Study of Manifold Geometry using Multiscale Non-Negative Kernel Graphs

TL;DR

This paper introduces a multiscale non-negative kernel graph framework to analyze the geometric structure of high-dimensional data, estimating density, dimension, and curvature effectively.

Contribution

It presents a novel multiscale graph construction method using NNK regression graphs for detailed geometric analysis of data manifolds.

Findings

Effective estimation of local data geometry on synthetic datasets

Outperforms baselines in manifold curvature estimation

Applicable to real-world high-dimensional data

Abstract

Modern machine learning systems are increasingly trained on large amounts of data embedded in high-dimensional spaces. Often this is done without analyzing the structure of the dataset. In this work, we propose a framework to study the geometric structure of the data. We make use of our recently introduced non-negative kernel (NNK) regression graphs to estimate the point density, intrinsic dimension, and the linearity of the data manifold (curvature). We further generalize the graph construction and geometric estimation to multiple scale by iteratively merging neighborhoods in the input data. Our experiments demonstrate the effectiveness of our proposed approach over other baselines in estimating the local geometry of the data manifolds on synthetic and real datasets.