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

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.

• The paper develops a predictive taxonomy of structural misspecification, showing that metric validity depends on latent-factor structure, encoder geometry, dimensionality, and sample size.
• MCC can report 0.83 for pure noise when the representation-to-sample ratio reaches 0.5, while overcomplete codes cause MCC, R², and DCI-D to produce false negatives or false positives.
• The authors recommend null-encoder baselines, matched-dimension controls, multiple metrics, and careful checks of effective dimensionality and m/n before claiming identifiability.

Motivation and central claim

Empirical validation of identifiability guarantees in representation learning rests almost entirely on a small set of scalar metrics—MCC (in Pearson and Spearman variants), $R^2$, and DCI-D—computed on synthetic benchmarks with known ground-truth factors. The standard protocol treats a high score as evidence that an encoder has recovered the true generative factors up to a specified equivalence class. This paper argues that this inference is only coherent under structural conditions that are rarely checked: each metric implicitly encodes assumptions about the data-generating process (DGP), the encoder geometry, and the sample size relative to representation dimensionality. When these encoded assumptions mismatch the actual evaluation setting—a condition the authors call structural misspecification—metrics produce systematic false positives and false negatives that persist even as samples $n \to \infty$ and optimization is perfect. Crucially, these failures occur both inside classical identifiability regimes (e.g., correlated factors permitted by theorems) and in post-hoc settings such as mechanistic interpretability, where the metrics are used most but theorem assumptions hold least.

Unlike prior work that observed metric disagreements empirically (Sepliarskaia et al., 2019), [(Elhage et al., 2022)-adjacent surveys such as 2403.xxxx], this paper's contribution is predictive: it characterizes when and why each failure occurs, deriving closed-form results for the dominant failure modes.

A two-axis taxonomy of misspecification

The framework separates assumptions along two orthogonal axes.

DGP axis (latent factor structure). Four classes are distinguished: independent factors; correlated factors where each retains a unique degree of freedom ($d_{\mathrm{eff}} = d$); single-factor deterministic constraints (e.g., $z_2 = z_1^3$); and multi-factor constraints (e.g., $V = IR$, Ohm's law, illustrated via a running resistor-circuit example with effective dimensionality $d_{\mathrm{eff}} = 2$ out of four recorded quantities). The latter two classes reduce effective dimensionality below $d$ and arise naturally when the abstraction level of the factor set is unknown a priori—an extension beyond what standard disentanglement benchmarks assume.

Encoder axis. Encoders are classified by equivalence class (permutation/rescaling $G_{\mathrm{perm}}$, elementwise nonlinear $G_{\mathrm{nl}}$, affine $G_{\mathrm{aff}}$) and by dimension ratio $n \to \infty$0. The paper extends the definition of identifiability to dimension mismatch: an encoder identifies factors in $n \to \infty$1 up to $n \to \infty$2 if some coordinate projection of $n \to \infty$3 recovers them. Overcomplete geometries include elementwise duplication, nonlinear duplication, overcomplete linear entanglement, and distributed codes (each factor spread across $n \to \infty$4 disjoint nonlinear coordinates, e.g., $n \to \infty$5). A null encoder independent of the data serves as the control baseline.

Each metric targets one equivalence class: MCC-P/S target elementwise recovery, $n \to \infty$6 measures linear (affine-class) explained variance, and DCI-D probes importance concentration without requiring one-to-one alignment. Using a metric outside its target class yields systematically wrong scores.

Failure mode I: correlation conflated with identifiability

Under correlated latents combined with a linearly entangled encoder, the paper derives a closed-form expression for MCC-P showing explicit dependence on the pairwise correlation $n \to \infty$7, violating a formal invariance property (P1). The result is striking: MCC approaches 1 at both extremes $n \to \infty$8 and $n \to \infty$9 even though the encoder remains entangled, making an affine-class representation indistinguishable from one recovered up to permutation and rescaling. For practically relevant positive correlations, MCC inflates monotonically, with the gap between disentangled and entangled encoders narrowing—and sharpening—as $d_{\mathrm{eff}} = d$0 grows. Conversely, DCI-D exhibits the opposite pathology: it collapses to near-zero under any nontrivial condition number $d_{\mathrm{eff}} = d$1 of the mixing matrix, producing false negatives for representations that preserve all information up to $d_{\mathrm{eff}} = d$2.

Failure mode II: undetectable multi-factor redundancy

When $d_{\mathrm{eff}} = d$3, the question is whether metrics distinguish lossless compression of redundant factors from lossy omission. The findings split cleanly by family:

• MCC-P/S report 1.0 regardless, matching only $d_{\mathrm{eff}} = d$4 pairs and ignoring omitted factors entirely—even at $d_{\mathrm{eff}} = d$5 under fully independent factors.
• $d_{\mathrm{eff}} = d$6 and DCI-D correctly detect single-factor redundancy: under a constraint like $d_{\mathrm{eff}} = d$7, both plateau near 1.0 at $d_{\mathrm{eff}} = d$8, satisfying faithfulness to effective dimensionality (P2), while declining steadily when informative factors are dropped.
• No pairwise metric detects multi-factor redundancy: when a dropped factor depends jointly on two others (all pairwise correlations possibly zero), regression-based metrics penalize a lossless encoder as if it were lossy. Higher-order statistics would be required.

This has direct consequences for mechanism discovery settings where deterministic physical constraints among factors are generic rather than exceptional.

Failure mode III: overcompleteness breaks comparability

Sweeping $d_{\mathrm{eff}} = d$9 reveals encoder-specific violations of overcompleteness invariance (P3). Only trivial elementwise duplication leaves all metrics stable. Three failures stand out:

Encoder geometry	Observed behavior
Distributed codes (disjoint subsets)	MCC-P drops from ~0.85 to ~0.65 as $z_2 = z_1^3$0 goes 2→10 despite full recoverability
Overcomplete linear entanglement	DCI-D inflates from ~0.42 to ~0.80 as $z_2 = z_1^3$1 goes 1.5→10 — a false positive
Nonlinear cross-factor codes	$z_2 = z_1^3$2 collapses

The implication is that overcomplete representations—which are the norm in post-hoc analysis of pretrained models, where $z_2 = z_1^3$3—cannot be evaluated by any single existing metric; matched-dimension controls or multi-metric reporting are necessary.

Failure mode IV: finite-sample false positives governed by $z_2 = z_1^3$4

Using a null encoder carrying no information about the factors, the paper shows that MCC violates insensitivity to uninformative encoders whenever $z_2 = z_1^3$5. The theoretical analysis is clean: under the null, sample correlations are approximately $z_2 = z_1^3$6, and Hungarian matching over $z_2 = z_1^3$7 candidates per column selects maxima scaling as $z_2 = z_1^3$8—a bound depending on $z_2 = z_1^3$9 and $V = IR$0 but not on $V = IR$1, matching the empirical observation that inflation varies with $V = IR$2 and not $V = IR$3. At $V = IR$4, MCC-P reports 0.83 for pure noise. DCI-D shows moderate inflation at high estimation ratios; $V = IR$5 is the most robust, remaining near zero across the entire $V = IR$6 grid, though it requires $V = IR$7 under nonlinear encoders. Unlike the previous three failures, this one vanishes asymptotically—but the paper notes that practical regimes routinely exceed safe ratios: probing Llama-3.2-8B ($V = IR$8) with hundreds of samples gives $V = IR$9, and even standard benchmarks with $d_{\mathrm{eff}} = 2$0, $d_{\mathrm{eff}} = 2$1 exceed the threshold. In sparse-autoencoder-based interpretability, where both $d_{\mathrm{eff}} = 2$2 and $d_{\mathrm{eff}} = 2$3 hold simultaneously, all four failure modes compound.

Practical guidance

The paper distills its analyses into four formal properties (correlation invariance, effective-dimensionality faithfulness, overcompleteness invariance, uninformative-encoder sensitivity) and a summary table showing no existing metric satisfies all four—MCC variants violate all of them; $d_{\mathrm{eff}} = 2$4 satisfies P1 and P4 but fails on overcompleteness; DCI-D partially satisfies everything except overcompleteness invariance. A practitioner checklist recommends: verifying $d_{\mathrm{eff}} = 2$5 before interpreting MCC; always reporting null-encoder baselines; preferring $d_{\mathrm{eff}} = 2$6 under correlated factors; and using multiple metrics plus matched-dimension controls in the overcomplete regime. A systematic review of 62 CRL/nonlinear-ICA papers (2020–2025) found MCC used in 25, $d_{\mathrm{eff}} = 2$7 in 9, and none of the newer MI- or conditional-independence-based metrics (MIG, T-MEX, InfoMEC)—which themselves exhibit additional pathologies in the appendix (NaNs below $d_{\mathrm{eff}} = 2$8, persistent inflation for MCC-RDC).

Limitations and open questions

The authors are explicit that their analysis uses synthetic constructed encoders by design, to isolate metric misspecification from optimization artifacts; whether these failures manifest identically in learned encoders is left open. The taxonomy does not cover stochastic encoders or discrete factors, both common in practice. The multi-factor redundancy result raises a concrete methodological question the paper does not resolve: what measurement instrument based on higher-order statistics can certify losslessness under synergistic constraints? Similarly, the proposed desiderata characterize validity domains but do not yield new metrics satisfying them.

Conclusion

This paper converts scattered empirical anecdotes about metric disagreement into a predictive theory: each identifiability metric's failure modes follow from its implicitly encoded assumptions about the DGP, encoder geometry, and sample regime, and can be anticipated analytically rather than discovered post hoc. The demonstration that every commonly used metric—including those validating identifiability theorems whose assumptions they contradict—produces structured errors implies that single-metric reports on synthetic benchmarks constitute insufficient evidence of identifiability, particularly in the overcomplete, sample-constrained regimes characteristic of interpretability research.