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A Wiener Process Perspective on Local Intrinsic Dimension Estimation Methods
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Analyzes parametric LID estimators (LIDL/FLIPD, ID-NF/ID-DM, NB) through the Wiener process and Fick's second law, deriving closed-form LIDL bias for flat manifolds with non-uniform densities.
Problem
Parametric local intrinsic dimension (LID) estimators built on noise-diffused data densities behave unpredictably when their assumptions (small diffusion time, flat manifold, nice density) fail; their errors are not well characterized analytically.
Approach
Views Gaussian noise addition as a Wiener process and applies Fick's second law to replace time derivatives of the diffused density with spatial Laplacians, yielding closed-form expressions for the LIDL estimator's bias as a function of the on-manifold density and dimension (flat manifolds, uniform/normal/hypercube/parallel-plane/intersecting cases).
Key results
- For a flat manifold with constant density the LIDL estimator is unbiased for all t>0 (beta_t = d-D).
- For a Gaussian density the LIDL bias is beta_t(x) = d-D + ((|x|^2/(sigma^2+t)^2) - d/(sigma^2+t)) t, giving large positive bias in low-probability regions.
- For a union of two parallel hyperplanes with density weights, an extra term ||v||^2/t appears whose effect is suppressed by a weight that decays exponentially in 1/t.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use density-model Laplacian estimates to improve LIDL at higher diffusion times under non-uniform densities | LIDL · Higher diffusion times with non-uniform densities | LID estimation accuracy | Helped |
Surprising findings
- Authors suspect the original LIDL paper plotted curves using squared delta by mistake, since their analytic curves differ from the published ones.
- The paper reports no experiments on real datasets because the bias term is not known to be computable for arbitrary datasets.
Limitations
- Assumes flat manifolds; curvature is not handled.
- No experiments on real datasets; the bias term is not computable for arbitrary data.
- Analysis only covers LIDL in detail; NB and holistic methods (ID-NF, ID-DM) are only discussed qualitatively.
Experimental setup
Metrics: LIDL estimate bias as a function of diffusion time t
Baselines: LIDL FLIPD ID-NF ID-DM NB
Builds on: LIDL FLIPD ID-NF ID-DM NB
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