Study of Manifold Geometry using Multiscale Non-Negative Kernel Graphs
This work addresses the need for better understanding of data manifold geometry in machine learning, but it appears incremental as it builds on existing graph-based methods.
The authors tackled the problem of analyzing the geometric structure of high-dimensional data by proposing a framework that uses non-negative kernel regression graphs to estimate point density, intrinsic dimension, and manifold curvature, demonstrating effectiveness on synthetic and real datasets.
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.