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Who Guards the Guardians? The Challenges of Evaluating Identifiability of Learned Representations

Shruti Joshi, Théo Saulus, Wieland Brendel, Philippe Brouillard, Dhanya Sridhar, Patrik Reizinger

TL;DR

This work introduces a taxonomy separating DGP assumptions from encoder geometry, uses it to characterise the validity domains of existing metrics, and releases an evaluation suite for reproducible stress testing and comparison.

Abstract

Identifiability in representation learning is commonly evaluated using standard metrics (e.g., MCC, DCI, R^2) on synthetic benchmarks with known ground-truth factors. These metrics are assumed to reflect recovery up to the equivalence class guaranteed by identifiability theory. We show that this assumption holds only under specific structural conditions: each metric implicitly encodes assumptions about both the data-generating process (DGP) and the encoder. When these assumptions are violated, metrics become misspecified and can produce systematic false positives and false negatives. Such failures occur both within classical identifiability regimes and in post-hoc settings where identifiability is most needed. We introduce a taxonomy separating DGP assumptions from encoder geometry, use it to characterise the validity domains of existing metrics, and release an evaluation suite for reproducible stress testing and comparison.

Who Guards the Guardians? The Challenges of Evaluating Identifiability of Learned Representations

TL;DR

This work introduces a taxonomy separating DGP assumptions from encoder geometry, uses it to characterise the validity domains of existing metrics, and releases an evaluation suite for reproducible stress testing and comparison.

Abstract

Identifiability in representation learning is commonly evaluated using standard metrics (e.g., MCC, DCI, R^2) on synthetic benchmarks with known ground-truth factors. These metrics are assumed to reflect recovery up to the equivalence class guaranteed by identifiability theory. We show that this assumption holds only under specific structural conditions: each metric implicitly encodes assumptions about both the data-generating process (DGP) and the encoder. When these assumptions are violated, metrics become misspecified and can produce systematic false positives and false negatives. Such failures occur both within classical identifiability regimes and in post-hoc settings where identifiability is most needed. We introduce a taxonomy separating DGP assumptions from encoder geometry, use it to characterise the validity domains of existing metrics, and release an evaluation suite for reproducible stress testing and comparison.
Paper Structure (54 sections, 3 theorems, 62 equations, 37 figures, 3 tables)

This paper contains 54 sections, 3 theorems, 62 equations, 37 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A taxonomy for metric (mis)specification
  3. Factor dependencies reduce effective dimensionality
  4. Encoder Structure and Dimension Mismatch
  5. Metrics as Measurement Instruments
  6. Correlated and entangled latent factors lead to both false positives and false negatives
  7. Metrics cannot detect multi-factor redundancy
  8. Metrics cannot compare overparametrised encoders
  9. High representation-to-sample ratio increases risk of false positives
  10. Conclusion
  11. Practitioner Checklist
  12. Related Work
  13. Metric Usage Review
  14. Taxonomy
  15. Data Classes
  16. ...and 39 more sections

Key Result

Proposition 1

Under $\mathbf{D}_{\rho}$ + E3, $\mathrm{MCC}$-P depends explicitly on $\rho$, violating prop:rho-invariance. Moreover, at both extremes $\rho \to +1$ and $\rho \to -1$, $\mathrm{MCC}{}(\rho) \to 1$, despite an entangled encoder.

Figures (37)

  • Figure 1: Every identifiability metric fails under at least one common evaluation setting. We test four desiderata (\ref{['prop:rho-invariance', 'prop:deff-sensitivity', 'prop:oc-invariance', 'prop:false-positive']}) using controlled synthetic encoders that isolate metric behaviour from optimisation artefacts. (P1) Latent correlation: $\mathrm{MCC}$ conflates correlation with identifiability (FP $\uparrow$ to $0.97$); $\mathrm{DCI}$-D penalises it (FN $\downarrow$). (P2) Factor dropping: $\mathrm{DCI}$-D reports perfect disentanglement even when $9$ of $10$ factors are lost. (P3) Overcompleteness: $\mathrm{MCC}$ inflates for entangled encoders; $\mathrm{DCI}$-D deflates for disentangled ones. (P4) Null encoder: all metrics inflate as $m/n$ grows, with $\mathrm{MCC}$ scaling in the order of $\sqrt{2\log m / n}$. Only $R^2$ is robust across (P1), (P3), and (P4), but shares the (P2) limitation. No single metric is trustworthy across all settings.
  • Figure 2: $\mathrm{MCC}$ conflates correlation with identifiability. Under E3, $\mathrm{MCC}$ increases with $\rho$ and approaches the score of the perfectly disentangled encoder E1 at high correlation, despite the encoder remaining entangled. $\mathrm{DCI}$ better separates E1 from E3 but collapses to near-zero scores making it hard to distinguish from a non-identifiable encoder. The bias sharpens with increasing $d$. See \ref{['fig:apx-exp03-apx']} for all metrics.
  • Figure 3: Regression-based metrics detect single-factor redundancy.Left ($\mathbf{D}_{\perp}$): every dropped factor is informative; $R^2$ follows $m/d$ and $\mathrm{DCI}$-D declines steadily. $\mathrm{MCC}$-P/S report $1.0$ even at $m{=}1$ (false positive). Right ($\mathbf{D}_{ f}$): at $m{=}9 d_{\mathrm{eff}}$, the dropped factor is redundant ($z_2 = z_1^3$); $R^2$ and $\mathrm{DCI}$-D plateau near $1.0$, correctly recognising ossless compression, then decline as informative factors are removed. $\mathrm{MCC}$-P/S remain at $1.0$ throughout both panels and cannot distinguish the two.
  • Figure 4: Sweeping $m/d$ reveals encoder-specific violations of \ref{['prop:oc-invariance']}.E5 (elementwise linear duplication) is the only encoder for which all metrics remain stable. $\mathrm{DCI}$-D increases for entangled E7 as $m/d$ grows. $R^2$ collapses for nonlinear E6. $\mathrm{MCC}$-P decreases for disjoint E8. $d{=}5$, $n{=}1000$.
  • Figure 5: Null-encoder scores reveal that $m/n$ (columns), not $m/d$ (rows), governs false-positive inflation. Each cell shows the metric score of a random encoder E9 that carries no information about ${\mathbf{z}}$; any score above $0$ is a false positive. See \ref{['apx:mcc-false-positive']} for the theoretical analysis and \ref{['fig:apx-exp15-gauss']} for the Gaussian null (nearly identical).
  • ...and 32 more figures

Theorems & Definitions (8)

  • Definition 1: Identifiability up to $\mathcal{G}$
  • Definition 2: Effective dimensionality
  • Definition 3: Identifiability under dimension mismatch
  • Proposition 1: $\mathrm{MCC}$ produces false positives under correlation
  • Proposition 2: Dropped factors are invisible to $\mathrm{DCI}$
  • proof
  • Proposition 3: Functional dependence decreases $D$ under a perfect encoder
  • proof