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Register a new userMatrix positivity preservers in fixed dimension. I
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A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with non-negative Taylor coefficients. Despite the great deal of interest generated by this theorem, a characterization of functions preserving positivity for matrices of fixed dimension is not known. In this paper, we provide a complete description of polynomials of degree $N$ that preserve positivity when applied entrywise to matrices of dimension $N$. This is the key step for us then to obtain negative lower bounds on the coefficients of analytic functions so that these functions preserve positivity in a prescribed dimension. The proof of the main technical inequality is representation theoretic, and employs the theory of Schur polynomials. Interpreted in the context of linear pencils of matrices, our main results provide a closed-form expression for the lowest critical value, revealing at the same time an unexpected spectral discontinuity phenomenon. Tight linear matrix inequalities for Hadamard powers of matrices and a sharp asymptotic bound for the matrix-cube problem involving Hadamard powers are obtained as applications. Positivity preservers are also naturally interpreted as solutions of a variational inequality involving generalized Rayleigh quotients. This optimization approach leads to a novel description of the simultaneous kernels of Hadamard powers, and a family of stratifications of the cone of positive semidefinite matrices.
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Given $Isubsetmathbb{C}$ and an integer $N>0$, a function $f:Itomathbb{C}$ is entrywise positivity preserving on positive semidefinite (p.s.d.) matrices $A=(a_{jk})in I^{Ntimes N}$, if the entrywise application $f[A]=(f(a_{jk}))$ of $f$ to $A$ is p.s.d. for all such $A$. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic [Duke Math. J. 1942, 1959]. In fixed dimension $N$, results akin to work of Horn and Loewner [Trans. AMS 1969] show the first $N$ nonzero Maclaurin coefficients of a positivity preserver $f$ are positive; and the last $N$ coefficients are also positive if $I$ is unbounded. However, little was known about the other coefficients: the only examples to date for unbounded domains $I$ were absolutely monotonic, so work in all dimensions; and for bounded $I$ examples of non-absolutely monotonic preservers were very few (and recent). In this paper, we completely characterize the sign patterns of the Maclaurin coefficients of positivity preservers in fixed dimension $N$, over bounded and unbounded domains $I$. In particular, the above Horn-type conditions cannot be improved upon. This also yields the first polynomials which preserve positivity on p.s.d. matrices in $I^{Ntimes N}$ but not in $I^{(N+1)times (N+1)}$. We obtain analogous results for real exponents using the Harish-Chandra-Itzykson-Zuber formula. We then go from qualitative bounds, which suffice to understand all possible sign patterns, to exact quantitative bounds. As an application, we extend our previous qualitative and quantitative results to understand preservers of total non-negativity in fixed dimension - including their sign patterns. We deduce several further applications, including extending a Schur polynomial conjecture by Cuttler-Greene-Skandera to obtain a novel characterization of weak majorization for real tuples.
A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenbergs 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.
A (special case of a) fundamental result of Horn and Loewner [Trans. Amer. Math. Soc. 1969] says that given an integer $n geq 1$, if the entrywise application of a smooth function $f : (0,infty) to mathbb{R}$ preserves the set of $n times n$ positive semidefinite matrices with positive entries, then the first $n$ derivatives of $f$ are non-negative on $(0,infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and further used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In parallel, in recent works with Belton-Guillot-Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, real analyt
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.
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
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