Improving Metric Dimensionality Reduction with Distributed Topology
We propose a novel approach to dimensionality reduction combining techniques of metric geometry and distributed persistent homology, in the form of a gradient-descent based method called DIPOLE. DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. By fixing an initial embedding method (we use Isomap), DIPOLE can also be viewed as a full dimensionality-reduction pipeline. This framework is based on the strong theoretical and computational properties of distributed persistent homology and comes with the guarantee of almost sure convergence. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets, both visually and in terms of precise quantitative metrics.
Code (1)
Tasks
Dimensionality ReductionSimilar Papers 제목 키워드 기반
FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bu…
Dimensionality ReductionDiRe-RAPIDS: Topology-faithful dimensionality reduction at scale
Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We sh…
Dimensionality ReductionFeature Space Topology Control via Hopkins Loss
Feature space topology refers to the organization of samples within the feature space. Modifying this topology can be beneficial in machine learning applications, including dimensionality reduction, generative modeling, …
Dimensionality ReductionTransfer LearningSurvey: Geometric Foundations of Data Reduction
This survey is written in summer, 2016. The purpose of this survey is to briefly introduce nonlinear dimensionality reduction (NLDR) in data reduction. The first two NLDR were respectively published in Science in 2000 in…
Dimensionality ReductionSurveyProbabilistic Geometric Principal Component Analysis with application to neural data
Dimensionality reduction is critical across various domains of science including neuroscience. Probabilistic Principal Component Analysis (PPCA) is a prominent dimensionality reduction method that provides a probabilisti…
Dimensionality Reduction