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2 results

  • 4 - Subspaces, Rank, and Nearest-Subspace Classification

  • Jeffrey A. Fessler, University of Michigan, Ann Arbor, Raj Rao Nadakuditi, University of Michigan, Ann Arbor
  • Book:
    Linear Algebra for Data Science, Machine Learning, and Signal Processing
    Published online:
    01 November 2024
    Print publication:
    16 May 2024, pp 96-142

  • Summary

    An important operation in signal processing and machine learning is dimensionality reduction. There are many such methods, but the starting point is usually linear methods that map data to a lower-dimensional set called a subspace. When working with matrices, the notion of dimension is quantified by rank. This chapter reviews subspaces, span, dimension, rank, and nullspace. These linear algebra concepts are crucial to thoroughly understanding the SVD, a primary tool for the rest of the book (and beyond). The chapter concludes with a machine learning application, signal classification by nearest subspace, that builds on all the concepts of the chapter.

  • Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms

  • Part of
  • Ángel Chávez, Stephan Ramon Garcia, Jackson Hurley
  • Journal:
    Canadian Mathematical Bulletin / Volume 67 / Issue 2 / June 2024
    Published online by Cambridge University Press:
    06 November 2023, pp. 447-457
    Print publication:
    June 2024
    • Article
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  • We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to extend the main theorem in Chávez, Garcia, and Hurley (2023, Canadian Mathematical Bulletin 66, 808–826) from exponent $d\geq 2$ to $d \geq 1$. Our proofs are much simpler than the originals: they do not require Lewis’ framework for group invariance in convex matrix analysis. This clarification puts the entire theory on simpler foundations while extending its range of applicability.