arXiv research

SDP achieves Bayes error rate in synchronization and block models.

SDP achieves optimal error in noisy phase synchronization.

Motivated by the task of clustering either $d$ variables or $d$ points into $K$ groups, we investigate efficient algorithms to solve the Peng-Wei (P-W) $K$-means semi-definite programming (SDP) relaxation. The P-W SDP has been shown in the literature to have good statistical properties in a variety of settings, but rem…

A new method solves diagonally constrained SDPs quickly and accurately.

We consider the problem of estimating the discrete clustering structures under the Sub-Gaussian Mixture Model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation …

Optimal neural network approximation for Wasserstein gradient direction via convex optimization.

In this paper, we analyze different preconditionings designed to enhance robustness of pure-pixel search algorithms, which are used for blind hyperspectral unmixing and which are equivalent to near-separable nonnegative matrix factorization algorithms. Our analysis focuses on the successive projection algorithm (SPA), …

Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the pe…

The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP) solution to the problem of fitting the SBM, derived as a relaxation of the MLE. W…

New algorithm improves clustering accuracy without sacrificing scalability.

The well-known Influence Maximization (IM) problem has been actively studied by researchers over the past decade, with emphasis on marketing and social networks. Existing research have obtained solutions to the IM problem by obtaining the influence spread and utilizing the property of submodularity. This paper is based…

New SDP method certifies neural network robustness across all classes efficiently.

AMP algorithms can be efficiently simulated by SDPs even with corrupted data.

A number of statistical estimation problems can be addressed by semidefinite programs (SDP). While SDPs are solvable in polynomial time using interior point methods, in practice generic SDP solvers do not scale well to high-dimensional problems. In order to cope with this problem, Burer and Monteiro proposed a non-conv…

This paper describes a fast algorithm for recovering low-rank matrices from their linear measurements contaminated with Poisson noise: the Poisson noise Maximum Likelihood Singular Value thresholding (PMLSV) algorithm. We propose a convex optimization formulation with a cost function consisting of the sum of a likeliho…

Geometric technique determines exactness of SDP robustness certificate.

Simplifies solving noisy SDPs for low rank matrix recovery problems.

New algorithm solves large cardinality-constrained clustering problems.

Paper presents a randomized algorithm for SPCA with high probability approximation.

We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ are both in the community and $A_{ij} \sim Q$ otherwise, for …

A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few hundreds variables (in the underlying combinatorial problem). On the other hand…

The framework of Integral Quadratic Constraints (IQC) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to a semi-definite program (SDP). In the case of over-relaxed Alternating Direction Method of Multipliers (ADMM), an explicit and closed form solution to this SDP was …

The paper studies SDP feasibility and sos ranks for specific polynomials.

We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size $n$ by $k$ such that $X = YY^*$ is the SDP variable. The advantages of such formulation are twofold: the di…

The framework of Integral Quadratic Constraints of Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). Followup work by Nishihara et al. (2015) applies this technique to the entire family of over-relaxed Alternating…

A new method for clustering heterogeneous data using likelihood-adjusted SDP.

New algorithm estimates mean in high dimensions with nearly-linear time, robust to corrupted data.

A fast algorithm for $K$-means clustering using subsampled SDP.

Optimizes sparse mean-reverting portfolios for higher returns.

New algorithm reduces kernel optimization complexity.

Unified framework for hard affine SDP constraints in vRKHSs.

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to add…

Paper shows Burer-Monteiro method can solve SDPs in polynomial time under smoothed analysis.

Bundle method solves low rank SDP problems without full matrix construction.

Paper uses SDP for community detection with side information.

Paper improves robustness of SDP algorithms with nonconvex loss functions.

New algorithm optimizes tessellated kernels for larger datasets and improved performance.

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the She…

Sharp condition found for Burer-Monteiro method to work for MaxCut-type SDPs.

New method solves stochastic optimization problems with affine constraints.

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this p…

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP…

Unified framework guarantees exactness of asymmetric low-rank SDP learning.

A new Wasserstein $K$-means method for clustering probability distributions.

We propose an SDP relaxation for the Gromov-Wasserstein distance, providing globally optimal solutions.

We propose a semidefinite programming (SDP) algorithm for community detection in the stochastic block model, a popular model for networks with latent community structure. We prove that our algorithm achieves exact recovery of the latent communities, up to the information-theoretic limits determined by Abbe and Sandon (…

New scalable Lipschitz bounds improve neural network robustness analysis.

Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model of two equal-sized clusters. The s…