Related papers: Reduced Basis Greedy Selection Using Random Traini…
In this note, we develop fast and deterministic dimensionality reduction techniques for a family of subspace approximation problems. Let $P\subset \mathbbm{R}^N$ be a given set of $M$ points. The techniques developed herein find an $O(n…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
The effectiveness of dimensionality reduction with quadratic manifolds hinges on the choice of a reduced basis and the associated quadratic correction terms. Existing approaches typically rely on subspaces spanned by the leading principal…
This thesis presents recent advances in model order reduction methods with the primary aim to construct online-efficient reduced surrogate models for parameterized multiscale phenomena and accelerate large-scale PDE-constrained parameter…
Greedy bases are those bases where the Thresholding Greedy Algorithm (introduced by S. V. Konyagin and V. N. Temlyakov) produces the best possible approximation up to a constant. In 2017, Bern\'a and Blasco gave a characterization of these…
This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate…
Centrality measures, quantifying the importance of vertices or edges, play a fundamental role in network analysis. To date, triggered by some positive approximability results, a large body of work has been devoted to studying centrality…
Greedy Sampling Methods (GSMs) are widely used to construct approximate solutions of Configuration Optimization Problems (COPs), where a loss functional is minimized over finite configurations of points in a compact domain. While effective…
Setcover greedy algorithm is a natural approximation algorithm for test set problem. This paper gives a precise and tighter analysis of performance guarantee of this algorithm. The author improves the performance guarantee $2\ln n$ which…
Kernel based regularized interpolation is a well known technique to approximate a continuous multivariate function using a set of scattered data points and the corresponding function evaluations, or data values. This method has some…
Nonlinear kernels can be approximated using finite-dimensional feature maps for efficient risk minimization. Due to the inherent trade-off between the dimension of the (mapped) feature space and the approximation accuracy, the key problem…
We target time-dependent partial differential equations (PDEs) with heterogeneous coefficients in space and time. To tackle these problems, we construct reduced basis/ multiscale ansatz functions defined in space that can be combined with…
We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation…
The seminar assignment problem is a variant of the generalized assignment problem in which items have unit size and the amount of space allowed in each bin is restricted to an arbitrary set of values. The problem has been shown to be…
Many problems in signal processing and machine learning can be formalized as weak submodular optimization tasks. For such problems, a simple greedy algorithm (\textsc{Greedy}) is guaranteed to find a solution achieving the objective with a…
This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $\epsilon$-greedy exploration under the online setting. This problem setting is motivated by the successful deep…
A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in [Balabanov and Nouy 2019, Part I]. Following this framework, we here construct…
Kernel methods are versatile tools for function approximation and surrogate modeling. In particular, greedy techniques offer computational efficiency and reliability through inherent sparsity and provable convergence. Inspired by the…
We consider the problem of sparse atomic optimization, where the notion of "sparsity" is generalized to meaning some linear combination of few atoms. The definition of atomic set is very broad; popular examples include the standard basis,…