IAN: Iterated Adaptive Neighborhoods for manifold learning and dimensionality estimation

TL;DR

This paper introduces IAN, an adaptive neighborhood algorithm for manifold learning that improves local structure estimation and dimensionality analysis in limited, non-uniform data scenarios.

Contribution

The paper presents a novel iterative adaptive neighborhood method that adjusts local neighborhoods based on geometry, enhancing manifold learning and dimension estimation.

Findings

Outperforms standard k-NN in manifold tasks

Effectively identifies outliers and non-pure manifold regions

Improves accuracy in geodesic and dimensionality computations

Abstract

Invoking the manifold assumption in machine learning requires knowledge of the manifold's geometry and dimension, and theory dictates how many samples are required. However, in applications data are limited, sampling may not be uniform, and manifold properties are unknown and (possibly) non-pure; this implies that neighborhoods must adapt to the local structure. We introduce an algorithm for inferring adaptive neighborhoods for data given by a similarity kernel. Starting with a locally-conservative neighborhood (Gabriel) graph, we sparsify it iteratively according to a weighted counterpart. In each step, a linear program yields minimal neighborhoods globally and a volumetric statistic reveals neighbor outliers likely to violate manifold geometry. We apply our adaptive neighborhoods to non-linear dimensionality reduction, geodesic computation and dimension estimation. A comparison against standard algorithms using, e.g., k-nearest neighbors, demonstrates their usefulness. Code for our algorithm will be available at https://github.com/dyballa/IAN