Interpretable intrinsic dimension estimation through componentwise calibration of distance and angle

• Practical noise, sample-amplitude heterogeneity, and high candidate dimensions can distort geometric signals or obscure which component determines the estimate.The proposed formulation addresses these conditions with generic-order ratios, mean-direction profiling, and stable Bessel evaluation.Source: §1 · pp. 2–3
• Componentwise calibration separates distance, angular, and combined discrepancy curves, showing agreement between signals and which term shifts the minimum.The formulation also permits interchangeable components when their dimension-indexed reference laws have computable discrepancies.Source: §1 · p. 3
• Closed-form KL calibration admits generic-order Gride ratios, bringing their robustness to neighborhood-relative noise into the objective.The contribution derives the KL divergence for generic-order ratios.Source: §1 · p. 4
• Two sampling regimes motivate Profiled angular calibration, which aligns mean direction before matching concentration; Full matches both von Mises parameters.The Gaussian scale-mixture control tests profiling under sample-amplitude heterogeneity, and removing known amplitudes restores Full calibration by removing the angular-reference mismatch.Source: §1 · p. 4
• Across four pretrained CNNs, Gride–Profiled layer-wise profiles share the rise-and-fall course of two-nearest-neighbor and maximum-likelihood estimators; at the Profiled peak, Full estimates are 100–214 dimensions lower.On natural images, amplitude-reducing normalization moves angular location toward the reference.Source: §1 · p. 4

2.1. Local geometric estimators and their calibration

• TWO-NN uses the first two neighbor distances, while Gride extends the ratio to generic orders; larger orders trade locality for robustness to short-scale noise.Angular estimators use cosine-similarity moments or simplex geometry, while DANCo combines von Mises angle modeling with distance evidence.Source: §2.1 · p. 5

2.2. Intrinsic dimension in learned representations

• Finite-sample corrections include rescaling local estimates and combining neighborhood sizes; distribution matching compares statistics with finite-sample reference distributions.Prior work links layer-wise ID profiles to representation properties, but rising nearest-neighbor estimates can reflect distance ratios and representation expansion rather than increasing underlying dimension [30].Source: §2.1 · p. 5, §2.2 · p. 6

3.1. Local statistics and DANCo calibration

• DANCo calibrates first-neighbor distance and angular statistics against simulated candidate-dimension references, combining their discrepancies into one objective.Its angular model uses a von Mises distribution, with mean direction locating angles and concentration describing their spread.Source: §3.1 · pp. 7–8

3.2. Componentwise model-based calibration

• Separate distance and angular discrepancy curves expose each component’s influence on the combined minimum, with MiND–Full as the original DANCo objective.The four variants change the distance statistic between MiND and Gride and angular matching between Full and Profiled.Source: §3.2 · p. 10, Figure 1 · p. 10

3.3. Generic-order distance calibration

• Generic-order Gride ratios replace the first-neighbor distance statistic to reduce sensitivity to short-scale noise, with a closed-form KL divergence for calibration.For integer neighbor orders, the divergence reduces to a finite sum of digamma terms.Source: §3.3 · pp. 11–12

3.4. Angular concentration and profiled divergence

• Two angular regimes distinguish concentration from mean direction, motivating profiling that aligns location while retaining concentration matching.In the thin-shell regime, angular location can also shift with sample amplitude, while profiling removes the location penalty from the angular discrepancy.Source: §3.4 · pp. 14–16

3.5. Numerical evaluation

• Scaled Bessel evaluation keeps angular objectives finite through candidate dimension 400, and precomputed reference surfaces make full evaluation practical.At sample size 500, ambient dimension 400, and ten neighbors, evaluation through candidate 400 finishes in well under one second on both CPU and GPU.Source: §3.5 · pp. 16–17

4.1. Experimental design and metrics

• The experiments compare clean-manifold accuracy, spacing-scaled noise, a controlled d=70 GSM embedded in D=100, natural images, and CNN activations, with layer-wise candidates reaching 400.The GSM holds dimension and reference replicate fixed while varying sample amplitude.Source: §4 · p. 17, §4.1 · p. 18, §4.5 · p. 25

4.2. Clean-manifold accuracy

• 6.33% MPE makes MiND–Full the clean-manifold benchmark leader across 24 manifolds; TWO-NN follows at 11.12% and MiND–ML at 13.58%.The benchmark uses 20 data replicates per estimator; MiND–Full's median time excludes shared reference construction.Source: §4.2 · p. 19, Table 1 · p. 20
• On 14 matched clean manifolds, MiND–Full reaches 8.6 ± 0.7% MPE, below MiND distance-only at 10.1 ± 0.2% and Full angle-only at 15.9 ± 0.7%.MiND–Profiled also beats its corresponding component-only objectives, with 9.7 ± 0.2% MPE versus 10.1 ± 0.2% for MiND distance-only and 12.2 ± 0.3% for Profiled angle-only.Source: §4.2 · pp. 20–21

4.3. Robustness to neighborhood-relative noise

• At η=0.4, Gride–Full's MPE is 17.64% versus 27.74% for MiND–Full; on clean data, MiND–Full leads 6.75% to 9.39%.Gride-based combined objectives degrade more slowly as neighborhood-relative noise increases.Source: §4.3 · p. 21, Figure 3 · p. 21

4.4.1. Known-dimension Gaussian scale mixture

• At σS=0.25, MiND–Full estimates 22.80 ± 1.76 with 67.43% MPE; MiND–Profiled stays between 62.47 and 73.70 across nonzero amplitude levels.Profiling removes the angular-location penalty; removing known simulated amplitudes attributes the Full shortfall to amplitude heterogeneity.Source: §4.4.1 · pp. 22–23

4.4.2. Natural images and normalization

• CIFAR-10 and ImageNet mean directions fall below their reference ranges and below one, alongside larger Full–Profiled separations and distinct Full estimates, including 18.78 versus 15.80 on CIFAR-10 bird.The observed pattern is consistent with angular-location mismatch, which makes the Full estimate depend on the distance statistic.Source: §4.4.2 · pp. 23–24
• For ImageNet koala, contrast normalization moves ν̂ from 0.934 to 1.080 and narrows the MiND–Full/MiND–Profiled gap from 20.1 to 2.2 dimensions.The corresponding estimates shift from 24.9/45.0 to 72.2/70.0; normalization moves angular location toward the reference range in each CIFAR-10 and ImageNet class reported.Source: §4.4.2 · p. 25

4.5. Layer-wise profiles in convolutional neural networks

• Across four CNNs, Gride–Profiled, TWO-NN, and MLE rise through intermediate layers and decline toward the output; the largest median Full–Profiled separation occurs at the Profiled peak.At those peaks, ν̂ is 0.80–0.92 and Gride–Full reports 17–37 dimensions.Source: §4.5 · pp. 25–26

5. Discussion and conclusion

• The conclusion links generic-order Gride ratios to neighborhood-relative noise and angular-location profiling to amplitude heterogeneity.In high-dimensional thin-shell settings, amplitude shifts angular location, making profiling relevant there as well.Source: §5 · p. 27
• Profiling raises MiND from 22.80 ± 1.76 to 66.67 ± 1.81 against generating dimension 70; removing known amplitudes restores MiND–Full calibration.Full and Profiled are complementary: retain angular location when references match finite-sample geometry, and align it when location mismatch obscures concentration evidence.Source: §5 · p. 27
• Across four CNNs, the largest median Full–Profiled separation occurs at the Profiled peak, where Full estimates are 100–214 dimensions lower.On CIFAR-10 and ImageNet, amplitude-reducing normalization moves angular location toward the reference range and narrows the separation.Source: §5 · p. 28
• The component framework can accommodate other statistics when dimension-indexed reference laws and computable discrepancies are available.Suggested extensions include dependence-aware weights, adaptive neighbor orders, and amplitude-aware angular references.Source: §5 · p. 28

CRediT authorship contribution statement; Funding; Declaration of competing interest; in the manuscript preparation process

• The authors report their contribution roles, NSTC grant support, no competing interests, and use of Claude and OpenAI Codex for manuscript preparation.They state that the funder had no role in the study and that the authors reviewed and take responsibility for the published content.Source: CRediT authorship contribution statement · p. 28, Funding · p. 28, Declaration of competing interest · p. 29

Data and code availability

• The code is planned for public release on Zenodo upon acceptance; datasets are public, but raw images are not redistributed.Processed results and access instructions are to be included in the archive.Source: Data and code availability · p. 29

Supplementary Material: Interpretable intrinsic dimension estimation through

• The supplement follows the notation and method definitions established in the main paper.It defines d, D, and m as generating, ambient, and candidate dimension, and mmax as the candidate bound.Source: Supplementary Material: Interpretable… · p. 36
• The supplement uses k1 < k2 for Gride neighbor orders and states conventions for convergence in probability, asymptotic order, the identity matrix, and i.i.d.These conventions specify Yd p−→c, h(x) = O{g(x)}, Id, and independent and identically distributed.Source: Supplementary Material: Interpretable… · p. 36
• Its supporting sections provide derivations, computational and numerical-stability details, experimental designs, and extended evidence with component-level interpretations.Source: Supplementary Material: Interpretable… · p. 36

S1.1 Gride Kullback–Leibler divergence

• The generic-order Gride calibration derives KL divergence from beta-prime log moments plus a reference-side expectation, available in closed form for integer neighbor orders or by quadrature.When the fitted distance laws match (γ_m = 1), the divergence is zero.Source: S1.1 Gride Kullback–Leibler divergence · pp. 37–39

S1.2 Angular limits and the amplitude mechanism

• Mean angle is regime-dependent: local symmetry gives π/2 for every d ≥ 2, whereas fixed-N, high-dimensional samples with common amplitudes converge to π/3.The local-symmetry result applies to independent uniform directions; the π/3 limit holds for fixed N as d grows, including uniform unit-ball samples.Source: S1.2 Angular limits and the amplitude… · pp. 39, 41
• When both selected-neighbor amplitudes are below the center amplitude, the limiting angle is below π/3, even when the neighbor amplitudes differ.This amplitude-dependent limit explains why amplitude heterogeneity can shift angular location away from the common-amplitude reference.Source: S1.2 Angular limits and the amplitude… · pp. 39, 41
• In a GSM sample at σ_S = 0.25, mean direction decreases across centered-norm deciles from 1.031 to 0.668 radians.Figure S1 reports per-center mean direction by centered-norm decile, with within-decile standard deviations.Source: Figure S1 · p. 42, S1.2 Angular limits and the amplitude… · p. 42

S1.3 Profiled circular von Mises divergence

• Profiling aligns the mean directions, removing exactly the angular-location penalty while retaining the concentration comparison.For a fixed nonzero offset and positive estimated concentration, the penalty increases with reference concentration; across candidates it depends on both offset and concentration.Source: S1.3 Profiled circular von Mises… · p. 43

S1.4 Adequacy of the circular von Mises approximation

• Under local symmetry, the exact angle density is symmetric about π/2 for every dimension and cannot express the observed data–reference difference in angular location.The von Mises family is fitted separately to pooled observed and reference neighbor angles to assess descriptive fit; these fits differ from the aggregated calibration statistics.Source: S1.4 Adequacy of the circular von… · p. 43
• Outside-range mass was at most 5.0 × 10−8, and CDF gaps ranged from 0.004 to 0.022 across four settings, supporting the pooled fits but not the aggregated calibration parameters.For raw ImageNet koala, the pooled fit had τ = 39.98, whereas calibration used ˆτ = 90.1; its dominant data–reference discrepancy was in angular location.Source: S1.4 Adequacy of the circular von… · p. 44, Figure S2 · p. 44

S2.1 Algorithm

• The algorithm fits distance ratios first, conditionally computes angular statistics, evaluates reference discrepancies across candidate dimensions, and minimizes each component and combined curve.It returns the distance estimate alone when that estimate is at most five or the circular resultant vector is zero.Source: S2.1 Algorithm · pp. 44–45

S2.2 Computational complexity

• Observed-data time is dominated by exact neighbor search O(N^2D) and angle construction O(Nk^2D), while candidate-loop cost depends on the reference source.Precomputed spline references reduce online candidate evaluation to O(mmax); reused Monte Carlo references incur their simulation cost before timing.Source: S2.2 Computational complexity · pp. 45–46

S2.3 Numerical implementation and safeguards

• Exponentially scaled Bessel evaluation keeps angular KL objectives finite and positive through candidate 400, avoiding ordinary evaluation's overflow near candidate 278 and negative values from 284.Fractional refinement is restricted to [1, mmax], and undefined angular mean direction triggers a distance-only estimate.Source: S2.3 Numerical implementation and… · pp. 46–47

S2.4 Computational scaling by workload

• At mmax = 400, precomputed and reused references finish below 0.5 s on both backends, while newly generated Monte Carlo references take seconds and grow with the candidate bound.The measured workloads distinguish the cost of reference construction from the cost of evaluating candidates.Source: S2.4 Computational scaling by workload · p. 47

S3.1 Common settings

• Synthetic, noise, GSM, and image comparisons reuse Monte Carlo reference replicates across objectives and configurations, with independent fresh data replicates and MPE reported as replicate mean ± standard deviation.Clean MiND–Full MPE differs slightly across experiment designs because replicate counts and settings differ.Source: S3.1 Common settings · p. 48

S3.2 Clean-manifold benchmark

• The clean benchmark compares ten estimators on 24 manifolds with 2,500 points each across 20 matched data replicates; component comparisons use five reference replicates on 14 eligible manifolds.The other ten manifolds omit angular calibration when the preliminary MiND distance estimate is at most five.Source: S3.2 Clean-manifold benchmark · pp. 48–49

S3.3 Sensitivity sweeps

• Sensitivity sweeps vary neighborhood size k over 5, 10, 15, and 20, and sample size N over 625, 1,250, 2,500, and 5,000, using five replicates per setting.The neighborhood sweep fixes N = 2,500; the sample-size sweep fixes k = 10.Source: S3.3 Sensitivity sweeps · p. 49

S3.4 Neighborhood-relative noise

• The noise design evaluates η = 0, 0.05, 0.1, 0.2, and 0.4 with ten replicates of 2,500 observations per manifold, sharing clean data within each replicate across noise levels.This matching isolates the added noise as the changing factor.Source: S3.4 Neighborhood-relative noise · p. 49

S3.5 Gaussian scale mixture

• The GSM experiment embeds dimension-70 samples in 100 dimensions and varies lognormal amplitude across 30 replicates, then divides observations by known amplitudes in a paired control.The control retains latent Gaussian draws and paired references while recomputing neighbor searches, distance ratios, and angles.Source: S3.5 Gaussian scale mixture · p. 49

S3.6 Image data

• Image experiments use fixed samples from six MNIST, CIFAR-10, and ImageNet classes, recomputing observed statistics after transformations and averaging estimates over five reference replicates.The candidate range is m = 6 to 200; the reported reference mean-direction ranges are matched to sample size.Source: S3.6 Image data · pp. 49–50, Table S1 · p. 50

S3.7 CNN study

• The CNN study evaluates AlexNet, VGG-16, and ResNet-18/34 on seven ImageNet categories using three 500-image subsamples per architecture and category, with candidates up to 400.Gride–Profiled, TWO-NN, and MLE use common activation matrices after exact duplicate rows are removed.Source: S3.7 CNN study · p. 50

S4.1 Benchmark detail and reference estimators

• Across the 24-manifold clean benchmark, Table S2 gives estimates for ten estimators; FisherS fails on 19 replicates for M10d Cubic, with its reported estimate based on one success.The table reports each estimator's mean over successful replicates out of 20.Source: S4.1 Benchmark detail and reference… · p. 50, Table S2 · p. 51

S4.2 Clean component agreement

• All four combined objectives have lower mean MPE than their corresponding components on matched clean geometry, while combined objectives' mean signed errors lie within 2.6 percentage points of zero.Both distance-only objectives underestimate and both angle-only objectives overestimate; combined errors are closer to distance-only signed errors.Source: S4.2 Clean component agreement · p. 52, Table S3 · p. 52

S4.3 Sensitivity to neighborhood and sample size

• Gride–Full and Gride–Profiled track closely across neighborhood and sample-size sweeps, with a largest difference of 0.92 MPE points; TWO-NN is constant across k by construction.Profiled is lower at k = 5, 10 and N = 625, 1,250, 2,500; Full is lower at k = 15, 20 and N = 5,000.Source: S4.3 Sensitivity to neighborhood and… · p. 54, Table S4 · p. 54

S4.4 Neighborhood-relative noise

• At η = 0.4, Gride–Full has 23.4% MPE versus 37.6% for MiND–Full on the restricted 17-manifold subset, and Gride–Full overtakes MiND–Full between η = 0.05 and 0.1.The advantage persists through the largest evaluated noise level, consistent with wider neighbor orders spreading displacement over a longer baseline.Source: S4.4 Neighborhood-relative noise · p. 55, Table S5 · p. 55

S4.5 Profiling and distance drift under amplitude heterogeneity

• Across five replicates at σS = 0.25, where no candidate reference mean direction lies below observed, mean angular-location penalty rises from 1.28 to 14.76 nats over m = 6 to 100, pulling Full minimum below distance evidence.Source: S4.5 Profiling and distance drift… · p. 55

S4.6 Angular-reference transfer and normalization on images

• On CIFAR-10 and ImageNet, both normalizations move observed mean direction toward the reference range and narrow the Full–Profiled separation; raw koala and butterfly correlations are −0.60.Contrast normalization reduces the koala correlation between centered norm and per-center mean direction to −0.03; the observations support amplitude variation as a substantial contributor to image angular shift.Source: S4.6 Angular-reference transfer and… · p. 57, Table S8 · p. 57, Figure S7 · p. 58

S4.7 Layer-wise CNN profiles

• Across four CNNs, the largest median paired Full–Profiled difference coincides with the Gride–Profiled peak in every architecture, while output-layer paired differences are at most six dimensions.Mean direction rises from 0.78–0.87 at first pooling checkpoints to 1.04–1.11 at output layers.Source: S4.7 Layer-wise CNN profiles · pp. 58–59, Table S10 · p. 59