This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute.
Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al.\ \cite{kim2011exploiting} to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant i…
We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on the ranks. Applications include conditions for low-rank completions using the nuclear norm heuristic.
We propose a new algorithm to solve optimization problems of the form $\min f(X)$ for a smooth function $f$ under the constraints that $X$ is positive semidefinite and the diagonal blocks of $X$ are small identity matrices. Such problems often arise as the result of relaxing a rank constraint (lifting). In particular, many estimation tasks involving phases, rotations, orthonormal bases or permutations fit in this framework, and so do certain relaxations of combinatorial problems such as Max-Cut. The proposed algorithm exploits the facts that (1) such formulations admit low-rank solutions, and …
A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite rank of $M$, and it is an open question whether there exists an upper bound on this number as a function of the matrix size. We construct completely positive semidefinite matrices of size $4k^2+2k+2$ with complex completely positive semidefinite rank $2^k$ for any positive integer $k$. This shows that if such an upper bound exists, it has to be at least exp…
We study the convex hull of $SO(n)$, thought of as the set of $n\times n$ orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of $SO(n)$ is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.
Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, are extended to the case of symplectic eigenvalues. In this note, we will generalize Williamson's diagonal form for symmetric positive-definite matrices to the case of symmetric positive-semidefinite matrices, which allows us to define symplectic eigenvalues, and prove the trace minimization theorem in the new setting.
A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. If it is copositive, the algorithm can get a certificate for the copositivity. If it is not, the algorithm can get a point that refutes the copositivity. We show that the detection can be done by solving a finite number of semidefinite relaxat…
We introduce a notion of matrix valued Gram decompositions for correlation matrices whose study is motivated by quantum information theory. We show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra and thus, their size is exponential in terms of the rank of the correlation matrix. Using this we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank, recently derived in the literature. This fact also underlies and generalizes Tsirelson's seminal lower bound on the local …
A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre type semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for min…
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerden's conjecture for HPSD matrices, where the polynomial optimization problem i…
In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose a…
Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS opti…
We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expa…
This article considers the recovery of low-rank matrices via a convex nuclear-norm minimization problem and presents two null space properties (NSP) which characterize uniform recovery for the case of block-diagonal matrices and block-diagonal positive semidefinite matrices. These null-space conditions turn out to be special cases of a new general setup, which allows to derive the mentioned NSPs and well-known NSPs from the literature. We discuss the relative strength of these conditions and also present a deterministic class of matrices that satisfies the block-diagonal semidefinite NSP.
The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals $[a,b]$, $[0,\infty)$; and the strips $[a,b] \times \mathbb{R} \subset \mathbb{R}^2.$ A method for numerically computing such representations is also presented. The methodology is divided into two stages: (S1) diagonalizing the initial polynomial matrix based on the Schmüdgen's procedure \cite{Schmudgen09}; (S2) for each diagonal element of the resulting matrix,…
We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose non-edges determine the prescribed zeros. Each parametrization in this class is a polynomial map associated with a simplicial complex supported on cliques of the graph. The images of the maps are convex cones, and the maps can only be surjective onto the cone of zero-constrained positive semidefinite matrices when the associated graph is chordal and the simplicial complex is the clique complex of the graph. Our main result gives a semi-algebr…
In this paper, we propose iterative inner/outer approximations based on a recent notion of block factor-width-two matrices for solving semidefinite programs (SDPs). Our inner/outer approximating algorithms generate a sequence of upper/lower bounds of increasing accuracy for the optimal SDP cost. The block partition in our algorithms offers flexibility in terms of both numerical efficiency and solution quality, which includes the approach of scaled diagonally dominance (SDD) approximation as a special case. We discuss both the theoretical results and numerical implementation in detail. Our main…
This paper studies an optimization problem on the sum of traces of matrix quadratic forms on $m$ orthogonal matrices, which can be considered as a generalization of the synchronization of rotations. While the problem is nonconvex, the paper shows that its semidefinite programming relaxation can solve the original nonconvex problems exactly, under an additive noise model with small noise in the order of $O(-m^{1/4})$, where $m$ is the number of orthogonal matrices. This result can be considered as a generalization of existing results on phase synchronization.
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension $\gd(G)$ of a graph $G$, a recently studied graph parameter related to the low psd matrix completion problem. …
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.
Given a nonnegative matrix M with rational entries, we consider two quantities: the usual positive semidefinite (psd) rank, where the matrix is factored through the cone of real symmetric psd matrices, and the rational-restricted psd rank, where the matrix factors are required to be rational symmetric psd matrices. It is clear that the rational-restricted psd rank is always an upper bound to the usual psd rank. We show that this inequality may be strict by exhibiting a matrix with psd rank four whose rational-restricted psd rank is strictly greater than four.
We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study relationships between this cone and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters $α(G)$ and $χ(G)$, which are roughly obtained by allowing variables to be positive semid…
The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point. We develop a penalty formulation of the problem. We present calmness results for locally optim…
We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+^n$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set $\mathcal Q$ of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior a…
We provide formulas for Riemannian gradient and Levi-Civita connection for a family of metrics on fixed-rank matrix manifolds, based on nonconstant metrics on Stiefel manifolds.
We consider a structured estimation problem where an observed matrix is assumed to be generated as an $s$-sparse linear combination of $N$ given $n\times n$ positive-semidefinite matrices. Recovering the unknown $N$-dimensional and $s$-sparse weights from noisy observations is an important problem in various fields of signal processing and also a relevant pre-processing step in covariance estimation. We will present related recovery guarantees and focus on the case of nonnegative weights. The problem is formulated as a convex program and can be solved without further tuning. Such robust, non-B…
Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lag…
In this note, we use a natural desingularization of the conormal variety of the variety of n x n symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.
The set of matrices of given positive semidefinite rank is semialgebraic. In this paper we study the geometry of this set, and in small cases we describe its boundary. For general values of positive semidefinite rank we provide a conjecture for the description of this boundary. Our proof techniques are geometric in nature and rely on nesting spectrahedra between polytopes.
To demonstrate the discrepancy between second-order cone and semidefinite programming, Hamza Fawzi showed that the cone S_+^3 of symmetric positive semidefinite matrices of size 3 is not second-order cone representable (socr). A slice of S_+^3 is intersection of S_+^3 and a linear sub-space of the space S^3 of 3 \times 3 symmetric matrices. It is known that some slices of S_+^3 are socr, while some others are not. We classify socr slices of S_+^3 by showing that a slice of S_+^3 is socr if and if it has dimension at most 4 or is orthogonal to a non-zero singular matrix (where the orthogonality…
A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone of $n\times n$ symmetric matrices where all $k\times k$ principal submatrices are PSD. We call a matrix in this $k$-\emph{locally PSD}. In order to compare $S^{n,k}$ to the of PSD matrices, we study eigenvalues of $k$-{locally PSD} matrices. The key insight in this paper is that there is a convex cone $H(e_k^n)$ so that if $X \in \mathcal{S}^{n,k}$, then the vector of eigenvalues…
The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $n^2$ where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $\mathcal{O}(n)$ no greater than the number of edges in one of the graphs. The relaxation can be furthe…
One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called "dimension-free". Analyzing dimension-free problems has led to the development recently of a non-commutative (nc) real algebraic geometry (RAG) which, when combined with convexity, produces di…
We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to achieve a lower rank solution. This procedure is computationally efficient as it does not require projecting on the cone of positive-semidefinite matrices. Its performance in terms of objective improvement and rank reduction is tested over multiple graphs of large-scale Gset graph collection and over binary optimization problems from the Biq Mac collection.
We show that there is an isometry between the real ambient space of all Mueller matrices and the space of all Hermitian matrices which maps the Mueller matrices onto the positive semidefinite matrices. We use this to establish an optimality result for the filtering of Mueller matrices, which roughly says that it is always enough to filter the eigenvalues of the corresponding "coherency matrix". Then we further explain how the knowledge of the cone of Hermitian positive semidefinite matrices can be transferred to the cone of Mueller matrices with a special emphasis towards optimisation. In part…
In this work, we generalize the probability simplex constraint to matrices, i.e., $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_K = \mathbf{I}$, where $\mathbf{X}_i \succeq 0$ is a symmetric positive semidefinite matrix of size $n\times n$ for all $i = \{1,\ldots,K \}$. By assuming positive definiteness of the matrices, we show that the constraint set arising from the matrix simplex has the structure of a smooth Riemannian submanifold. We discuss a novel Riemannian geometry for the matrix simplex manifold and show the derivation of first- and second-order optimization related ingredients.
We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems, possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints. In particular, the positive semidefinite variable along with a group of linear constraints can be transformed into a variable on a smooth manifold via matrix factorization. The nonsmooth regularization and other general linear constraints are handled by the augmented Lagrangian method. Therefore, each subproblem can be solved by a semi…
The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.
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.
Solving semidefinite programs (SDP) in a short time is the key to managing various mathematical optimization problems. The matrix-completion primal-dual interior-point method (MC-PDIPM) extracts a sparse structure of input SDP by factorizing the variable matrices. In this paper, we propose a new factorization based on the inverse of the variable matrix to enhance the performance of MC-PDIPM. We also use multithreaded parallel computing to deal with the major bottlenecks in MC-PDIPM. Numerical results show that the new factorization and multithreaded computing reduce the computation time for SD…
Let $D$ be the set of $n\times n$ positive semidefinite matrices of trace equal to one, also known as the set of density matrices. We prove two results on the hardness of approximating $D$ with polytopes. First, we show that if $0 < ε< 1$ and $A$ is an arbitrary matrix of trace equal to one, any polytope $P$ such that $(1-ε)(D-A) \subset P \subset D-A$ must have linear programming extension complexity at least $\exp(c\sqrt{n})$ where $c > 0$ is a constant that depends on $ε$. Second, we show that any polytope $P$ such that $D \subset P$ and such that the Gaussian width of $P$ is at most twice …
We investigate the problem of finding inner ap-proximations of positive semidefinite (PSD) cones. We developa novel decomposition framework of the PSD cone by meansof conical combinations of smaller dimensional sub-cones. Weshow that many inner approximation techniques could besummarized within this framework, including the set of (scaled)diagonally dominant matrices, Factor-widthkmatrices, andChordal Sparse matrices. Furthermore, we provide a moreflexible family of inner approximations of the PSD cone, wherewe aim to arrange the sub-cones so that they are maximallyseparated from each other. I…
