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We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to…
The problem of recovering a low $n$-rank tensor is an extension of sparse recovery problem from the low dimensional space (matrix space) to the high dimensional space (tensor space) and has many applications in computer vision and graphics…
The Steiner tree problem is a well-known problem in network design, routing, and VLSI design. Given a graph, edge costs, and a set of dedicated vertices (terminals), the Steiner tree problem asks to output a sub-graph that connects all…
We are concerned with the tensor equations whose coefficient tensor is an M-tensor. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend…
Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…
The Sylvester's denumerant \( d(t; \boldsymbol{a}) \) is a quantity that counts the number of nonnegative integer solutions to the equation \( \sum_{i=1}^{N} a_i x_i = t \), where \( \boldsymbol{a} = (a_1, \dots, a_N) \) is a sequence of…
We propose and study a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. Using a convex relaxation based on tensorial lifting, and applying first-order proximal algorithms, these problems could be solved…
The objective of this paper is to investigate a new numerical method for the approximation of the self-diffusion matrix of a tagged particle process defined on a grid. While standard numerical methods make use of long-time averages of…
This paper considers the numerical solution of generalized Sylvester matrix equations, which arise in many scientific and engineering applications but remain challenging to solve efficiently, particularly when the coefficient matrices are…
Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To…
This study establishes consistency conditions and a general solution for a coupled system that consists of five two-sided Sylvester-like tensor equations in ten quaternion variables throughout the Einstein tensor product. Certain specific…
We present experimental and theoretical results on a method that applies a numerical solver iteratively to solve several non-negative quadratic programming problems in geometric optimization. The method gains efficiency by exploiting the…
Scalable Gaussian process (GP) inference is essential for sequential decision-making tasks, yet improving GP scalability remains a challenging problem with many open avenues of research. This paper focuses on iterative GPs, where iterative…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
Iterative sketching and sketch-and-precondition are well-established randomized algorithms for solving large-scale, over-determined linear least-squares problems. In this paper, we introduce a new perspective that interprets Iterative…
Nesterov's accelerated gradient descent method (AGD) is a seminal deterministic first-order method known to achieve the optimal order of iteration complexity for solving convex smooth optimization problems. Two distinct sequences of…
We propose a two-level nested preconditioned iterative scheme for solving sparse linear systems of equations in which the coefficient matrix is symmetric and indefinite with relatively small number of negative eigenvalues. The proposed…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems $\dot{x}(t) = A_0x(t)+A_1x(t-\tau)+B_0u(t)$. We propose a new algorithm for…
The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially…