Related papers: Clustering is semidefinitely not that hard: Nonneg…
Many clustering problems enjoy solutions by semidefinite programming. Theoretical results in this vein frequently consider data with a planted clustering and a notion of signal strength such that the semidefinite program exactly recovers…
We give the first approximation algorithm for mixed packing and covering semidefinite programs (SDPs) with polylogarithmic dependence on width. Mixed packing and covering SDPs constitute a fundamental algorithmic primitive with recent…
This paper develops new semidefinite programming (SDP) relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. The first class of problem…
In this paper, we propose a low-rank coordinate descent approach to structured semidefinite programming with diagonal constraints. The approach, which we call the Mixing method, is extremely simple to implement, has no free parameters, and…
We consider semidefinite programming (SDP) for the binary stochastic block model with equal-sized communities. Prior work of Hajek, Wu, and Xu proposed an SDP (sym-SDP) for the symmetric case where the intra-community edge probabilities are…
In this paper, we consider an SDP relaxation of the quadratic knapsack problem (QKP). After using the Burer-Monteiro factorization, we get a non-convex optimization problem, whose feasible region is an algebraic variety. Although there…
Clustering is widely used in unsupervised learning method that deals with unlabeled data. Deep clustering has become a popular study area that relates clustering with Deep Neural Network (DNN) architecture. Deep clustering method…
Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…
This paper introduces a new robust interior point method analysis for semidefinite programming (SDP). This new robust analysis can be combined with either logarithmic barrier or hybrid barrier. Under this new framework, we can improve the…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
We show {\it semidefinite programming} (SDP) feasibility problem is equivalent to solving a {\it convex hull relaxation} (CHR) for a finite system of quadratic equations. On the one hand, this offers a simple description of SDP. On the…
Many machine learning tasks, such as principal component analysis and low-rank matrix completion, give rise to manifold optimization problems. Although there is a large body of work studying the design and analysis of algorithms for…
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…
This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for…
Clustering is a widely used unsupervised learning technique involving an intensive discrete optimization problem. Associative Memory models or AMs are differentiable neural networks defining a recursive dynamical system, which have been…
Deep neural networks have proven to be highly effective when large amounts of data with clean labels are available. However, their performance degrades when training data contains noisy labels, leading to poor generalization on the test…
Graph-based variational methods have recently shown to be highly competitive for various classification problems of high-dimensional data, but are inherently difficult to handle from an optimization perspective. This paper proposes a convex…
Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods. However, when the dimension of the problem gets large, interior point methods become impractical in terms of both computational time and memory…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
We introduce a sketch-and-solve approach to speed up the Peng-Wei semidefinite relaxation of k-means clustering. When the data is appropriately separated we identify the k-means optimal clustering. Otherwise, our approach provides a…