Related papers: Exploiting Sparsity for Semi-Algebraic Set Volume …
Sparse tensor algebra is challenging to efficiently parallelize due to the irregular, data-dependent, and potentially skewed structure of sparse computation. We propose the first partitioning algorithm that provably load balances the…
We use convex relaxation techniques to produce lower bounds on the optimal value of subset selection problems and generate good approximate solutions. We then explicitly bound the quality of these relaxations by studying the approximation…
Analysis sparsity is a common prior in inverse problem or machine learning including special cases such as Total Variation regularization, Edge Lasso and Fused Lasso. We study the geometry of the solution set (a polyhedron) of the analysis…
Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
Many classification approaches first represent a test sample using the training samples of all the classes. This collaborative representation is then used to label the test sample. It was a common belief that sparseness of the…
Semidefinite relaxations are widely used to compute upper bounds on the objective of optimization problems involving noncommutative polynomials. Such optimization problems are prevalent in quantum information. We present an algorithm able…
The ability to resolve detail in the object that is being imaged, named by resolution, is the core parameter of an imaging system. Super-resolution is a class of techniques that can enhance the resolution of an imaging system and even…
Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations.…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
The mathematical theory of super-resolution developed recently by Cand\`{e}s and Fernandes-Granda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set…
We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other…
Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core…
Training neural network models with discrete (categorical or structured) latent variables can be computationally challenging, due to the need for marginalization over large or combinatorial sets. To circumvent this issue, one typically…
Deep Neural Networks are powerful tools for solving machine learning problems, but their training often involves dense and costly parameter updates. In this work, we use a novel Max-Plus neural architecture in which classical addition and…
We study sum of squares (SOS) relaxations to optimize polynomial functions over a set $V\cap R^n$, where $V$ is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents…
Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This…
The sparse group Lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level. In this paper, we develop an efficient augmented Lagrangian method for large-scale non-overlapping sparse…
We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative…
Recovering sparse signals from linear measurements has demonstrated outstanding utility in a vast variety of real-world applications. Compressive sensing is the topic that studies the associated raised questions for the possibility of a…