arXiv research

New regularizers tighten convex relaxation bounds for neural networks.

Convex relaxations improve CNNs with fixed weights.

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we con…

In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they lift the original $n {\times} n$-dimensional variable to an $n^2 {\times} n^2$-d…

New method improves neural network verification by considering multivariate input space of ReLU neurons.

Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation techniques to find a convex surrogate for the problem. Unfortunately, convex relaxat…

Improved neural network robustness certification through tighter convex relaxations.

Paper relaxes optimal transport using convex functions for data science.

Algorithm for exact partitioning of high-order models using convex tensor relaxation.

Paper improves understanding of noisy matrix completion using convex relaxation and nonconvex optimization.

Unified convex relaxation framework for neural network robustness verification.

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditi…

Two-stage nonconvex algorithm and convex relaxation both achieve optimal accuracy in noisy blind deconvolution.

A number of recent work studied the effectiveness of feature selection using Lasso. It is known that under the restricted isometry properties (RIP), Lasso does not generally lead to the exact recovery of the set of nonzero coefficients, due to the looseness of convex relaxation. This paper considers the feature selecti…

Bounds on chemical reaction network relaxation rates using convex analysis.

The paper develops sum-of-squares relaxations for computing $f$-divergences.

We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statist…

Paper solves graph matching problem using convex relaxation to the simplex.

Although many convex relaxations of clustering have been proposed in the past decade, current formulations remain restricted to spherical Gaussian or discriminative models and are susceptible to imbalanced clusters. To address these shortcomings, we propose a new class of convex relaxations that can be flexibly applied…

New approach to convex hulls for low-rank problems.

We suggest using the max-norm as a convex surrogate constraint for clustering. We show how this yields a better exact cluster recovery guarantee than previously suggested nuclear-norm relaxation, and study the effectiveness of our method, and other related convex relaxations, compared to other clustering approaches.

Proposes new convex relaxations for certifying spatial robustness of neural networks.

Graph alignment problem solved with convex relaxations for correlated matrices.

PEREGRiNN verifies safety of ReLU NNs by penalizing relaxation in a greedy manner.

The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, w…

The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loos…

Paper relaxes convexity assumptions in mean curvature flow results.

New convex relaxations solve sparse regression problems efficiently.

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor…

New convex method solves densest subgraph problem with high probability.

Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an l0-norm penalty. Exactly enforcing the l0-no…

We develop a convex relaxation method for analyzing neural network generalization.

New method improves signal estimation by convexifying $\ell_0$-norm constraints.

New method relaxes optimization problems to find solutions more reliably.

Joint sparsity regularization in multi-task learning has attracted much attention in recent years. The traditional convex formulation employs the group Lasso relaxation to achieve joint sparsity across tasks. Although this approach leads to a simple convex formulation, it suffers from several issues due to the loosenes…

SGD with mini-batches can solve convex low-rank matrix problems efficiently.

A nearly tight convex relaxation for sparse Naive Bayes features.

New method tackles nonlinear, infinite-dimensional signal processing problems.

We give a novel formal theoretical framework for unsupervised learning with two distinctive characteristics. First, it does not assume any generative model and based on a worst-case performance metric. Second, it is comparative, namely performance is measured with respect to a given hypothesis class. This allows to avo…

We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techn…

In this paper, we propose an unifying view of several recently proposed structured sparsity-inducing norms. We consider the situation of a model simultaneously (a) penalized by a set- function de ned on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in Lp-n…

Paper revisits set membership estimation for linear systems with relaxed disturbance bounds.

Renet improves Elastic Net by dynamically selecting between convex blending and refitting, enhancing prediction accuracy.

New conditions prevent gaps in optimal control problems.

Rank minimization (RM) is a wildly investigated task of finding solutions by exploiting low-rank structure of parameter matrices. Recently, solving RM problem by leveraging non-convex relaxations has received significant attention. It has been demonstrated by some theoretical and experimental work that non-convex relax…

New conditions ensure Dantzig-Wolfe relaxation matches rank-constrained optimization problems.

In applications such as recommendation systems and revenue management, it is important to predict preferences on items that have not been seen by a user or predict outcomes of comparisons among those that have never been compared. A popular discrete choice model of multinomial logit model captures the structure of the …

This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis} (Sparse PCA) problem, and the family of {\em Sum-of-Squares} (SoS, aka Lasserre/…