Abstract Let λ 1 , … , λ n denote the eigenvalues of a n × n matrix, ordered by nonincreasing absolute value, and let γ 1 ≥ ⋯ ≥ γ n denote the tropical eigenvalues of an associated n × n matrix, obtained by replacing every entry of the original matrix by its absolute value. We show that for all 1 ≤ k ≤ n , | λ 1 ⋯ λ k | ≤ C n , k γ 1 ⋯ γ k , where C n , k is a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1 .
We note that a different generalization of the Hadamard-Ostrowski-P 'olya theorem, dealing with the case of matrix polynomials, not relying on tropical eigenvalues, but thinking of the norm as a valuation'', appeared recently in @cite_4 , refining a result of @cite_24 . Tropical eigenvalues generally lead to tighter estimates in the case of structured or sparse matrices.
@cite_24 @cite_4
1856442002 1818090810 2122327334 2110355775
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups. We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and PA³lya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated. Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Grobner bases taking into account the valuation of K. Because of the use of the valuation, this theory is promising for stable computations over polynomial rings over a p-adic fields. We design a strategy to compute such tropical Grobner bases by adapting the Matrix-F5 algorithm. Two variants of the Matrix-F5 algorithm, depending on how the Macaulay matrices are built, are available to tropical computation with respective modifications. The former is more numerically stable while the latter is faster. Our study is performed both over any exact field with valuation and some inexact fields like Qp or Fq [[t]]. In the latter case, we track the loss in precision, and show that the numerical stability can compare very favorably to the case of classical Grobner bases when the valuation is non-trivial. Numerical examples are provided. Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that promote the individual structures are known, and allow for recovery using an orderwise optimal number of measurements (e.g., @math norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, using multiobjective optimization with these norms can do no better, orderwise, than exploiting only one of the structures, thus revealing a fundamental limitation in sample complexity. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation. Further, specializing our results to the case of sparse and low-rank matrices, we show that a nonconvex formulation recovers the model from very few measurements (on the order of the degrees of freedom), whereas the convex problem combining the @math and nuclear norms requires many more measurements, illustrating a gap between the performance of the convex and nonconvex recovery problems. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give sample complexity bounds for problems such as sparse phase retrieval and low-rank tensor completion.