More Data Cannot Break a Symmetry: Identifiability by Design

Jing Xu; Christopher Kanan

cs.LGAug 27, 2026

More Data Cannot Break a Symmetry: Identifiability by Design

Authors: Jing Xu, Christopher Kanan

Organizations: Department of Computer Science, University of Rochester

Abstract

Unsupervised representational alignment recovers a stimulus-by-stimulus correspondence from geometry alone, but the automorphism group of the stimulus geometry bounds what any such alignment can identify, before data exist. The obvious diagnostic for this degeneracy, the cheapest non-identity relabelling, ranks two published designs in the wrong order, because dense sampling creates near-duplicates whose transposition is nearly free. We turn this known invariance (Demetci et al., 2024) into a design-time diagnostic and intervention. In colour, where candidate geometries have closed form, we show that the failure is structural: sixty-four times the restart budget leaves a symmetric design unmoved while an asymmetric set at the same N recovers every time. Discriminating representational models and recovering a correspondence are essentially uncorrelated objectives (r = -0.02 over 3,000 subsets). Choosing nine colours by this diagnostic alone, without consulting any learned representation, moves all 93 model representations away from the degenerate point and cuts catastrophic alignment failures from 75% to 2% with the models, the layers, N and the solver all held fixed. The same risk arises wherever a regular design meets its candidate geometry's isometry group, including evenly spaced orientations, tones, or motion directions, and the check costs one function call before data collection.

Figures & tables

Figure 1: Under the HSV cylinder every rotation of the nine-colour set costs within

0.003

of the identity (the nominal design is exactly equally spaced; using the calibrated sRGB values supplied with the dataset introduces deviations of at most

0.003

); under the observers’ own judgements (rotation cost

0.26

) or under nine colours drawn from the colour solid, it does not. Lower panels: the two correspondences a zero cost cannot distinguish from the truth.

Figure 2: Computation rescues an optimisation problem and cannot rescue an identifiability one. The ring is unmoved by sixty-four times the search; the 93-colour displacement falls to

0.052

while exact accuracy stays at

0\%

, which is why displacement is the primary outcome.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Design	Field	Group	Catastrophic cost
Nine saturated hues	colour	$D_{9}$	$0.000$ (exact)
Twelve orientations	V1	$D_{12}$	$0.000$ (exact)
Twelve motion directions	MT	$D_{12}$	$0.000$ (exact)
Twelve tones within an octave	audition	$D_{12}$	$0.000$ (exact)
$5\times 5$ factorial grid	psychophysics	$D_{4}$	$0.000$ (exact)
THINGS objects ( $n=200$ )	object vision	—	$0.224$

Appendix

Table 1: Catastrophic cost across subfields. Zeros marked exact are achieved by the design’s own symmetry group and so are not search bounds. Each is conditional on its candidate geometry exactly as the colour result is: the orientation zero assumes orientation is an angle on a circle, the standard model rather than a fact. THINGS uses human similarity embeddings from the SPoSE model ( Hebart et al., 2020 ) .

Figure 3: Catastrophic cost against set size for three sampling schemes. Keeping stimuli on the hue circle at

S=V=1

(orange) leaves the cost at zero regardless of

N

; any departure from the invariant submanifold lifts it immediately. The published 93-colour set (star, upper right) falls on the curve at its own

N

: it is not special, just not on the circle. The matched-

N

contrast at

N=9

rules out the reading that the 93-colour success is simply more data. Bands show 10th–90th percentile over 40 random draws.

Figure 4: Design-time prescription. Before data collection, compute the catastrophic cost of the stimulus set under every candidate geometry and take the worst case

\Gamma(S)

. A zero means the correspondence cannot be identified under that geometry; the remedy is to vary the sampling away from the invariant submanifold, not to add more stimuli along it.

Figure 5: Four designs matched at nine stimuli, shown to the same 93 representations. Left: the closest any layer of each model comes to zero, models ranked within each design. The correspondence-optimised set moves every model further out, though the spread grows in proportion to the shift rather than beyond it (SD over median

0.30

against

0.29

for the ring), so the models are not made more separable. The geometry-discrimination set tracks the random draw almost exactly. Right: how often the recovered correspondence is catastrophic, with the design-time cost beside each bar.

Figure 6: Catastrophic cost across 93 model representations. Left: the published nine-colour set against 930 matched-size draws from the colour solid, the same representations and the same

N

in both cases. No representation sits at the degenerate point, and the solid draws sit further from it. Right: the closest any layer of each model comes to zero. The line marks the human observers’ catastrophic cost (

0.085

); five models have at least one layer below it.

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Representation Alignment Rests on Linear Structure

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Department of Electrical Engineering and Computer Science · Massachusetts Institute of Technology

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Open paper AskPDF arXiv

We investigate the Platonic Representation Hypothesis (PRH) through a tripartite statistical framework of representations: signal, bias, and noise. {1) Signal:} We propose that Platonic alignment arises from the universal relationship between objects and attributes, which is encoded linearly in representations according to the Linear Representation Hypothesis (LRH). We provide evidence that LRH helps explain PRH by extracting linear object-attribute features with sparse autoencoders and showing that these sparse representations often exhibit stronger cross-modal alignment than their dense counterparts. {2) Bias:} Models have different implicit biases due to the diverse architectures and training procedures used. We show that this difference can be partially mitigated. Centering and normalization consistently improve cross-model alignment. {3) Noise:} Finite-sample training leads to noise in representations. We provide evidence that representational noise is driven by data scarcity by revealing a strong and consistent positive correlation between word frequency and alignment in LLMs and text embedding models. Synthesizing signal, bias, and noise, we propose a statistical model that refines the Linear Representation Hypothesis and explains further phenomena related to the alignment of representations emerging from diverse modern AI architectures.

Abstract

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