Related papers: On Error Bounds for Rank-Constrained Affine Matrix…
Suppose that we observe entries or, more generally, linear combinations of entries of an unknown $m\times T$-matrix $A$ corrupted by noise. We are particularly interested in the high-dimensional setting where the number $mT$ of unknown…
The problem of ensuring constraints satisfaction on the output of machine learning models is critical for many applications, especially in safety-critical domains. Modern approaches rely on penalty-based methods at training time, which do…
Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can…
Let $\mathbf{P}=\{ p_1, p_2, \ldots p_n \}$ and $\mathbf{Q} = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\mathbf{A}$ represent the $m\times n$ pairwise distance matrix with $\mathbf{A}_{i,j} = d(p_i,…
Performative prediction is a framework accounting for the shift in the data distribution induced by the prediction of a model deployed in the real world. Ensuring rapid convergence to a stable solution where the data distribution remains…
We introduce a general random model of a combinatorial optimization problem with geometric structure that encapsulates both linear programming and integer linear programming. Let $Q$ be a bounded set called the feasible set, $E$ be an…
In this paper, we solve a maximization problem where the objective function is quadratic and convex or concave and the constraints set is the reachable value set of a convergent discrete-time affine system. Moreover, we assume that the…
We analyze the local convergence of proximal splitting algorithms to solve optimization problems that are convex besides a rank constraint. For this, we show conditions under which the proximal operator of a function involving the rank…
With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness…
Reachability analysis is a formal method to guarantee safety of dynamical systems under the influence of uncertainties. A substantial bottleneck of all reachability algorithms is the necessity to adequately tune specific algorithm…
We consider mappings, which are structure consisting of a single function (and possibly some number of unary relations) and address the problem of approximating a continuous mapping by a finite mapping. This problem is the inverse problem…
This paper presents rigorous forward error bounds for linear conic optimization problems. The error bounds are formulated in a quite general framework; the underlying vector spaces are not required to be finite-dimensional, and the convex…
We develop an efficient stochastic variance reduced gradient descent algorithm to solve the affine rank minimization problem consists of finding a matrix of minimum rank from linear measurements. The proposed algorithm as a stochastic…
Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…
The metric capacitated facility location is a well-studied problem for which, while constant factor approximations are known, no efficient relaxation with constant integrality gap is known. The question whether there is such a relaxation is…
The problem of recovering the configuration of points from their partial pairwise distances, referred to as the Euclidean Distance Matrix Completion (EDMC) problem, arises in a broad range of applications, including sensor network…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values,…
We consider a constrained optimization problem arising from the study of the Helmholtz equation in unbounded domains. The optimization problem provides an approximation of the solution in a bounded computational domain. In this paper we…
Modern machine learning architectures are often highly expressive. They are usually over-parameterized and can interpolate the data by driving the empirical loss close to zero. We analyze the convergence of Local SGD (or FedAvg) for such…