A Universal Nearest-Neighbor Estimator for Intrinsic Dimensionality

The manifold hypothesis suggests that data lies on manifolds with smaller intrinsic dimension (ID) than their ambient dimension. However there is no empirical agreement on the estimates for ID from different estimators for realistic datasets. Thus it is important to test ID estimators (IDEs) with targeted stressors. In this work, we consider the role of anisotropy. To this end, we propose HomID, a collection of homogeneous spaces with anisotropic embedding, for benchmarking ID estimators. We observe that methods that perform well on standard benchmarks systematically degrade on HomID under identical resource allocation. We further observe that anisotropic distortion of such benchmarks also results in performance degradation. Finally, we demonstrate that controlled anisotropic distortions induce systematic shifts in the distributions on which these methods rely, providing a concrete mechanism for the resulting estimation errors in two particular IDEs.

DANCo (Dimensionality from Angle and Norm Concentration) jointly calibrates nearest-neighbor distance and angular statistics and consistently reaches state-of-the-art accuracy on clean intrinsic-dimension (ID) benchmarks. Practical data, however, introduce neighborhood-relative noise and sample-amplitude heterogeneity that can distort these geometric signals. We reformulate DANCo componentwise, retaining separate distance and angular discrepancy curves so that the source of an estimate can be identified and interpreted. For the distance component, we derive a closed-form Kullback-Leibler divergence for the generic-order ratios of the generalized ratios ID estimator (Gride); when both angular parameters are matched (Full), Gride reduces mean percentage error from $27.7\%$ to $17.6\%$ at noise equal to $40\%$ of typical neighbor spacing on 24 manifolds. For the angular component, two sampling regimes motivate aligning mean direction while retaining concentration matching (Profiled). On a Gaussian scale mixture with generating dimension 70 embedded in 100 dimensions, profiling raises the Minimum Neighbor Distance (MiND) estimate from $22.8$ to $66.7$, while removing the known amplitudes restores MiND-Full to $71.9$; the control thus attributes the Full shortfall to amplitude heterogeneity. On CIFAR-10 and ImageNet, amplitude-reducing normalizations move angular location toward the references and narrow the Full-Profiled gap, an observational counterpart to the controlled mixture. Across four pretrained convolutional neural networks, Gride-Profiled, the two-nearest-neighbor estimator (TWO-NN), and the maximum-likelihood estimator (MLE) exhibit similar rise-and-fall profiles, while Full-Profiled differences identify the layers most sensitive to angular calibration.

While diffusion-based methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data, their statistical difficulty remains largely unexplored. We study estimation of the finite-scale population functional underlying FLIPD (Kamkari et al., 2024; arXiv:2406.03537), a diffusion-based local intrinsic dimension (LID) quantity defined through the logarithmic scale derivative of a Gaussian-smoothed density. Intuitively, Gaussian smoothing turns local dimension into a scale law: near a $d$-dimensional manifold, the kernel mass grows like $σ^d$, so differentiating with respect to the noise scale reveals the intrinsic exponent. Under a regular manifold model, we show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(σ^2)$. We then establish a minimax lower bound of order $(nσ^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2α/(2α+d)}$.