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We study an algorithm which has been proposed by Chinesta et al. to solve high-dimensional partial differential equations. The idea is to represent the solution as a sum of tensor products and to compute iteratively the terms of this sum.…
For the classical maximum coverage problem, the greedy algorithm achieves a worst-case $1-1/e$ approximation, which is optimal unless $\text{P} = \text{NP}$. The notion of coverage appears in a wide range of optimization tasks, where…
This study investigated typical performance of approximation algorithms known as belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problems on sparse biregular random graphs. After using the cavity…
For Schauder bases, Dilworth et al. introduced and characterized the partially greedy property, which is strictly weaker than the (almost) greedy property. Later, Berasategui et al. defined and studied the strong partially greedy property…
In this article, we derive a novel convergence estimate for the weak POD-Greedy method with multiple POD modes and variable greedy thresholds in terms of the entropy numbers of the parametric solution manifold. Combining the POD with the…
Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP)…
Submodular maximization with a cardinality constraint can model various problems, and those problems are often very large in practice. For the case where objective functions are monotone, many fast approximation algorithms have been…
In many high-frequency simulation workflows, eigenvalue tracking along a parameter variation is necessary. This can become computationally prohibitive when repeated time-consuming eigenvalue problems must be solved. Therefore, we employ a…
Identifying breakpoints in piecewise regression is critical in enhancing the reliability and interpretability of data fitting. In this paper, we propose novel algorithms based on the greedy algorithm to accurately and efficiently identify…
We consider the problem of studying the performance of greedy algorithm on sensor selection problem for stable linear systems with Kalman Filter. Specifically, the objective is to find the system parameters that affects the performance of…
In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal…
In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a $(2/3)$-approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval…
Optimization is important in machine learning problems, and quasi-Newton methods have a reputation as the most efficient numerical schemes for smooth unconstrained optimization. In this paper, we consider the explicit superlinear…
Graph-based approaches are empirically shown to be very successful for the nearest neighbor search (NNS). However, there has been very little research on their theoretical guarantees. We fill this gap and rigorously analyze the performance…
This paper is devoted to the theoretical study of the efficiency, namely, stability of some greedy algorithms. In the greedy approximation theory researchers are mostly interested in the following two important properties of an algorithm --…
We propose a new yet natural algorithm for learning the graph structure of general discrete graphical models (a.k.a. Markov random fields) from samples. Our algorithm finds the neighborhood of a node by sequentially adding nodes that…
The submodular maximization problem is widely applicable in many engineering problems where objectives exhibit diminishing returns. While this problem is known to be NP-hard for certain subclasses of objective functions, there is a greedy…
This paper examines the ability of greedy algorithms to estimate a block sparse parameter vector from noisy measurements. In particular, block sparse versions of the orthogonal matching pursuit and thresholding algorithms are analyzed under…
It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate…
Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the…