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Schur polynomials and matrix positivity preservers

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review

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Schur polynomials and matrix positivity preservers. / Belton, Alexander; Guillot, Dominique; Khare, Apoorva et al.
Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2016 Vancouver, Canada. DMTCS Proceedings, 2016. p. 155-166.

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review

Harvard

Belton, A, Guillot, D, Khare, A & Putinar, M 2016, Schur polynomials and matrix positivity preservers. in Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2016 Vancouver, Canada. DMTCS Proceedings, pp. 155-166. <https://fpsac2016.sciencesconf.org/113707>

APA

Belton, A., Guillot, D., Khare, A., & Putinar, M. (2016). Schur polynomials and matrix positivity preservers. In Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2016 Vancouver, Canada (pp. 155-166). DMTCS Proceedings. https://fpsac2016.sciencesconf.org/113707

Vancouver

Belton A, Guillot D, Khare A, Putinar M. Schur polynomials and matrix positivity preservers. In Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2016 Vancouver, Canada. DMTCS Proceedings. 2016. p. 155-166

Author

Belton, Alexander ; Guillot, Dominique ; Khare, Apoorva et al. / Schur polynomials and matrix positivity preservers. Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics: FPSAC 2016 Vancouver, Canada. DMTCS Proceedings, 2016. pp. 155-166

Bibtex

@inproceedings{9a719d500cbd4695ba886597223050a9,
title = "Schur polynomials and matrix positivity preservers",
abstract = "A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg{\textquoteright}s work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers. ",
author = "Alexander Belton and Dominique Guillot and Apoorva Khare and Mihai Putinar",
year = "2016",
month = may,
day = "25",
language = "English",
pages = "155--166",
booktitle = "Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics",
publisher = "DMTCS Proceedings",

}

RIS

TY - GEN

T1 - Schur polynomials and matrix positivity preservers

AU - Belton, Alexander

AU - Guillot, Dominique

AU - Khare, Apoorva

AU - Putinar, Mihai

PY - 2016/5/25

Y1 - 2016/5/25

N2 - A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg’s work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.

AB - A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg’s work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.

M3 - Conference contribution/Paper

SP - 155

EP - 166

BT - Proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics

PB - DMTCS Proceedings

ER -