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Preserving positivity for rank-constrained matrices

Dominique Guillot, Apoorva Khare, Bala Rajaratnam

Abstract

Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations for a fixed value of $n$ are difficult to obtain, and in fact are only known in the $2 \times 2$ case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping $n \times n$ positive semidefinite matrices of rank at most $l$ into positive semidefinite matrices of rank at most $k$ for $1 \leq l \leq n$ and $1 \leq k < n$. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner [Trans. Amer. Math. Soc. 136, 1969] can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.

Preserving positivity for rank-constrained matrices

Abstract

Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension . Such characterizations for a fixed value of are difficult to obtain, and in fact are only known in the case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping positive semidefinite matrices of rank at most into positive semidefinite matrices of rank at most for and . We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner [Trans. Amer. Math. Soc. 136, 1969] can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.

Paper Structure

This paper contains 12 sections, 40 theorems, 81 equations.

Table of Contents

  1. Introduction and main results
  2. Previous results and extensions
  3. Preserving positivity under rank constraints I: The rank $1$ case
  4. A three-step approach: functions preserving positivity of rank 1 matrices
  5. Preserving positivity of rank 1 matrices and Laplace transforms
  6. Two-sided extensions of power functions
  7. Preserving positivity under rank constraints II: The special rank $2$ case
  8. Preserving positivity under rank constraints III: The higher rank case
  9. Proof of the main Theorem \ref{['Tltok']}
  10. The regime $1 \leq k < l$
  11. The regime $l \leq k < 2l$
  12. Preserving positivity and absolutely monotonic functions

Key Result

Theorem A

Let $0 < R \leq \infty$, $I = [0, R)$ or $(-R,R)$, and $f: I \rightarrow \mathbb{R}$. Fix $1 \leq k < n$, and suppose $f$ admits at least $k$ nonzero derivatives at $0$.

Theorems & Definitions (92)

  • Definition 1.1
  • Remark 1.2
  • Theorem A: Rank $1$, fixed dimension
  • Theorem B: Higher rank, fixed dimension
  • Definition 1.3
  • Theorem C: Rank $2$, arbitrary dimension
  • Remark 1.4
  • Theorem 2.1: see Schoenberg Schoenberg42, Rudin Rudin59, Herz Herz63, Christensen and Ressel Christensen_et_al78
  • Theorem 2.2: Vasudeva, vasudeva79
  • Theorem 2.3
  • ...and 82 more