Interpretable intrinsic dimension estimation through componentwise calibration of distance and angle

Abstract

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.