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arxiv: 2605.01221 · v1 · submitted 2026-05-02 · 💻 cs.LG

Recognition: unknown

Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation

Genki Osada
Authors on Pith no claims yet

Pith reviewed 2026-05-09 14:06 UTC · model grok-4.3

classification 💻 cs.LG
keywords local intrinsic dimensionHessian spectral filteringdiffusion modelsmemorization detectionhigh-dimensional manifoldsStochastic Lanczos Quadraturetangent space estimation
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The pith

Spectral filtering on the log-density Hessian counts only tangent directions to estimate local intrinsic dimension even when noise fills most of high-dimensional space.

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

In high-dimensional data from diffusion models, most directions around a point are noise from the vast normal space, which overwhelms signals from the lower-dimensional tangent space that actually matters. Standard local intrinsic dimension estimators break down because they cannot separate this noise. The paper shows that the Hessian of the log-density has large eigenvalues tied to those normal directions and near-zero eigenvalues tied to the tangent directions. By explicitly cutting off the large eigenvalues, the method counts the remaining zero-curvature directions to produce a stable dimension estimate. This works at linear cost in dimension using Stochastic Lanczos Quadrature and detects memorization effects in trained models.

Core claim

Local Hessian Spectral Dimension is computed by applying a cutoff to the eigenvalues of the log-density Hessian at each point, discarding the large values associated with normal directions and retaining only the near-zero values that mark the tangent space, all without building the full Hessian matrix.

What carries the argument

Spectral filtering of the log-density Hessian, which isolates the near-zero eigenvalues corresponding to zero-curvature tangent directions by removing large eigenvalues from normal directions.

If this is right

  • The estimator remains stable in ambient dimensions where previous local intrinsic dimension methods degrade due to noise.
  • Tracking the filtered dimension during training reveals when diffusion models begin to memorize specific samples rather than learn the underlying distribution.
  • Linear scaling with dimension allows the method to run on large-scale models without computing or storing full Hessian matrices.
  • Experiments on both synthetic manifolds and real diffusion training data show improved robustness compared to unfiltered approaches.

Where Pith is reading between the lines

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

  • If the spectral gap persists across different data modalities, the same filtering idea could be applied to other Hessian-based tasks such as identifying flat regions in loss landscapes.
  • The linear-time implementation opens the possibility of using local dimension estimates as a routine diagnostic during the training of any high-dimensional generative model.
  • Applying the cutoff adaptively rather than with a fixed threshold might reduce sensitivity to the precise choice of spectral separation point.

Load-bearing premise

The log-density Hessian shows a clear gap that separates large eigenvalues of normal directions from near-zero eigenvalues of tangent directions, so a simple cutoff can be applied without losing signal.

What would settle it

A dataset or model where the eigenvalues of the log-density Hessian lack any consistent gap between large and near-zero values, so that different cutoff choices produce wildly varying dimension estimates on the same points.

Figures

Figures reproduced from arXiv: 2605.01221 by Genki Osada.

Figure 1
Figure 1. Figure 1: Synthetic Manifold: Moon (left), Funnel (center), and L 1+2+3 (right). The L 1+2+3 dataset consists of 3D cube, 2D square, and 1D line submanifolds. The overlaid colors indicate the LID estimated by our LHSD, accurately recovering the underlying manifold dimensionalities. leading to severe degradation in estimation accuracy. To address this, approaches utilizing deep generative mod￾els, particularly diffus… view at source ↗
Figure 2
Figure 2. Figure 2: LHSD Filter Behavior (Eq. (9)). Filter responses f(λ) with varying parameters are overlaid on the Hessian spectrum of L 900 ⊂ R 3072 dataset. Increasing c shifts the cutoff κ rightward, while increasing p steepens the transition. The cutoff consistently falls within the spectral gap (noise scale t = 0.04). becomes isotropic and the spectral gap collapses; the spec￾tral collapse and its detectability are ex… view at source ↗
Figure 3
Figure 3. Figure 3: Selection of t on L 900 ⊂ R 3072 for filter with c = 0.1 and p = 4. (a): Transition mass M(t) identifies a “safe zone” (blue bar): the valley where M(t) ≈ 0 between two collision peaks. See view at source ↗
Figure 4
Figure 4. Figure 4: LID estimation by LHSD on (Left) L 900 ⊂ R 3072 with MAE 11.53, and (Right) F 10+80+200 ⊂ R 3072 with MAE 5.19. lies within the spectral gap ( view at source ↗
Figure 7
Figure 7. Figure 7 view at source ↗
Figure 6
Figure 6. Figure 6: Robustness to noise scale t. LHSD (blue) maintains consistently low estimation error (MAE) across a wide range of noise scales on all datasets. settings: (1) the original 3D space (D = 3), and (2) a high￾dimensional embedding (D = 784) obtained by nonlinearly mapping the manifolds into the Fashion-MNIST space (see App. G.2). Previous studies reported that existing diffusion￾based methods fail significantly… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of generated samples with the lowest and highest LID values estimated by LHSD. SVHN, CIFAR-10 (‘airplane’ class), and LAION-Aesthetics are shown from top to bottom. LHSD successfully distinguishes between visually simple images (e.g., flat backgrounds, single objects) and complex ones (e.g., dense textures, cluttered scenes) across different resolutions and domains. (a) Training (b) Memorized ge… view at source ↗
Figure 10
Figure 10. Figure 10: The PNG complexity distributions (left) for the Mem￾orized and Non-memorized sets overlap significantly, ruling out image complexity as a confounding factor. Under this controlled condition, the Estimated LID (right) shows a distinct separation, demonstrating that LHSD can detect memorized samples by cap￾turing their low-intrinsic dimensionality. Aesthetics dataset (Rombach et al., 2022).4 For the latter,… view at source ↗
Figure 11
Figure 11. Figure 11: Visualization of Hessian spectral separation computed on 20 randomly sampled data points for each dataset. In all shown cases, the noise scale t is chosen such that the filter cutoff κ (dashed line) is correctly positioned within the spectral gap between tangent and normal components. D. Noise-Scale Selection via Transition Mass The accuracy of LHSD relies on the alignment between the spectral filter’s tr… view at source ↗
Figure 12
Figure 12. Figure 12: Selection of t on F 10+25+50 ⊂ R 100 (Top) and F 10+80+200 ⊂ R 1024 (Bottom). (a),(e): Transition mass M(t) identifies a “safe zone” (blue bar). (b)–(d) and (f)–(h): Cases with t selected outside the safe zone. See Figs. 11b and 11e for selected safe setting. E. Spectral Gap and Collapse A prerequisite for LHSD is the existence of a spectral gap separating the tangent and normal eigenvalues. We empiricall… view at source ↗
Figure 13
Figure 13. Figure 13: illustrates this phenomenon on the F 900 ⊂ R 3072 and F 10+25+50 ⊂ R 100 datasets. For F 900 ⊂ R 3072, a distinct spectral gap persists at t = 0.60, allowing the filter to isolate tangent components, whereas at t = 0.65, the clusters collide, closing the spectral gap. Similarly, for F 10+25+50 ⊂ R 100, the separation holds at t = 0.5 but collapses at t = 0.6. Importantly, the diagnostic M(t) reveals that … view at source ↗
Figure 14
Figure 14. Figure 14: Sensitivity analysis of LHSD with respect to Lanczos steps m. Top panels (a-d) show the MAE across noise scales t for varying m. The bottom panel (e) compares the inference time. Increasing m improves accuracy up to a saturation point around m = 5, beyond which computational cost increases without significant gain. 19 view at source ↗
Figure 15
Figure 15. Figure 15: Histograms of LID estimated by LHSD. The peaks of the distributions closely align with the ground-truth dimensions of the underlying submanifolds (e.g., peaks at 10, 80, and 200 for L 10+80+200), demonstrating the accuracy of the estimation. J. Details of Runtime Measurement Experiments Stochastic FLIPD (FLIPD-Hutch). The official implementation of FLIPD (Kamkari et al., 2024b) is a deterministic algorith… view at source ↗
Figure 16
Figure 16. Figure 16: Noise scale sensitivity: LHSD (ours) on Moon. 22 view at source ↗
Figure 17
Figure 17. Figure 17: Noise scale sensitivity: FLIPD on Moon. 23 view at source ↗
Figure 18
Figure 18. Figure 18: Noise scale sensitivity: NB on Moon. 24 view at source ↗
Figure 19
Figure 19. Figure 19: Noise scale sensitivity: LHSD (ours) on Funnel. 25 view at source ↗
Figure 20
Figure 20. Figure 20: Noise scale sensitivity: FLIPD on Funnel. 26 view at source ↗
Figure 21
Figure 21. Figure 21: Noise scale sensitivity: NB on Funnel. 27 view at source ↗
Figure 22
Figure 22. Figure 22: Noise scale sensitivity: LHSD (ours) on L 1+2+3 . 28 view at source ↗
Figure 23
Figure 23. Figure 23: Noise scale sensitivity: FLIPD on L 1+2+3 . 29 view at source ↗
Figure 24
Figure 24. Figure 24: Noise scale sensitivity: NB on L 1+2+3 . 30 view at source ↗
read the original abstract

While diffusion models enable new approaches for estimating Local Intrinsic Dimension (LID), existing methods fail in high-dimensional spaces where noise from vast normal directions overwhelms the tangent signal. We propose Local Hessian Spectral Dimension (LHSD), which resolves this by applying spectral filtering to the log-density Hessian, explicitly cutting off large eigenvalues associated with normal directions to count zero-curvature tangent directions. Implemented using Stochastic Lanczos Quadrature (SLQ), LHSD avoids full Hessian construction, achieving linear scalability with dimension $D$. Experiments on synthetic and real data confirm LHSD's superior robustness and its utility in detecting memorization in large-scale diffusion models.

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

2 major / 1 minor

Summary. The paper proposes Local Hessian Spectral Dimension (LHSD) to estimate local intrinsic dimension (LID) in high-dimensional spaces, particularly for diffusion models where noise from normal directions overwhelms tangent signals. LHSD applies spectral filtering to the log-density Hessian, cutting off large eigenvalues associated with normal directions to count near-zero eigenvalues corresponding to tangent directions. It is implemented using Stochastic Lanczos Quadrature (SLQ) to achieve linear scalability with dimension D without constructing the full Hessian. Experiments on synthetic and real data are presented to show superior robustness and utility in detecting memorization in large-scale diffusion models.

Significance. If the central claims hold, LHSD could provide a robust and scalable tool for LID estimation in challenging high-D regimes. The avoidance of full Hessian via SLQ is a practical strength for high dimensions. The application to memorization detection in diffusion models suggests broader impact in machine learning model analysis. Credit is due for addressing scalability explicitly.

major comments (2)
  1. [§3] §3 (LHSD construction): The central construction filters the Hessian spectrum by cutting off large eigenvalues (normal directions) to count near-zero ones (tangent space). This requires a detectable gap between the two clusters, but no derivation shows the gap is guaranteed or how to locate the cutoff without supervision. In practice, finite-sample Hessian estimates from diffusion models or high-D data can have overlapping or noisy spectra due to curvature, sampling variance, or manifold non-flatness; a fixed or heuristic cutoff then either undercounts tangent directions or leaks normal noise.
  2. [§4] §4 (Experiments): The abstract claims superior robustness and utility in detecting memorization, but supplies no derivations, cutoff selection details, error analysis, or quantitative results. This makes it difficult to verify whether the math and experiments support the claims, particularly under the weakest assumption that a simple cutoff can be applied without bias.
minor comments (1)
  1. [Abstract] Abstract: The claim of 'linear scalability with dimension D' would be strengthened by an explicit complexity statement or comparison to baseline runtimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address the major comments point by point below, acknowledging where clarifications and expansions are needed. We propose targeted revisions to strengthen the presentation of the method and experiments.

read point-by-point responses

  1. Referee: [§3] §3 (LHSD construction): The central construction filters the Hessian spectrum by cutting off large eigenvalues (normal directions) to count near-zero ones (tangent space). This requires a detectable gap between the two clusters, but no derivation shows the gap is guaranteed or how to locate the cutoff without supervision. In practice, finite-sample Hessian estimates from diffusion models or high-D data can have overlapping or noisy spectra due to curvature, sampling variance, or manifold non-flatness; a fixed or heuristic cutoff then either undercounts tangent directions or leaks normal noise.

    Authors: We agree that the spectral gap is a key assumption and that a general guarantee cannot be derived without additional conditions on the data. Section 3 of the manuscript motivates the separation from the geometry of a manifold embedded in high dimensions, where normal directions produce large positive eigenvalues of the log-density Hessian while tangent directions yield near-zero eigenvalues. In the revision we will explicitly state the required assumptions (locally approximately flat manifold, sufficient sampling density relative to curvature, and bounded noise) and add a formal discussion of when the gap is expected to exist. For cutoff location we will describe the practical heuristic employed (sorting eigenvalues and selecting the threshold at the largest gap in the log-eigenvalue spectrum, or via a simple percentile rule calibrated on synthetic data) and include a sensitivity analysis showing robustness to moderate overlap. We will also discuss failure modes when the gap is absent. revision: yes

  2. Referee: [§4] §4 (Experiments): The abstract claims superior robustness and utility in detecting memorization, but supplies no derivations, cutoff selection details, error analysis, or quantitative results. This makes it difficult to verify whether the math and experiments support the claims, particularly under the weakest assumption that a simple cutoff can be applied without bias.

    Authors: Section 4 and the appendix already contain quantitative results: error rates on synthetic manifolds with known ground-truth dimensions, comparisons against MLE and PCA baselines, and metrics for memorization detection on diffusion models. Derivations for the Hessian filtering and SLQ estimator appear in Section 3 and the supplementary material. Nevertheless, we accept that the cutoff procedure and error analysis are not presented with sufficient prominence or detail. In the revision we will add an explicit subsection on cutoff selection, include bounds on the SLQ approximation error, and report additional quantitative experiments on cutoff sensitivity and bias under controlled spectral overlap. These additions will make the empirical support easier to verify. revision: yes

Circularity Check

0 steps flagged

No circularity: LHSD introduces independent spectral filtering mechanism via SLQ

full rationale

The paper's core contribution is a new estimator LHSD that applies explicit spectral cutoff to the log-density Hessian eigenvalues to isolate tangent-space directions. This filtering step is presented as a novel construction, not derived from or equivalent to any fitted parameter, prior self-result, or input data by definition. Implementation via Stochastic Lanczos Quadrature is a computational technique for avoiding full Hessian construction, with no indication that the dimension count reduces to a self-referential fit or renamed known result. Experiments on synthetic/real data serve as external validation rather than internal tautology. No load-bearing self-citations or uniqueness theorems from the same authors are invoked to force the method. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The approach rests on domain assumptions about Hessian eigenvalue structure in high-dimensional densities; no free parameters or invented entities are explicitly quantified in the abstract.

axioms (1)
  • domain assumption The Hessian of the log-density has large eigenvalues associated with normal directions overwhelmed by noise and near-zero eigenvalues for tangent directions.
    This separation is required for the spectral cutoff to isolate intrinsic dimension.
invented entities (1)
  • Local Hessian Spectral Dimension (LHSD) no independent evidence
    purpose: Robust estimator of local intrinsic dimension via filtered Hessian spectrum.
    Newly proposed method; no independent evidence of correctness provided in abstract.

pith-pipeline@v0.9.0 · 5388 in / 1300 out tokens · 33190 ms · 2026-05-09T14:06:53.178279+00:00 · methodology

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