On the sign patterns of entrywise positivity preservers in fixed dimension

Khare, Apoorva; Tao, Terence

doi:10.1353/ajm.2021.0049

Cited by 10 publications

(43 citation statements)

References 31 publications

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“…Once again, this set differs vastly from the one in Theorem 6.3. (3) In a fixed dimension p ≥ 2, there are polynomial preservers with negative coefficients, as obtained in [3,39]. This line of investigation uncovered unexpected connections between positivity and Schur polynomials, which preceded the findings in Section 4.1: see [6,38].…”

Section: Preservers Of Total Positivity On Arbitrary Domainsmentioning

confidence: 61%

Preservers of totally positive kernels and Polya frequency functions

Belton,

Guillot,

Khare

et al. 2021

Preprint

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Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Pólya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial transforms, we unveil an ubiquitous separation between discrete and continuous spectra of such inner fractional powers. Classical works of Schoenberg, Karlin, Hirschman, and Widder are completed by our classification. Concepts of probability theory and group representation theory naturally enter into the picture.

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Section: Preservers Of Total Positivity On Arbitrary Domainsmentioning

confidence: 61%

Preservers of totally positive kernels and Polya frequency functions

Belton,

Guillot,

Khare

et al. 2021

Preprint

Self Cite

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“…A recent variant is the following lemma, shown by evaluating f [−] at matrices (tu j u k ) n j,k=1 and using the invertibility of "generic" generalized Vandermonde matrices. Lemma 3.9 (Belton-Guillot-Khare-Putinar [11] and Khare-Tao [89]). Let n ≥ 1 and 0 < ρ ≤ ∞.…”

Section: By Considering Fmentioning

confidence: 99%

“…Remarkably, until 2016 not a single example was known of a polynomial positivity preserver with a negative coefficient. Then, in quick succession, the two papers [11,89] provided a complete understanding of the sign patterns of entrywise polynomial preservers of P N . The goal of this chapter is to discuss some of the results in these works.…”

Section: Entrywise Polynomials Preserving Positivity In Fixed Dimensionmentioning

confidence: 99%

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A Panorama of Positivity. I: Dimension Free

Belton¹,

Guillot²,

Khare³

et al. 2019

Trends in Mathematics

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This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.

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“…We restrict here to a brief comparison of Vybíral's Theorem 1.2 with basic results in [1,7] about entrywise polynomial maps that preserve positivity on P n for fixed n. These latter say that for real matrices in P n with entries in (0, ǫ) (resp. (ǫ, ∞)) for any ǫ > 0, if an entrywise polynomial preserves positivity on such matrices of rank one, then its first (resp.…”

Section: Further Ramificationsmentioning

confidence: 99%

Sharp nonzero lower bounds for the Schur product theorem

Khare

2019

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By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product M • N of two positive semidefinite matrices M, N is again positive. Vybíral (2019) improved on this by showing the uniform lower bound M • M ≥ En/n for all n × n real or complex correlation matrices M , where En is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 1999] and to positive definite functions. Vybíral then asked if one can obtain similar uniform lower bounds for higher entrywise powers of M , or when N = M, M . A natural third question is to obtain a tighter lower bound that need not vanish, over infinitedimensional Hilbert spaces as well. In this short note, we affirmatively answer all three questions by extending and refining Vybíral's result. As a special case, the above bound of En/n can be improved to En/rk(M ). In addition, our lower bounds -which we show are tracial Cauchy-Schwartz inequalities -are sharp. We end with some consequences for positive definite functions on groups, metric spaces, and Hilbert spaces. Notation:(1) A positive semidefinite matrix is a complex Hermitian matrix with non-negative eigenvalues. Denote the space of such n × n matrices by P n = P n (C). ( 2) The Loewner ordering on n × n complex matrices is the partial order such that M ≥ N if and only if M − N ∈ P n . (3) We say that a matrix in P n is a real/complex correlation matrix if it has all diagonal entries 1, and all entries real/complex respectively. (4) The Schur product of two (possibly rectangular) m × n complex matrices A = (a ij ), B = (b ij ) equals the m × n matrix A • B with (i, j) entry a ij b ij . (5) Given a fixed integer n ≥ 1, let e = e(n) := (1, . . . , 1) T ∈ C n ; and E n := ee T ∈ P n is the matrix of all ones. (6) Given a matrix M n×n and a subset J ⊂ {1, . . . , n}, let M J×J denote the principal submatrix of M corresponding to the rows and columns indexed by J.

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On the sign patterns of entrywise positivity preservers in fixed dimension

Cited by 10 publications

References 31 publications

Preservers of totally positive kernels and Polya frequency functions

Preservers of totally positive kernels and Polya frequency functions

A Panorama of Positivity. I: Dimension Free

Sharp nonzero lower bounds for the Schur product theorem

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