# Computationally Efficient Learning of Statistical Manifolds

arXiv: https://arxiv.org/abs/2103.11773 · HTML: https://mlsanity.com/paper/2103.11773/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 69 references that are in the map.
- First posted on arXiv 22 Feb 2021.

## Builds on (most cited first, 7 of 7)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [ANN-Benchmarks: A Benchmarking Tool for Approximate Nearest Neighbor Algorithms](https://mlsanity.com/paper/1807.05614/) (2018)
- [Efficient Manifold and Subspace Approximations with Spherelets](https://mlsanity.com/paper/1706.08263/) (2017)
- [Perturbation Bounds for Procrustes, Classical Scaling, and Trilateration, with Applications to Manifold Learning](https://mlsanity.com/paper/1810.09569/) (2018)
- [Integrated Principal Components Analysis](https://mlsanity.com/paper/1810.00832/) (2018)
- [A slice tour for finding hollowness in high-dimensional data](https://mlsanity.com/paper/1910.10854/) (2019)

## Abstract (arXiv)

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of probability distributions rather than vector-valued variables are available or when data size is large. We resolve this problem by proposing a new method for approximation in statistical manifold learning. The novelty of our approximation is the strongly consistent distance estimators based on independent and identically distributed samples from probability distributions. By exploiting the connection between Hellinger/total variation distance for discrete distributions and the L2/L1 norm, we demonstrate that the proposed distance estimators, combined with approximate nearest neighbor searching, could largely improve the computational efficiency with little to no loss in the accuracy of manifold embedding. The result is robust to different manifold learning algorithms and different approximate nearest neighbor algorithms. The proposed method is applied to learning statistical manifolds of electricity usage. This application demonstrates how underlying structures in high dimensional data, including anomalies, can be visualized and identified, in a way that is scalable to large datasets.