Best $\ell_1$-approximation of nonnegative polynomials by sums of squares

平方和锥与非负电路和锥的几何研究

Geometrical Study of the Cone of Sums of Squares plus Sums of Nonnegative Circuits

本文从几何角度研究SOS与SONC之和构成的凸锥，证明其正常性、非封闭性，并给出希尔伯特定理类比的新证明，表明其为两者的真超集。

Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate the analysis of optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS based certificates remain valid: First, for polynomials, which are nonnegative over the $n$…

Nonnegative sum-symmetric matrices, optimal-score partitions, and optimal resource allocation

The main result of the note describes certain optimal-score partitions, which can be interpreted as optimal resource allocations. This result is based on the fact that any nonnegative square matrix whose column sums are the same as the corresponding row sums can be represented as the sum of circuit matrices.

关于非平方和的非负多项式的奇数次幂

On odd powers of nonnegative polynomials that are not sums of squares

本文系统研究了奇数次幂非平方和的“固执”非负多项式，提出了实孤立零点的SOS-不变量，并据此证明了6次非负三元形式极射线多项式及更高次形式的固执性，同时揭示了非固执多项式构成内部为严格正多项式的凸锥。

矩、平方和与热带化

Moments, Sums of Squares, and Tropicalization

本文用热带化刻画半代数集上矩锥与伪矩锥的组合结构，借此区分非负多项式与平方和，并揭示平方和逼近的局限。

Sums of Squares: A Real Projective Story

The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety. We survey the rich history of this area and focus on recent developments highlighting unexpected links to complex algebraic geometry and commutative algebra.

Sums of Separable and Quadratic Polynomials

We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial, and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials, but negative already for bivariate quarti…

Recovering Sparse Nonnegative Signals via Non-convex Fraction Function Penalty

Many real world practical problems can be formulated as $\ell_{0}$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative signals to underdetermined linear systems. They have been widely applied in signal and image processing, machine learning, pattern recognition and computer vision. Unfortunately, this $\ell_{0}$-minimization problem with nonnegativity constraint is computational and NP-hard because of the discrete and discontinuous nature of the $\ell_{0}$-norm. In this paper, we replace the $\ell_{0}$-norm with a non-convex fraction function, and study th…

Equivalence and Strong Equivalence between Sparsest and Least $\ell_1$-Norm Nonnegative Solutions of Linear Systems and Their Application

Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that l1-minimization is efficient for solving some classes of l0-minimization problems. From a mathematical point of view, however, the understanding of the relationship between l0- and l1-minimization remains incomplete. In this paper, we further discuss several theoretical questions associated with these two problems. For instance, how to completely characterize the uniqueness of least l1-…

Refined Asymptotics for Multigraded Sums of Squares

To prove that a polynomial is nonnegative on R^n one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables -- with k>=4 fixed -- those that are SOS occupy a vanishingly small fraction of tho…

Sums of hermitian squares and the BMV conjecture

Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtai…

Strong nonnegativity and sums of squares on real varieties

Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic property is equivalent to nonnegativity for nonsingular real varieties. Moreover, for singular varieties, we reprove and generalize obstructions of Gouveia and Netzer to the convergence of the theta body hierarchy of convex bodies approximating the convex hull of a real variety.

Hyperbolic polynomials, interlacers, and sums of squares

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we write as a linear slice of the cone of nonnegative polynomials. In particular, this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials. Usi…

Sums of Squares on the Hypercube

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube C={0,1}^n, a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globa…

An Approach to Constrained Polynomial Optimization via Nonnegative Circuit Polynomials and Geometric Programming

In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third author. On the other hand, we investigate geometric programming methods for constrained polynomial optimization problems, which were recently developed by Ghasemi and Marshall. We show that the combination of both results yields a new method to solve certain classes of constrained polynomial optimization problems. We test the new method experimentally and compa…

有限半代数集上有理数域上的有效正定定理

An Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets

本文研究了有限半代数集上多项式非负性的有效表示方法，提出基于有理数平方和的构造，并给出了度数和位复杂度的界，改进了多项式优化中的矩/平方和层次结构结果。

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that $S(k,\ell) = \mbox{tr}(A_k^* B_\ell)$. The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern. In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.

完美图的平方和刻画

A Sum of Squares Characterization of Perfect Graphs

该研究给出完美图的代数刻画，证明图为完美图当且仅当关联非负多项式是平方和，还得到非平方和非负多项式族、刻画特殊图类并将完美图经典结果转化为平方和证明命题。

Computing the norm of nonnegative matrices and the log-Sobolev constant of Markov chains

We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In particular, exploiting the Birkoff--Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norm…

Efficient Tuning-Free $\ell_1$-Regression of Nonnegative Compressible Signals

In compressed sensing the goal is to recover a signal from as few as possible noisy, linear measurements. The general assumption is that the signal has only a few non-zero entries. The recovery can be performed by multiple different decoders, however most of them rely on some tuning. Given an estimate for the noise level a common convex approach to recover the signal is basis pursuit denoising. If the measurement matrix has the robust null space property with respect to the $\ell_2$-norm, basis pursuit denoising obeys stable and robust recovery guarantees. In the case of unknown noise levels, …