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We present a novel, practical, and provable approach for solving diagonally constrained semi-definite programming (SDP) problems at scale using accelerated non-convex programming. Our algorithm non-trivially combines acceleration motions…
Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The Dynamically…
A family of symmetric matrices $A_1,\ldots, A_d$ is SDC (simultaneous diagonalization by congruence, also called non-orthogonal joint diagonalization) if there is an invertible matrix $X$ such that every $X^T A_k X$ is diagonal. In this…
Block coordinate descent methods and stochastic subgradient methods have been extensively studied in optimization and machine learning. By combining randomized block sampling with stochastic subgradient methods based on dual averaging, we…
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic…
Singular Value Decomposition (SVD) is a fundamental matrix factorization technique in linear algebra, widely applied in numerous matrix-related problems. However, traditional SVD approaches are hindered by slow panel factorization and…
A widely used method for solving SOS (Sum Of Squares) decomposition problem is to reduce it to the problem of semi-definite programs (SDPs) which can be efficiently solved in theory. In practice, although many SDP solvers can work out some…
Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not well captured by current order…
In downlink multiuser multiple-input multiple-output (MIMO) systems, block diagonalization (BD) is a practical linear precoding scheme which achieves the same degrees of freedom (DoF) as the optimal linear/nonlinear precoding schemes.…
The research reported in this paper addresses the fundamental task of separation of locally moving or deforming image areas from a static or globally moving background. It builds on the latest developments in the field of robust principal…
The linearly constrained convex composite programming problems whose objective function contains two blocks with each block being the form of nonsmooth+smooth arises frequently in multiple fields of applications. If both of the smooth terms…
In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the…
In this paper we develop random block coordinate gradient descent methods for minimizing large scale linearly constrained separable convex problems over networks. Since we have coupled constraints in the problem, we devise an algorithm that…
We propose a novel factorization of a non-singular matrix $P$, viewed as a $2\times 2$-blocked matrix. The factorization decomposes $P$ into a product of three matrices that are lower block-unitriangular, upper block-triangular, and lower…
The singular value decomposition (SVD) is a powerful tool in modern numerical linear algebra, which underpins computational methods such as principal component analysis (PCA), low-rank approximations, and randomized algorithms. Many…
Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes…
We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant…
We study a block-structured class of convex-concave saddle-point problems in which both the primal and dual variables admit natural separable decompositions. Motivated by large-scale applications where a full update on either side can be…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's…