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Operationalizing Quantized Disentanglement

Vitoria Barin-Pacela, Kartik Ahuja, Simon Lacoste-Julien, Pascal Vincent

TL;DR

The paper tackles unsupervised disentanglement in the presence of nonlinear diffeomorphisms by leveraging quantized identifiability, which rests on axis-aligned independent discontinuities in the latent density. It introduces Cliff, a regularizer that enforces axis-aligned density cliffs through univariate and bivariate criteria based on kernel density estimates of standardized latent factors, plus a collapse-prevention term. Empirically, Cliff outperforms baselines like IOSS and HFS on synthetic data, a balls-rendered dataset, and Shapes3D, evidencing improved latent axis alignment and disentanglement scores under nonlinear distortions. The work broadens the practicality of unsupervised disentanglement by reducing global density assumptions and remains compatible with diverse model architectures.

Abstract

Recent theoretical work established the unsupervised identifiability of quantized factors under any diffeomorphism. The theory assumes that quantization thresholds correspond to axis-aligned discontinuities in the probability density of the latent factors. By constraining a learned map to have a density with axis-aligned discontinuities, we can recover the quantization of the factors. However, translating this high-level principle into an effective practical criterion remains challenging, especially under nonlinear maps. Here, we develop a criterion for unsupervised disentanglement by encouraging axis-aligned discontinuities. Discontinuities manifest as sharp changes in the estimated density of factors and form what we call cliffs. Following the definition of independent discontinuities from the theory, we encourage the location of the cliffs along a factor to be independent of the values of the other factors. We show that our method, Cliff, outperforms the baselines on all disentanglement benchmarks, demonstrating its effectiveness in unsupervised disentanglement.

Operationalizing Quantized Disentanglement

TL;DR

The paper tackles unsupervised disentanglement in the presence of nonlinear diffeomorphisms by leveraging quantized identifiability, which rests on axis-aligned independent discontinuities in the latent density. It introduces Cliff, a regularizer that enforces axis-aligned density cliffs through univariate and bivariate criteria based on kernel density estimates of standardized latent factors, plus a collapse-prevention term. Empirically, Cliff outperforms baselines like IOSS and HFS on synthetic data, a balls-rendered dataset, and Shapes3D, evidencing improved latent axis alignment and disentanglement scores under nonlinear distortions. The work broadens the practicality of unsupervised disentanglement by reducing global density assumptions and remains compatible with diverse model architectures.

Abstract

Recent theoretical work established the unsupervised identifiability of quantized factors under any diffeomorphism. The theory assumes that quantization thresholds correspond to axis-aligned discontinuities in the probability density of the latent factors. By constraining a learned map to have a density with axis-aligned discontinuities, we can recover the quantization of the factors. However, translating this high-level principle into an effective practical criterion remains challenging, especially under nonlinear maps. Here, we develop a criterion for unsupervised disentanglement by encouraging axis-aligned discontinuities. Discontinuities manifest as sharp changes in the estimated density of factors and form what we call cliffs. Following the definition of independent discontinuities from the theory, we encourage the location of the cliffs along a factor to be independent of the values of the other factors. We show that our method, Cliff, outperforms the baselines on all disentanglement benchmarks, demonstrating its effectiveness in unsupervised disentanglement.

Paper Structure

This paper contains 36 sections, 3 theorems, 25 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A brief overview of quantized identifiability
  3. Related work
  4. A new regularizer: Cliff Alignment (Cliff)
  5. Univariate criterion -- Encouraging cliffs in the marginals
  6. Bivariate criterion -- Encouraging independent cliffs
  7. Preventing the collapse to Diracs
  8. Total loss
  9. Computational complexity
  10. Experiments
  11. Synthetic data
  12. Balls dataset
  13. Disentanglement benchmarks
  14. Conclusion
  15. Summary of theorems from "On the Identifiability of Quantized Factors"
  16. ...and 21 more sections

Key Result

Theorem 1

barin-pacela24a Grid structure preservation and recovery theorem. Let $h: \mathcal{S}\subset \mathbb{R}^d \rightarrow \mathcal{S}' \subset \mathbb{R}^d$ be a diffeomorphism, where both $\mathcal{S}$ and $\mathcal{S}'$ are open connected subsets of $\mathbb{R}^d$. Suppose we have an axis-aligned grid If $s_{i}=-1$, then:

Figures (6)

  • Figure 1: a. Joint PDF $p(z_1,z_2)$ with a cliff at $z_1=0$, parallel (aligned) to the $z_2$ axis. b. Marginal PDF of $z_1$, $p(z_1)$, displayed on the left axis; magnitude of the gradient ${\partial p(z_1)}/{\partial z_1}$ on the right axis (on a different scale). The magnitude of the gradient is high at the cliff, which is a point of sharp change in the marginal density. c. Marginal PDF of $z_2$, $p(z_2)$, and its repective gradient magnitude; no cliffs observed along this axis.
  • Figure 2: a. Joint PDF with $p(z_1, z_2)$ with a cliff parallel to the diagonal, therefore not axis-aligned -- this is the same density as in Figure \ref{['fig:derivatives']}, but rotated by $45^{\circ}$. b. Marginal PDF of $z_1$, $p(z_1)$ and its respective gradient magnitude ${\partial p(z_1)}/{\partial z_1}$. c. Marginal PDF of $z_2$, $p(z_2)$, and its respective gradient magnitude. No cliffs observed in the marginals because this cliff is not axis-aligned. Note that while there are some bumps in the gradient magnitude, the scale of this magnitude is small, especially when compared to the cliff from Figure \ref{['fig:derivatives']}.
  • Figure 3: Synthetic data: True factors $z$ (a) are mapped through observed variables $x$ through a nonlinear mixing function. The nonlinearity is manifested through distortions. We learn a decoder $g$ that yields the reconstructed factor $z'=g(x)$. Our method, Cliff (c) matches the true factors (a) almost perfectly and obtains a much more straight and axis-aligned representation (MCC of 94.1 $\pm$ 0.9) than IOSS (d) (MCC of 91.6 $\pm$ 0.8) .
  • Figure 4: Figure from barin-pacela24a. Recovery of quantized factors. Left: The true (continuous) latent factors $Z_1$ and $Z_2$ are not independent, but their joint probability density $p_Z$ has independent discontinuities: sharp changes in the density that are aligned with the axes and form a grid. Middle: The factors get warped and entangled by the diffeomorphism $f$ into observations $X$, but the discontinuities in their density survive in the observed space. Right: We can learn a diffeomorphism $g$ that yields a density $p_{Z'}$ having axis-aligned discontinuities. This suffices to recover a grid whose cells match the initial grid's cells (up to possible permutation and axis reversal). Pink cell example: the points $Z'$ in cell $(3,2)$ originated from the points $Z$ in cell $(3,2)$. To construct these cells, the quantization of each continuous factor to an integer depends on thresholds based on the location of the discontinuities. The quantizations of $Z'_1$ and $Z'_2$ match precisely the quantizations of $Z_1$ and $Z_2$, up to possible permutation and axis reversal. This summarizes the identifiability of quantized factors under diffeomorphisms.
  • Figure 5: Loss landscape. Top row: Univariate criterion (encouraging cliffs in the marginals), minimized at $0^{\circ}$ or $90^{\circ}$. Bottom row: Bivariate criterion (encouraging independent cliffs): avoids $\theta_1 = \theta_2 = 0^{\circ}$ and $\theta_1 = \theta_2 = 90^{\circ}$; instead, the minimum is at $(0^{\circ}, 90^{\circ})$ and its multiples.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition A.1
  • Theorem 1
  • Corollary 1
  • Theorem 2