相关论文: Analysis of Linear Difference Schemes in the Spars…
We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse…
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform…
We demonstrate a construction of error-correcting codes from graphs by means of $k$-resolving sets, and present a decoding algorithm which makes use of covering designs. Along the way, we determine the $k$-metric dimension of grid graphs…
We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local $L^1$-error between the exact and numerical solutions is $\mathcal{O}(\Delta x^{2/(19+d)})$,…
Motivated by distributed machine learning settings such as Federated Learning, we consider the problem of fitting a statistical model across a distributed collection of heterogeneous data sets whose similarity structure is encoded by a…
Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $u\in C^{4}(\bar{\Omega})$ is needed to preserve $\mathcal{O}(h^{2})$…
For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value of various problems. However, there has not been as much success in…
Sparse linear regression is a central problem in high-dimensional statistics. We study the correlated random design setting, where the covariates are drawn from a multivariate Gaussian $N(0,\Sigma)$, and we seek an estimator with small…
Sparse coding and dictionary learning are popular techniques for linear inverse problems such as denoising or inpainting. However in many cases, the measurement process is nonlinear, for example for clipped, quantized or 1-bit measurements.…
We consider the problem of learning a graph modeling the statistical relations of the $d$ variables from a dataset with $n$ samples $X \in \mathbb{R}^{n \times d}$. Standard approaches amount to searching for a precision matrix $\Theta$…
We generalize the reduction mechanism for linear programming problems and semidefinite programming problems from [arXiv:1410.8816] in two ways 1) relaxing the requirement of affineness and 2) extending to fractional optimization problems.…
We present a novel feature selection technique, Sparse Linear Centroid-Encoder (SLCE). The algorithm uses a linear transformation to reconstruct a point as its class centroid and, at the same time, uses the $\ell_1$-norm penalty to filter…
In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor…
Regularized methods have been widely applied to system identification problems without known model structures. This paper proposes an infinite-dimensional sparse learning algorithm based on atomic norm regularization. Atomic norm…
We review connections between coding-theoretic objects and sparse learning problems. In particular, we show how seemingly different combinatorial objects such as error-correcting codes, combinatorial designs, spherical codes, compressed…
This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci.…
We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to…
We present a method for electronic structure calculations that retains all of the advantages of real space and addresses the inherent inefficiency of a regular grid, which has equal precision everywhere. The computations are carried out on…
Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…