# On Manifold Hypothesis: Hypersurface Submanifold Embedding Using Osculating Hyperspheres

arXiv: https://arxiv.org/abs/2202.01619 · HTML: https://mlsanity.com/paper/2202.01619/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 21 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 3 of 3)

- [Generalizing from a Few Examples: A Survey on Few-Shot Learning](https://mlsanity.com/paper/1904.05046/) (2019)
- [Unsupervised and Supervised Principal Component Analysis: Tutorial](https://mlsanity.com/paper/1906.03148/) (2019)
- [Johnson-Lindenstrauss Lemma, Linear and Nonlinear Random Projections, Random Fourier Features, and Random Kitchen Sinks: Tutorial and Survey](https://mlsanity.com/paper/2108.04172/) (2021)

## Built on by (most cited first, 2 of 2)

- [Variational autoencoders with latent high-dimensional steady geometric flows for dynamics](https://mlsanity.com/paper/2410.10137/) (2024)
- [Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations](https://mlsanity.com/paper/2605.02279/) (2026)

## Abstract (arXiv)

Consider a set of n data points in the Euclidean space ℝd. This set is called dataset in machine learning and data science. Manifold hypothesis states that the dataset lies on a low-dimensional submanifold with high probability. All dimensionality reduction and manifold learning methods have the assumption of manifold hypothesis. In this paper, we show that the dataset lies on an embedded hypersurface submanifold which is locally (d-1)-dimensional. Hence, we show that the manifold hypothesis holds at least for the embedding dimensionality d-1. Using an induction in a pyramid structure, we also extend the embedding dimensionality to lower embedding dimensionalities to show the validity of manifold hypothesis for embedding dimensionalities 1, 2, dots, d-1. For embedding the hypersurface, we first construct the d nearest neighbors graph for data. For every point, we fit an osculating hypersphere Sd-1 using its neighbors where this hypersphere is osculating to a hypothetical hypersurface. Then, using surgery theory, we apply surgery on the osculating hyperspheres to obtain n hyper-caps. We connect the hyper-caps to one another using partial hyper-cylinders. By connecting all parts, the embedded hypersurface is obtained as the disjoint union of these elements. We discuss the geometrical characteristics of the embedded hypersurface, such as having boundary, its topology, smoothness, boundedness, orientability, compactness, and injectivity. Some discussion are also provided for the linearity and structure of data. This paper is the intersection of several fields of science including machine learning, differential geometry, and algebraic topology.