Geometric Interpretation of the Robustness of Taxicab Singular Value Decomposition of High-Dimensional Sparse Data

Abstract

This paper examines the fundamental properties of projections in L2-normed (Euclidean) and L1-normed (Taxicab) spaces through two complementary frameworks : projection operators and distance minimization. The problem may be viewed as the most elementary regression problem, namely the projection of one point onto another. Its relevance stems from the fact that any SVD-like matrix decomposition can be interpreted as a pair of simultaneous regressions of the rows and columns of a data matrix within either Euclidean or Taxicab geometry. Whereas orthogonality (L2-conjugacy) is the key concept in Euclidean geometry, L1-conjugacy, characterized by the sign function, assumes an analogous role in Taxicab geometry. This contrast is further illustrated by the relationship between the classical Pythagorean theorem and its Taxicab counterpart. The primary goal of this study is to compare three decompositions : the classical singular value decomposition (SVD), the Taxicab singular value decomposition (TSVD), and the L1-min SVD. We also provide a geometric explanation for the robustness of TSVD in the analysis of extremely sparse, high-dimensional datasets by comparing the volumes of hyperspheres inscribed in hypercubes under the two geometries.