Related papers: Adaptive almost full recovery in sparse nonparamet…
We consider sparsity-based techniques for the approximation of high-dimensional functions from random pointwise evaluations. To date, almost all the works published in this field contain some a priori assumptions about the error corrupting…
This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…
We consider a nonparametric additive model of a conditional mean function in which the number of variables and additive components may be larger than the sample size but the number of nonzero additive components is "small" relative to the…
We consider the problem of recovering the support of a sparse signal using noisy projections. While extensive work has been done on the dense measurement matrix setting, the sparse setting remains less explored. In this work, we establish…
We investigate structured sparsity methods for variable selection in regression problems where the target depends nonlinearly on the inputs. We focus on general nonlinear functions not limiting a priori the function space to additive…
We consider the model {eqnarray*}y=X\theta^*+\xi, Z=X+\Xi,{eqnarray*} where the random vector $y\in\mathbb{R}^n$ and the random $n\times p$ matrix $Z$ are observed, the $n\times p$ matrix $X$ is unknown, $\Xi$ is an $n\times p$ random noise…
Statistical studies of astronomical data sets, in particular of cataloged properties for discrete objects, are central to astrophysics. One cannot model those objects' population properties or incidences without a quantitative understanding…
This paper presents Sparse Gradient Descent as a solution for variable selection in convex piecewise linear regression, where the model is given as the maximum of $k$-affine functions $ x \mapsto \max_{j \in [k]} \langle a_j^\star, x…
In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric…
We consider the problem of exact support recovery of sparse signals via noisy measurements. The main focus is the sufficient and necessary conditions on the number of measurements for support recovery to be reliable. By drawing an analogy…
Given $m$ $d$-dimensional responsors and $n$ $d$-dimensional predictors, sparse regression finds at most $k$ predictors for each responsor for linear approximation, $1\leq k \leq d-1$. The key problem in sparse regression is subset…
Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable…
We consider the problem of estimating the support of a vector $\beta^* \in \mathbb{R}^{p}$ based on observations contaminated by noise. A significant body of work has studied behavior of $\ell_1$-relaxations when applied to measurement…
A new sparse semiparametric model is proposed, which incorporates the influence of two functional random variables in a scalar response in a flexible and interpretable manner. One of the functional covariates is included through a…
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 consider the following signal recovery problem: given a measurement matrix $\Phi\in \mathbb{R}^{n\times p}$ and a noisy observation vector $c\in \mathbb{R}^{n}$ constructed from $c = \Phi\theta^* + \epsilon$ where $\epsilon\in…
Sparse modeling has been widely and successfully used in many applications such as computer vision, machine learning, and pattern recognition. Accompanied with those applications, significant research has studied the theoretical limits and…
The model interpretation is essential in many application scenarios and to build a classification model with a ease of model interpretation may provide useful information for further studies and improvement. It is common to encounter with a…
We consider the problem of random sampling for band-limited functions. When can a band-limited function $f$ be recovered from randomly chosen samples $f(x_j), j\in \mathbb{N}$? We estimate the probability that a sampling inequality of the…