the equation, constraining coefficients to functions of a single estimated variable.Source: C SPARSITY IN UNMIXING MATRIX:… · p. 14
  • Each generating latent variable is related to exactly one estimated latent variable, and each estimated variable is associated with at most one generating variable.Estimated variables with no generating-variable association encode only noise; Table 1 gives results for the two-parameter family members considered.Source: C SPARSITY IN UNMIXING MATRIX:… · p. 14, Table 1 · p. 14

D NETWORK ARCHITECTURE

The estimating model maps data to latent space through an inverse network built from affine coupling blocks. Its implementation uses MLP or convolutional coupling functions, with different network architectures for the artificial-data and EMNIST experiments.

D NETWORK ARCHITECTURE

  • The estimating model g is implemented in reverse as g−1, mapping data space to latent space.The network uses affine coupling blocks, with two affine coupling functions per block.Source: D NETWORK ARCHITECTURE · p. 15
  • Affine coupling functions concatenate scale and translation outputs, use a 2 tanh clamp on scale, and can be implemented with MLPs or convolutional networks.Convolutional splits are along channels; downsampling multiplies channels by four while halving image width and height.Source: D NETWORK ARCHITECTURE · p. 15
  • The artificial-data and EMNIST networks have 4,480 and 2,620,192 learnable parameters, respectively.Table 2 and Table 3 are identified as the architectures for the artificial-data and EMNIST experiments.Source: D NETWORK ARCHITECTURE · p. 15, Table 2 · p. 15, Table 3 · p. 15

D.1 NOTE ON OPTIMIZATION METHOD

In the experiments, mixture-component means and variances are updated from minibatch data transformed to latent space. These quantities depend on the model parameters through the inverse mapping rather than being directly learned parameters.

D.1 NOTE ON OPTIMIZATION METHOD

  • At each iteration, each mixture component’s mean and variance are updated from the latent-space transformations of its minibatch members.The update uses all data points belonging to that component in minibatch D.Source: §D.1 · p. 15
  • The component means and variances depend on g−1 and therefore change with model parameters θ, even though they are not directly parameterized by θ.Their values are functions of the transformed minibatch data.Source: §D.1 · p. 15
  • The authors expect that learnable component means and variances updated alongside the model weights would work equally well.Source: §D.1 · p. 15

E FIGURES FROM THE ARTIFICIAL DATA EXPERIMENTS

Artificial-data figures show that GIN recovers the two-dimensional latent structure with sufficient data, but not with limited data; it also succeeds in one case outside the theory’s sufficient conditions. EMNIST figures visualize temperature-reduced samples and the effects of the 22 most significant variables.

E FIGURES FROM THE ARTIFICIAL DATA EXPERIMENTS

  • With 100,000 data points and five mixture components, GIN reconstructs the generating latent space and assigns importance to only two of ten variables.Source: Figure 6 · p. 16
  • With 10,000 data points, GIN fails to recover the ground-truth latent variables: the second variable splits across two reconstructed variables and three variables appear important.The first variable is well approximated, but the importance spectrum no longer reflects the generating space’s two dimensions.Source: Figure 7 · p. 16
  • With three mixture components—insufficient for identifiability according to the theory—GIN nevertheless reconstructs the ground-truth latent variables.The figure suggests the theory’s conditions are sufficient but not necessary.Source: Figure 8 · p. 16

F EMNIST FIGURES

  • Temperature-reduced EMNIST samples use the 22 most significant variables, hold the others at their means, and preserve digit-shape variability while eliminating image noise.Each row shares a latent code while columns condition on different classes, yielding consistent digit style across rows.Source: Figure 9 · p. 17
  • Figures 10–15 vary one latent variable from −2 to +2 standard deviations across columns, hold the others at their means, and show affected image areas in rightmost heatmaps.Each row is conditioned on a different digit label; the heatmaps use absolute pixel differences between −1 and +1 standard deviations.Source: Figure 10 · p. 18, Figure 11 · p. 19, Figure 12 · p. 20
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