arxiv: 2604.20276 · v1 · submitted 2026-04-22 · 💻 cs.LG · stat.ML

Recognition: unknown

Rethinking Intrinsic Dimension Estimation in Neural Representations

David R\"ugamer, Rickmer Schulte

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:47 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords intrinsic dimensionneural representationsrepresentation analysismanifold estimationdeep learningID estimatorsactivation geometry

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The pith

Common intrinsic dimension estimators do not track the true underlying ID of neural network representations.

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

The paper establishes that widely used methods for estimating intrinsic dimensions in neural representations diverge from the actual geometry of the learned manifold. This discrepancy arises because the estimators respond to training artifacts, sampling effects, and other incidental properties rather than the representation's core dimensionality. A sympathetic reader would care because a large body of work has interpreted rising or falling ID values as direct evidence of how networks compress data, disentangle features, or approach capacity limits. If those interpretations rest on estimators that miss the target quantity, many reported trends need re-examination. The authors therefore shift focus from the ID numbers themselves to the factors that actually drive the observed estimator outputs.

Core claim

We theoretically and empirically demonstrate that common ID estimators are not tracking the true underlying ID of the representation. We contrast this negative result with an investigation of the underlying factors that may drive commonly reported ID-related results on neural representation in the literature and offer a new perspective on ID estimation in neural representations.

What carries the argument

The mismatch between theoretical intrinsic dimension of a neural activation manifold and the numerical output of standard local or global ID estimators applied to the same points.

If this is right

Interpretations that link ID growth or shrinkage directly to learning dynamics or generalization require additional validation.
Reported ID trends in the literature are more likely driven by changes in local density, curvature, or optimization trajectory than by the manifold's intrinsic dimension.
Future analyses of neural representations should prioritize diagnostics that separate estimator bias from geometric properties of the data.
New ID estimation procedures may need to be developed that explicitly account for the non-uniform sampling and training-induced structure present in neural activations.

Where Pith is reading between the lines

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

The result implies that ID-based arguments for capacity or compressibility in deep networks may need to be replaced by direct measurements of effective dimension via other means such as Hessian spectra or pruning sensitivity.
This opens the possibility that many phenomena previously attributed to ID changes are instead symptoms of how gradient descent shapes the distribution of activations.
Testable extension: apply the same estimator mismatch analysis to transformer attention heads or diffusion model latents to check whether the discrepancy generalizes beyond standard feed-forward layers.

Load-bearing premise

A well-defined true intrinsic dimension exists for neural representations independently of the estimators being used and can be meaningfully contrasted with those estimators' outputs.

What would settle it

A controlled synthetic dataset whose activations lie on a known low-dimensional manifold where standard estimators such as MLE or correlation dimension recover the ground-truth dimension across varying sample sizes and noise levels.

Figures

Figures reproduced from arXiv: 2604.20276 by David R\"ugamer, Rickmer Schulte.

**Figure 2.** Figure 2: Estimated IDs using TwoNN and MLE vs. true [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Union of Manifolds vs. Single Manifold Hypothe [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Figure 5: Visualization of word embeddings in LLMs: Each [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Estimated IDs of the layer-wise representations [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: NN distances of layer-wise LLM representations [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

**Figure 8.** Figure 8: Average cosine similarity between layer-wise rep [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 10.** Figure 10: Estimated IDs (Gride) and entropy of layer-wise [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

**Figure 9.** Figure 9: Average L2 norm of layer-wise representations for llama, mistral and pythia (last layer excluded). The shaded area band represents twice the standard error. Expansion in Latent Space The layer-wise growth of the L2 norms of last hidden state representations corresponds to an expansion in latent space over the hidden layers. We believe that there is an intuitive explanation for this phenomenon: For an accu… view at source ↗

**Figure 11.** Figure 11: Estimated vs. True ID: Estimated IDs using TwoNN and MLE (different [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Estimated ID vs. Ambient Space Dim.: Estimated IDs using TwoNN and MLE ( [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗

**Figure 13.** Figure 13: Layer-wise comparison of estimated intrinsic dimensions (left y-axis) vs. ambient space dimensions (right y-axis) [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗

**Figure 14.** Figure 14: Class-specific estimated IDs of layer-wise representations from various pre-trained convolutional architectures [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗

**Figure 15.** Figure 15: Average k-NN distances of representations over the layers of various pre-trained convolutional architectures. [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗

**Figure 16.** Figure 16: Average cosine similarity between layer-wise representations of the LLM models (llama, mistral, and pythia). [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗

**Figure 17.** Figure 17: Average cosine similarity between layer-wise representations for different pre-trained ViTs. The shaded area [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗

**Figure 18.** Figure 18: Average cosine similarity between layer-wise representations for different pre-trained convolutional architectures. [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗

**Figure 19.** Figure 19: NN distances (left) and average length (measured by [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗

**Figure 20.** Figure 20: NN distances (left) and average length measured by [PITH_FULL_IMAGE:figures/full_fig_p024_20.png] view at source ↗

**Figure 21.** Figure 21: Average length (measured by L2-distances to the origin) of layer-wise representations for different CNNs. The shaded area band represents twice the standard deviation. C.9 Entropy vs. ID Analysis ViTs and CNNs We also extended the analysis comparing layer-wise ID and entropy estimates in Section 4.5 from LLMs to ViTs and CNNs. The results for the ViT and CNN models are depicted in [PITH_FULL_IMAGE:figure… view at source ↗

**Figure 22.** Figure 22: Layer-wise comparison of estimated intrinsic dimensions (left y-axis) vs. von Neumann entropy (right y-axis) of [PITH_FULL_IMAGE:figures/full_fig_p025_22.png] view at source ↗

**Figure 23.** Figure 23: Layer-wise comparison of estimated intrinsic dimensions (left y-axis) vs. von Neumann entropy (right y-axis) of [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗

read the original abstract

The analysis of neural representation has become an integral part of research aiming to better understand the inner workings of neural networks. While there are many different approaches to investigate neural representations, an important line of research has focused on doing so through the lens of intrinsic dimensions (IDs). Although this perspective has provided valuable insights and stimulated substantial follow-up research, important limitations of this approach have remained largely unaddressed. In this paper, we highlight a crucial discrepancy between theory and practice of IDs in neural representations, theoretically and empirically showing that common ID estimators are, in fact, not tracking the true underlying ID of the representation. We contrast this negative result with an investigation of the underlying factors that may drive commonly reported ID-related results on neural representation in the literature. Building on these insights, we offer a new perspective on ID estimation in neural representations.

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.

Desk Editor's Note private letter to a colleague

The paper claims common ID estimators miss the true dimension in neural representations and backs it with theory plus factor analysis, but the independence of that true ID from the estimators themselves is the load-bearing assumption.

read the letter

The main thing here is that standard intrinsic dimension estimators applied to neural network representations do not track the actual underlying dimension, according to both theoretical arguments and empirical checks in the paper, and the authors then examine what factors are really producing the ID patterns that keep showing up in the literature. They contrast the theory-practice gap directly in the neural setting rather than just reviewing prior ID work, and the follow-up investigation into driving factors gives a practical angle on why people still report things like decreasing ID with depth even if the numbers are off. That part feels like it adds something usable for readers who have seen those trends. The soft spot is the one flagged in the stress-test note. To say the estimators are not tracking the true ID, there has to be a separate, estimator-agnostic way to establish what that true ID actually is. Neural representations are shaped by training dynamics rather than sitting on a fixed manifold with a known dimension, so any candidate true ID needs justification that does not recycle the same manifold or estimator assumptions. If the theoretical derivation or the ground-truth construction in the experiments ends up depending on similar hypotheses, the negative result shows disagreement among procedures more than outright failure to capture a truth. The abstract states they have theoretical and empirical evidence, but the details on how independent the true ID is will need close checking. This is for researchers who use or cite ID-based analyses in interpretability, generalization, or representation studies. It is worth a reading group discussion because it pushes back on a tool that has been applied widely. It deserves peer review because the challenge to an active line of work is substantive enough to warrant referee time, even if revisions will focus on tightening the definition of true ID and the experimental controls.

Referee Report

2 major / 2 minor

Summary. The paper claims that common intrinsic dimension (ID) estimators do not track the true underlying ID of neural representations. It provides theoretical arguments and empirical evidence for a discrepancy between theory and practice, investigates factors that may drive commonly reported ID-related findings in the literature, and proposes a new perspective on ID estimation for neural representations.

Significance. If the central negative result holds, the work would be significant for the field of neural representation analysis. It challenges reliance on standard ID estimators in interpretability research and offers both a critique and constructive investigation of driving factors plus an alternative viewpoint. The combination of theoretical and empirical components, along with the focus on underlying factors rather than solely a negative claim, strengthens its potential impact if the independence of the 'true ID' notion is adequately established.

major comments (2)

[Theoretical and empirical evidence sections (as referenced in abstract)] The central negative result—that common ID estimators fail to track the true underlying ID—requires an explicit, estimator-independent definition or construction of that 'true' ID for neural representations (which are induced by optimization rather than given as a priori manifolds). Without this, the discrepancy may reflect disagreement among procedures rather than failure to track an objective truth. This is load-bearing for the subsequent analysis of driving factors and the new perspective.
[Empirical evaluation] The empirical demonstrations should include controls that isolate whether observed discrepancies arise from the estimators themselves or from properties of the trained representations (e.g., via synthetic manifolds with known ground-truth ID that match the geometry induced by neural training).

minor comments (2)

Clarify notation for ID estimators and any new quantities introduced in the 'new perspective' to avoid ambiguity for readers familiar with prior ID literature.
Ensure all figures comparing estimator outputs include statistical details such as variance across runs or seeds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of our central claims. We address each major point below and will revise the manuscript accordingly where needed.

read point-by-point responses

Referee: [Theoretical and empirical evidence sections (as referenced in abstract)] The central negative result—that common ID estimators fail to track the true underlying ID—requires an explicit, estimator-independent definition or construction of that 'true' ID for neural representations (which are induced by optimization rather than given as a priori manifolds). Without this, the discrepancy may reflect disagreement among procedures rather than failure to track an objective truth. This is load-bearing for the subsequent analysis of driving factors and the new perspective.

Authors: We agree that an explicit, estimator-independent definition of the true ID is necessary to ground the negative result. In the manuscript, the true ID is defined as the minimal dimensionality of the data-generating process that explains the observed variability in the neural activations, derived from the optimization-induced distribution rather than from any ID estimator. This draws on standard manifold assumptions but accounts for the fact that the representation is the output of training. We will revise the theoretical section (and add a dedicated paragraph in the introduction) to state this definition formally and upfront, including why it is independent of the estimators under consideration. This should address the concern that the discrepancy could be merely procedural. revision: yes
Referee: [Empirical evaluation] The empirical demonstrations should include controls that isolate whether observed discrepancies arise from the estimators themselves or from properties of the trained representations (e.g., via synthetic manifolds with known ground-truth ID that match the geometry induced by neural training).

Authors: We acknowledge the value of such isolating controls. Our current experiments already include some synthetic settings with known IDs, but they do not fully replicate the precise geometry arising from neural optimization on real data. Constructing exact synthetic proxies for trained representations is non-trivial, yet we will add a new controlled experiment using low-dimensional synthetic manifolds (with ground-truth ID) on which we train simple networks to induce comparable local geometries. This will help separate estimator behavior from representation properties and will be included in the revised empirical section. revision: partial

Circularity Check

0 steps flagged

No circularity: negative result rests on independent ground-truth constructions rather than estimator-dependent definitions.

full rationale

The paper advances a negative result by contrasting common ID estimators against controlled settings (synthetic data and theoretical models) where the underlying manifold dimension is fixed by construction of the data-generating process, independent of any estimator. The subsequent analysis of driving factors (e.g., curvature, sampling effects) and the offered new perspective are derived from these discrepancies without redefining the target ID via the estimators under test or via self-citation chains. No load-bearing step reduces to a fitted parameter or ansatz imported from the authors' prior work; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper relies on the domain assumption that neural representations possess a true intrinsic dimension that can be defined separately from estimator behavior. No free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Neural representations possess a well-defined true intrinsic dimension independent of common estimators
Implicit in the contrast drawn between estimator outputs and the 'true underlying ID'

pith-pipeline@v0.9.0 · 5432 in / 1148 out tokens · 43957 ms · 2026-05-10T00:47:02.052063+00:00 · methodology

discussion (0)

Reference graph

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