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arxiv: 2606.19268 · v2 · pith:BEM4L6EGnew · submitted 2026-06-17 · 🧮 math.ST · cs.CG· stat.TH

Patnaik-Pearson intrinsic dimension for internal representations of neural networks

Tom Hadfield This is my paper

Pith reviewed 2026-07-03 23:39 UTC · model grok-4.3

classification 🧮 math.ST cs.CGstat.TH
keywords intrinsic dimensionneural networkstransformerspower lawempirical spectral densitytoken embeddingsBERT
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The pith

The Patnaik-Pearson dimension measures intrinsic dimension in neural network representations and aligns critical tail exponents with HTSR and SETOL when weight matrices follow Pareto spectra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the Patnaik-Pearson dimension as an intrinsic dimension estimator for data manifolds, drawing on the TwoNN method and spectral analyses from HTSR and SETOL. It establishes properties of the estimator and demonstrates that, for weight matrices whose empirical spectral density follows a power-law distribution, the critical values of the tail exponent match those from the earlier HTSR and SETOL frameworks. The work then tracks how this dimension changes under standard neural network operations and applies the measure to token embeddings in BERT-base and DeepSeek-R1-Distill-Qwen-1, observing its evolution across layers.

Core claim

For weight matrices whose Empirical Spectral Density follows a Pareto distribution, the Patnaik-Pearson dimension relates directly to HTSR and SETOL analysis such that the critical values of the tail exponent coincide between the two approaches.

What carries the argument

The Patnaik-Pearson dimension, an intrinsic dimension estimator constructed from the TwoNN method combined with Patnaik-Pearson statistics, applied to weight matrices and token embeddings treated as data manifolds.

If this is right

  • The dimension of the initial token embedding manifold can be computed directly for transformer models.
  • The Patnaik-Pearson dimension evolves in measurable ways as embeddings pass through successive layers.
  • The coincidence of critical tail exponents holds specifically under the Pareto assumption for spectral densities.
  • Numerical evaluation on real models like BERT-base confirms the dimension can be tracked layer by layer.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the dimension stabilizes or changes predictably across layers, it may indicate fixed points in how networks compress or expand manifolds.
  • The approach could extend to other architectures by treating attention weights or activations as additional manifolds.
  • Discrepancies in non-Pareto cases might highlight when spectral methods lose their direct connection to intrinsic dimension estimates.

Load-bearing premise

The empirical spectral density of the weight matrices follows a Pareto power-law distribution.

What would settle it

Empirical computation on a weight matrix whose spectral density deviates from Pareto form, showing that the Patnaik-Pearson critical tail exponent no longer matches the HTSR or SETOL value.

Figures

Figures reproduced from arXiv: 2606.19268 by Tom Hadfield.

Figure 1
Figure 1. Figure 1: 1000 points in R 2 with Patnaik-Pearson dimension 1.514, TwoNN dimension 1.942. This suggests that the Patnaik-Pearson dimension may be thought of as a “global” measure of dimension, whereas the TwoNN dimension captures local dimensionality. 0 200 400 600 800 1000 actual dimension 0 200 400 600 800 intrinsic dimension Patnaik-Pearson and TwoNN for a solid ball of varying dimension embedded in 1000-dimensio… view at source ↗
Figure 2
Figure 2. Figure 2: Patnaik-Pearson and TwoNN dimension estimates for a solid ball of dimension varying between [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Patnaik-Pearson and TwoNN dimension estimates for a solid ball of dimension varying between [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Uniform distribution : numerical tests of (29), (a) for 10 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Uniform distribution : numerical tests of (31), (a) for 10 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Marchenko-Pastur : numerical tests of (30), [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Marchenko-Pastur : numerical tests of (32), [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) ν(α,d) d as α varies, for d increasing from 100 to 1,000,000, together with ν∞(α). (b) Difference of ν(α,d) d from conjectured limit as α varies, for d increasing from 100 to 1,000,000. and the result follows. For 2 > α > 1, then E(Y ) is finite, but E(Y 2 ) is infinite. By Kolmogorov’s SLLN [8], 1 d X d i=1 λi a.s → E(λ) < ∞, 1 d X d i=1 λ 2 i a.s → ∞ Hence 1 d ν(α, d) →0 as d →∞. For 1 > α, then both… view at source ↗
Figure 6
Figure 6. Figure 6: Numerical tests of (38, 41), for α = 1.5 and α = 2.5 and increasing d. So for α ̸= 1, 2, then X d k=1 λk ≈ d s 1 − s [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical tests of (40, 43), for α = 1.5 and α = 2.5 and increasing d. So, for α > 2, limd → ∞ 1 d ν(α, d) = C(α), and, further, for large d then ln( 1 d ν(α, d) − C(α)) = ln(C(α)) − [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Numerical tests of (39) and (42), for α = 1.0 and α = 2.0, for increasing d Tail exponent α ν(α, d) for large d 1 d ν(α, d) for large d α > 2 C(α)d C(α) α = 2 4d ln(d) 4 ln(d) 2 > α > 1 −C(α)d 2  α−1 α  −C(α)d −  2−α α  α = 1 (ln(d))2 1 d (ln(d))2 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Product hypotheses (50). 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 nu/d (X) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 nu/d Pareto : nu/d (XTX) compared to nu/d (XT) * nu/d (X) as nu/d (X) varies N, d randomly chosen between 500 and 1000, with N > d nu/d (XTX) nu/d (XT) * nu/d (X) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 nu/d (W) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 nu/d (softmax(W)) Pareto : actual and estimated nu/d (softm… view at source ↗
Figure 8
Figure 8. Figure 8: Product hypotheses (52). 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 nu/d (X) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 nu/d Pareto : nu/d (XTX) compared to nu/d (XT) * nu/d (X) as nu/d (X) varies N, d randomly chosen between 500 and 1000, with N > d nu/d (XTX) nu/d (XT) * nu/d (X) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 nu/d (W) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 nu/d (softmax(W)) Pareto : actual and estimated nu/d (softm… view at source ↗
Figure 9
Figure 9. Figure 9: (a) Product hypothesis (50) is not satisfied by [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Product hypothesis (52) is not satisfied by [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Patnaik-Pearson dimension for XW XT using (50). (b) Patnaik-Pearson dimension of Attention(Q, K, V ), using (52, 53) 4.6 Activation functions : ReLU We investigate the effect of the ReLU activation function. For X an N × d data manifold, define (ReLU(X))ij = ( Xij : Xij ≥ 0 0 : Xij < 0 , (56) [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Patnaik-Pearson dimension for XW XT using (52). (b) Patnaik-Pearson dimension of Attention(Q, K, V ), using (54, 55) 0.0 0.2 0.4 0.6 0.8 1.0 nu/d X 0.0 0.2 0.4 0.6 0.8 1.0 nu/d ReLU(X) nu/d ReLU(X) vs nu/d X N = 1000, d = 1000 nu/d ReLU(X) nu/d X approx nu/d ReLU(X) 1 2 3 4 5 6 alpha(X) 1 2 3 4 5 6 alpha(ReLU(X)) alpha(ReLU(X)) vs alpha(X) N = 1000, d = 1000 alpha(ReLU(X)) alpha(X) approx alpha(ReLU(X… view at source ↗
Figure 11
Figure 11. Figure 11: The effect of ReLU on Patnaik-Pearson dimension. Approximation given by (57). [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sum of two matrices X1 + X2 : heavy-tails dominate. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sum of two matrices X1 + X2 : heavy-tails dominate. 4.7 Addition, Interpolation and Concatenation For X1 and X2 both N × d, define X1 + X2 as the usual matrix addition. As shown in [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Interpolation between two data manifolds [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Interpolation between two data manifolds [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Concatenation : Patnaik-Pearson dimension of [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Normalisation : Patnaik-Pearson dimension for [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 15
Figure 15. Figure 15: Normalisation : Patnaik-Pearson dimension for [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Patnaik-Pearson dimension and nu/d for sampled BERT token embeddings [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 16
Figure 16. Figure 16: Patnaik-Pearson dimension and nu/d for sampled BERT token embeddings [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Layerwise evolution of Patnaik-Pearson dimension for BERT. [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: Layerwise evolution of Patnaik-Pearson dimension for BERT. [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Deepseek token embeddings: Patnaik-Pearson dimension and nu/d for samples of token [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 18
Figure 18. Figure 18: Deepseek token embeddings: Patnaik-Pearson dimension and nu/d for samples of token [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Layerwise evolution of the Patnaik-Pearson dimension of Deepseek embeddings. [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 19
Figure 19. Figure 19: Layerwise evolution of the Patnaik-Pearson dimension of Deepseek embeddings. [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
read the original abstract

We define a new measure of intrinsic dimension of a data manifold, which we call the Patnaik-Pearson dimension, and apply this to internal representations of neural networks, in particular transformers. The inspiration for this comes from the HTSR and SETOL work of Martin, Mahoney and Hinrichs, combined with the TwoNN intrinsic dimension estimator of Facco et al. We prove various properties of this intrinsic dimension estimator. Treating weight matrices of neural networks as data manifolds, for weight matrices whose Empirical Spectral Density follows a Pareto (Power Law) distribution, we relate the Patnaik-Pearson dimension to the HTSR and SETOL analysis, and show that critical values of the tail exponent coincide for the two approaches. Using a combination of theoretical and numerical techniques, we study the behaviour of the Patnaik-Pearson dimension of a data manifold under the transformations typical to neural networks. We apply this machinery to the BERT-base and DeepSeek-R1-Distill-Qwen-1 models, to investigate first the Patnaik-Pearson dimension of the initial data manifold of token embeddings, and second the evolution of the Patnaik-Pearson dimension as token embeddings pass through the layers of the model. Code and notebooks used for the numerical results presented here is available at https://github.com/tdhadfield/PatnaikPearson

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the Patnaik-Pearson intrinsic dimension estimator for data manifolds, drawing from HTSR/SETOL and the TwoNN method. It proves properties of the estimator, relates the dimension to HTSR/SETOL analysis specifically for weight matrices whose empirical spectral density follows a Pareto distribution (showing coincidence of critical tail exponents), examines the estimator's behavior under typical neural network transformations, and applies it to study token embeddings and their evolution through layers in BERT-base and DeepSeek-R1-Distill-Qwen-1 models. Code is provided for the numerical results.

Significance. If the conditional relation holds and the applications are robust, the work offers a new intrinsic dimension tool that connects to heavy-tailed spectral regularization frameworks, potentially aiding analysis of representation geometry in transformers. The explicit conditioning on the Pareto ESD and the availability of reproducible code are strengths.

minor comments (3)
  1. The abstract states that the relation to HTSR/SETOL holds for weight matrices with Pareto ESD, but the applications focus on token embeddings; clarify in the introduction or §3 whether the Pareto condition is checked or relevant for the BERT/DeepSeek experiments.
  2. The manuscript mentions proving 'various properties' of the estimator; ensure the main text explicitly lists these properties with references to the relevant theorems or propositions.
  3. Minor notation inconsistencies may exist between the Patnaik-Pearson definition and the TwoNN baseline; verify consistency in the methods section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. No specific major comments are provided in the report, so there are no individual points to address point-by-point. We are happy to make any minor changes the editor may request based on the overall summary.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the Patnaik-Pearson dimension by combining the TwoNN estimator with concepts from HTSR/SETOL, proves independent properties of the new estimator, and derives the relation between critical tail exponents only conditionally on the explicit assumption that ESD follows a Pareto distribution. No step equates the new dimension to a fitted input or prior result by construction, and no self-citation chain is load-bearing for the central claims. The applications to BERT and DeepSeek are presented as separate empirical uses of the defined machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents exhaustive enumeration; the ledger is therefore provisional. The new dimension is constructed from the TwoNN estimator and the Pareto assumption on spectral density; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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