In this largely expository article, we highlight the significance of various types of ‘dimension’ for obtaining uniform convergence results in probability theory and we demonstrate how these results lead to certain notions of generalization for classes of binary-valued and realvalued functions. We also present new results on the generalization ability of certain types of artificial neural netwo...

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. Œe result is a practical scalable algorithm that applies to real world data. Œe UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of...

A dimension invariance property of finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation ...

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...