Related papers: On one extremal property of a regular simplex

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one…

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far.…

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in $L^1$ and $L^\infty$ norms depending on the…

Optimization problems over permutation matrices appear widely in facility layout, chip design, scheduling, pattern recognition, computer vision, graph matching, etc. Since this problem is NP-hard due to the combinatorial nature of…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

Given $d\geq 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$ of $\mathbf{x}\in\mathbb{R}^d$ satisfying $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$ with denominator $1\leq q < T$ decays to the…

In numerous substitution models for the $\l_{0}$-norm minimization problem $(P_{0})$, the $\l_{p}$-norm minimization $(P_{p})$ with $0<p<1$ have been considered as the most natural choice. However, the non-convex optimization problem…

In this paper, we describe the randomized QLP (RQLP) algorithm and its enhanced version (ERQLP) for computing the low rank approximation to $A$ of size $m\times n$ efficiently such that $A\approx QLP$, where $L$ is the rank-$k$…

Suppose that $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, is an open set satisfying the corkscrew condition with an $n$-dimensional ADR boundary, $\partial \Omega$. In this note, we show that if harmonic functions are…

We present a short, direct proof of the uniform convexity of L^p spaces for 1<p<\infty.

We study the Lambert series $\mathscr{L}_q(s,x) = \sum_{k=1}^\infty k^s q^{k x}/(1-q^k)$, for all $s \in \mathbb{C}$. We obtain the complete asymptotic expansion of $\mathscr{L}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields…

We study the convergence rate of the proximal-gradient homotopy algorithm applied to norm-regularized linear least squares problems, for a general class of norms. The homotopy algorithm reduces the regularization parameter in a series of…

We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.

Dinh D\~ung and T. Ullrich have proven a multivariate Whitney's theorem for the local anisotropic polynomial approximation in $L_p(Q)$ for $1 \le p \le \infty$, where $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the…

For $V : \mathbb{R}^d \to \mathbb{R}$ coercive, we study the convergence rate for the $L^1$-distance of the empiric minimizer, which is the true minimum of the function $V$ sampled with noise with a finite number $n$ of samples, to the…

This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly…

We prove sharp $\ell^{p}L^{p}$ decoupling inequalities for $2$ quadratic forms in $4$ variables. We also recover several previous results (arXiv:1409.1634, arXiv:1501.07224, arXiv:1609.02022, arXiv:1609.04107) in a unified way.

We obtain new equitightness and $C([0,T];L^p(\mathbb{R}^N))$-convergence results for finite-difference approximations of generalized porous medium equations of the form $$ \partial_tu-\mathfrak{L}[\varphi(u)]=g\qquad\text{in…

Let $P$ be an $m$-homogeneous polynomial in $n$-complex variables $x_1, \dotsc, x_n$. Clearly, $P$ has a unique representation in the form \begin{equation*} P(x)= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \,…

We present an iterative support shrinking algorithm for $\ell_{p}$-$\ell_{q}$ minimization~($0 <p < 1 \leq q < \infty $). This algorithm guarantees the nonexpensiveness of the signal support set and can be easily implemented after being…