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 (2) their rank-restricted versions are smooth optimization problems on a Riemannian manifold. Combining insights from both the Riemannian and the convex geometries of the problem, we characterize when second-order critical points of the smooth problem reveal KKT points of the semidefinite problem. We compare against state of the art, mature software and find that, on certain interesting problem instances, what we call the staircase method is orders of magnitude faster, is more accurate and scales better. Code is available.
Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and two new, block-diagonal hierarchies are proposed. They have the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. Our construction is applied to the stable set problem and experimental results for Paley graphs are reported.
We study a class of quadratically constrained quadratic programs (QCQPs), called {\em diagonal QCQPs\/}, which contain no off-diagonal terms $x_j x_k$ for $j \ne k$, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-time…
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.
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.
We study semidefinite programs with diagonal constraints. This problem class appears in combinatorial optimization and has a wide range of engineering applications such as in circuit design, channel assignment in wireless networks, phase recovery, covariance matrix estimation, and low-order controller design. In this paper, we give an algorithm to solve this problem to $\varepsilon$-accuracy, with a run time of $\widetilde{\mathcal{O}}(m/\varepsilon^{3.5})$, where $m$ is the number of non-zero entries in the cost matrix. We improve upon the previous best run time of $\widetilde{\mathcal{O}}(m/…
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most $k$. We show that this problem is $\NP$-hard for any fixed integer $k\ge 2$. Equivalently, for $k\ge 2$, it is $\NP$-hard to test membership in the rank constrained elliptope $\EE_k(G)$, i.e., the set of all partial matrices with off-diagonal entries specified at the edges of $G$, that can be completed to a positive semidefinite matrix of rank at most $k$. Additionally, we show that deciding membership in the conve…
In this paper, we study randomized and cyclic coordinate descent for convex unconstrained optimization problems. We improve the known convergence rates in some cases by using the numerical semidefinite programming performance estimation method. As a spin-off we provide a method to analyse the worst-case performance of the Gauss-Seidel iterative method for linear systems where the coefficient matrix is positive semidefinite with a positive diagonal.
The joint approximate diagonalization of non-commuting symmetric matrices is an important process in independent component analysis. This problem can be formulated as an optimization problem on the Stiefel manifold that can be solved using Riemannian optimization techniques. Among the available optimization techniques, this study utilizes the Riemannian Newton's method for the joint diagonalization problem on the Stiefel manifold, which has quadratic convergence. In particular, the resultant Newton's equation can be effectively solved by means of the Kronecker product and the vec and veck oper…
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,…
Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables and the number of constraints increases. To improve scalability, SDPs can be approximated with lower and upper bounds through the use of structured subsets (e.g., diagonally-dominant and scaled-diagonally dominant matrices). Meanwhile, any underlying sparsity or symmetry structure may be leveraged to form an equivalent SDP with smaller posi…
We develop a practical approach to semidefinite programming (SDP) that includes the von Neumann entropy, or an appropriate variant, as a regularization term. In particular we solve the dual of the regularized program, demonstrating how a carefully chosen randomized trace estimator can be used to estimate dual gradients effectively. We also introduce specialized optimization approaches for common SDP, specifically SDP with diagonal constraint and the problem of the determining the spectral projector onto the span of extremal eigenvectors. We validate our approach on such problems with applicati…
Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer approximations, with the only inner approximations available being linear programming based methods proposed by Bundfuss and Dür and also Yıldırım, and a semidefinite programming based method proposed by Lasserre. In this paper, we propose the use of the cone of nonnegative scaled diagonally dominant matrices as a natural inner approximation to the complete…
The Gram dimension $\gd(G)$ of a graph $G$ is the smallest integer $k\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists). For any fixed $k$ the class of graphs satisfying $\gd(G) \le k$ is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is $K_{k+1}$ for $k\le 3$ and that there are two mini…
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…
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.
A Dual Riemannian Augmented Lagrangian Method for Low-Rank SDPs with Unit Diagonal
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Nanjing Research Institute of Electronics Technology（南京电子技术研究所）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
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…
We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction. Specifically, we show if an orthogonal projection map satisfies certain invariance conditions, restricting to its range yields an equivalent primal-dual pair over a lower-dimensional symmetric cone---namely, the cone-of-squares of a Jordan subalgebra of symmetric matrices. We present a simple algorithm for minimizing the rank of this projection and hence the dimension of this subalgebra. We also show that minimizing rank optimizes the direct-sum decomposition of the algebra into simple ideals, y…
National University of Singapore（新加坡国立大学）；Institute of High Performance Computing, Agency for Science, Technology and Research（新加坡科技研究局高性能计算研究所）；Singapore University of Technology and Design（新加坡科技设计大学）
Given a matrix $X\in \mathbb{R}_+^{m\times n}$ with nonnegative entries, a Positive Semidefinite (PSD) factorization of $X$ is a collection of $r \times r$-dimensional PSD matrices $\{A_i\}$ and $\{B_j\}$ satisfying $X_{ij}= \mathrm{tr}(A_i B_j)$ for all $\ i\in [m],\ j\in [n]$. PSD factorizations are fundamentally linked to understanding the expressiveness of semidefinite programs as well as the power and limitations of quantum resources in information theory. The PSD factorization task generalizes the Non-negative Matrix Factorization (NMF) problem where we seek a collection of $r$-dimension…
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…
In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices $X,Y \in \mathbb{R}^{n,m}$, find the matrix $A \in \mathbb{R}^{n,n}$ that minimizes the Frobenius norm of $AX-Y$ and which is such that $A+A^T$ is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where $A$ is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As…
Riemannian Gradient Descent for Gaussian Mixture Models with unknown diagonal covariances
Centrale Lyon, INSA Lyon, Université Lyon 1, Université Jean Monnet, CNRS, ICJ UMR5208（里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、让·莫内大学、法国国家科学研究中心、ICJ UMR5208）；Institut Universitaire de France (IUF)（法兰西学院）
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…
In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, …
Structured Semidefinite Programming for Recovering Structured Preconditioners
Simons Institute（西蒙斯研究所）；Stanford University（斯坦福大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）；New York University（纽约大学）；Broad Institute of MIT and Harvard（麻省理工学院和哈佛大学布罗德研究所）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
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…
There has been a lot of interest recently in proving lower bounds on the size of linear programs needed to represent a given polytope P. In a breakthrough paper Fiorini et al. [Proceedings of 44th ACM Symposium on Theory of Computing 2012, pages 95-106] showed that any linear programming formulation of maximum-cut must have exponential size. A natural question to ask is whether one can prove such strong lower bounds for semidefinite programming formulations. In this paper we take a step towards this goal and we prove strong lower bounds for a certain class of SDP formulations, namely SDPs over…
University of Liège（列日大学）；Mines ParisTech（巴黎高等矿业学院）；INRIA-Lille（里尔法国国家信息与自动化研究所）
Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fix…
Scalable Second-order Riemannian Optimization for $K$-means Clustering
Department of Statistics, University of Illinois Urbana-Champaign（统计系，伊利诺伊大学厄巴纳-香槟分校）；Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign（电气与计算机工程系，伊利诺伊大学厄巴纳-香槟分校）；Department of Mathematics, University of Southern California（数学系，南加州大学）
Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model for sparse PCA which can impose orthogonality and sparsity simultaneously. A Riemannian proximal method is proposed in the work of Chen et al. for the efficient solution of this optimization problem. In this paper, two acceleration schemes are introduced. First and foremost, we extend the FISTA method from the Euclidean space to the Riemannian man…
This paper presents the first certifiably correct algorithm for distributed pose-graph optimization (PGO), the backbone of modern collaborative simultaneous localization and mapping (CSLAM) and camera network localization (CNL) systems. Our method is based upon a sparse semidefinite relaxation that we prove provides globally-optimal PGO solutions under moderate measurement noise (matching the guarantees enjoyed by state-of-the-art centralized methods), but is amenable to distributed optimization using the low-rank Riemannian Staircase framework. To implement the Riemannian Staircase in the dis…
The so-called Burer-Monteiro method is a well-studied technique for solving large-scale semidefinite programs (SDPs) via low-rank factorization. The main idea is to solve rank-restricted, albeit non-convex, surrogates instead of the SDP. Recent works have shown that, in an important class of SDPs with elegant geometric structure, one can find globally optimal solutions to the SDP by finding rank-deficient second-order critical points of an unconstrained Riemannian optimization problem. Hence, in such problems, the Burer-Monteiro approach can provide a scalable and reliable alternative to inter…
A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems, and study its properties. When all the diagonal entries are given, the algorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if…
Semidefinite programs are generally challenging to solve due to their high dimensionality. Burer and Monteiro developed a non-convex approach to solve linear SDP problems by applying its low rank property. Their approach is fast because they used factorization to reduce the problem size. In this paper, we focus on solving the SDP relaxation of a graph equipartition problem, which involves an additional semidefinite upper bound constraint over the traditional linear SDP. By applying the factorization approach, we get a non-convex problem with an additional non-smooth spectral inequality constra…
Given a subset $\mathbf{S}=\{A_1, \dots, A_m\}$ of $\mathbb{S}^n$, the set of $n \times n$ real symmetric matrices, we define its {\it spectrahull} as the set $SH(\mathbf{S}) = \{p(X) \equiv (Tr(A_1 X), \dots, Tr(A_m X))^T : X \in \mathbfΔ_n\}$, where ${\bf Δ}_n$ is the {\it spectraplex}, $\{ X \in \mathbb{S}^n : Tr(X)=1, X \succeq 0 \}$. We let {\it spectrahull membership} (SHM) to be the problem of testing if a given $b \in \mathbb{R}^m$ lies in $SH(\mathbf{S})$. On the one hand when $A_i$'s are diagonal matrices, SHM reduces to the {\it convex hull membership} (CHM), a fundamental problem i…
The current bottleneck of globally solving mixed-integer (non-convex) quadratically constrained problem (MIQCP) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We propose a cutting surface procedure based on multiple diagonal perturbations to derive strong convex quadratic relaxations for nonconvex quadratic problem with separable constraints. Our resulting relaxation does not use significantly more variables than the original problem, in contrast to many other relaxations based on lifting. The corresponding sepa…
An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with…
