Related papers: Error localization of best L1 polynomial approxima…
In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be…
In order to realize analog compressed sensing, the paper considers approximate proximal operators of the $\ell_1$ and minimax concave penalty (MCP) regularization functions. Specifically, we propose to realize the approximate functions by…
Compressed sensing is a new methodology for constructing sensors which allow sparse signals to be efficiently recovered using only a small number of observations. The recovery problem can often be stated as the one of finding the solution…
Compressed sensing aims at reconstructing sparse signals from significantly reduced number of samples, and a popular reconstruction approach is $\ell_1$-norm minimization. In this correspondence, a method called orthonormal expansion is…
We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ…
The nearest-centroid classifier is a simple linear-time classifier based on computing the centroids of the data classes in the training phase, and then assigning a new datum to the class corresponding to its nearest centroid. Thanks to its…
This paper studies the problem of recovering a signal vector and the corrupted noise vector from a collection of corrupted linear measurements through the solution of a l1 minimization, where the sensing matrix is a partial Fourier matrix…
In this paper, we study the support recovery conditions of weighted $\ell_1$ minimization for signal reconstruction from compressed sensing measurements when multiple support estimate sets with different accuracy are available. We identify…
We consider the unconstrained $L_2$-$L_p$ minimization: find a minimizer of $\|Ax-b\|^2_2+\lambda \|x\|^p_p$ for given $A \in R^{m\times n}$, $b\in R^m$ and parameters $\lambda>0$, $p\in [0,1)$. This problem has been studied extensively in…
The best uniform polynomial approximation of the checkmark function $f(x)=|x-\alpha |$ is considered, as $\alpha$ varies in $(-1,1)$. For each fixed degree $n$, the minimax error $E_n (\alpha)$ is shown to be piecewise analytic in $\alpha$.…
We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a…
We consider the problem of finding an approximate solution to $\ell_1$ regression while only observing a small number of labels. Given an $n \times d$ unlabeled data matrix $X$, we must choose a small set of $m \ll n$ rows to observe the…
We study the proximal gradient descent (PGD) method for $\ell^{0}$ sparse approximation problem as well as its accelerated optimization with randomized algorithms in this paper. We first offer theoretical analysis of PGD showing the bounded…
We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\lambda|x|}$, where $\lambda >0$, $\log |x|$ and $|x|^{\alpha}$, where $-1 < \alpha < 1$. We also give…
We present a detailed analysis of the unconstrained $\ell_1$-weighted LASSO method for recovery of sparse data from its observation by randomly generated matrices, satisfying the Restricted Isometry Property (RIP) with constant $\delta<1$,…
We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $\mathbb{P}_\Lambda:={\rm span} \{y\mapsto y^\nu \,: \, \nu\in \Lambda\}$ with $\Lambda\subset \mathbb{N}_0^d$ over the…
We address the problem of compressed sensing (CS) with prior information: reconstruct a target CS signal with the aid of a similar signal that is known beforehand, our prior information. We integrate the additional knowledge of the similar…
Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space. This…
This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we…
The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using…