On Manifold Hypothesis: Hypersurface Submanifold Embedding Using Osculating Hyperspheres
This work addresses a foundational problem in machine learning and data science by providing a theoretical proof for the manifold hypothesis, which is crucial for dimensionality reduction and manifold learning methods.
The paper tackles the problem of validating the manifold hypothesis by proving that a dataset in Euclidean space lies on an embedded hypersurface submanifold of dimensionality d-1, and extends this to lower dimensions using an induction method, with the result showing the hypothesis holds for embedding dimensionalities {1, 2, ..., d-1}.
Consider a set of $n$ data points in the Euclidean space $\mathbb{R}^d$. 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 $S^{d-1}$ 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.