Projections in L2 and L1 normed spaces and SVDs

TL;DR

This paper compares projections and singular value decompositions in L2 and L1 normed spaces, highlighting differences in orthogonality, conjugacy, and geometric properties like the Pythagorean theorem.

Contribution

It introduces a comparative analysis of projections and SVDs in L2 and L1 spaces, emphasizing their geometric and algebraic differences.

Findings

L2 projections rely on orthogonality, while L1 projections depend on sign functions.

The Pythagorean theorem has an L1 analogue in Taxicab geometry.

Different SVD variants are compared in terms of their properties and applications.

Abstract

We compare the essential properties of projections in the L2 and L1 normed spaces by two methods: Projection operators and by minimization of the distance. In Euclidean geometry the orthogonality (L2- conjugacy) plays central role; while in L1 normed space L1- conjugacy (the sign function) plays central role. Furthermore, this fact appears in the Pythagorean Theorem and its Taxicab analogue. We also compare three singular value decompositions: SVD, Taxicab SVD and L1min-SVD.