arXiv subjects

Heping Wang

Publications and source records attributed to Heping Wang.

At least 19 recordsLinked to original sources

arXiv2026

WiFi Sensing via Reservoir Computing

Practical WiFi sensing must handle clock-asynchronous links, cross-domain variation, and post-deployment updating under the limited compute budget of access point (AP), router, and embedded Internet-of-Things platforms such as ESP-class devices. Reservoir computing (RC) is attractive in this setting because its temporal encoder can remain fixed while only a lightweight readout needs to be optimized and updated. To address these deployment challenges under tight compute budgets, we present ReWiS, a WiFi-sensing-oriented reservoir framework that transforms channel state information (CSI) into structured micro-Doppler streams with common, antenna-specific, and differential motion cues, encodes them with a graph-coupled reservoir, and adapts to a new domain after deployment by freezing the reservoir and fine-tuning only a compact readout with a few labeled target samples. On a large-scale WiFi sensing benchmark, ReWiS achieves 89.2% in-domain macro-F1 and 82.0% mean cross-domain macro-F1 with only a 0.72M trainable readout, improves to 88.5\% after lightweight post-deployment adaptation, and remains competitive with recent deep baselines evaluated under the same protocol, which achieve 87.5%-89.2% mean cross-domain macro-F1, while requiring lower optimization cost and lower CPU latency. These results indicate that ReWiS provides a practical reservoir-based design for deployable WiFi sensing, with further potential for low-power hardware realization.

eess.SP↗

arXiv2026

A Note About Algebraic $(s, t)$-Weak Tractability Of Linear Tensor Product Problems In The Worst-Case Setting

This paper is devoted to discussing the linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate algebraic $(s, t)$-weak tractability (ALG-$(s, t)$-WT) under the absolute error criterion in the case ${\lambda}_1 > 1$, where ${\lambda}_1$ is the square of the univariate maximal singular value. We solve the problem by giving the necessary and sufficient conditions for ALG-$(s, t)$-WT on univariate singular values and fill the gap left open.

math.NA↗

arXiv2025

A note about exponential tractability of linear weighted tensor product problems in the worst-case setting

This paper is devoted to discussing the weighted linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential $(s, t)$-weak tractability (EXP-$(s, t)$-WT) with $\max(s,t)<1$ and exponential uniform weak tractability (EXP-UWT) under the absolute or normalized error criterion. We solve the problem by filling the remaining gaps left open on EXP-tractability. That is, we obtain necessary and sufficient conditions for EXP-$(s, t)$-WT with $\max(s, t) < 1$ and for EXP-UWT.

cs.CC↗

arXiv2025

Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds

For $1\le p,q\le \infty$, the Nikolskii factor for a diffusion polynomial $P_{\bf a}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\bf a})=\frac{\|P_{\bf a}\|_{q}}{\|P_{\bf a}\|_{p}},\ \ P_{\bf a}({\bf x})=\sum_{k:\lambda_{k}\leq n}a_{k}\phi_{k}({\bf x}),$$ where ${\bf a}=\{a_k\}_{\lambda_k\le n}$, and $\{(\phi_k,-\lambda_k^2)\}_{k=0}^\infty$ are the eigenpairs of the Laplace-Beltrami operator $\Delta_{\mathbb M}$ on a closed smooth Riemannian manifold $\mathbb M$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,\sigma^{2})$ coefficients and obtain the exact orders. For $1\leq p<q<\infty$, the average Nikolskii factor is of order $n^{0}$ (i.e., constant), as compared to the worst case bound of order $n^{d(1/p-1/q)}$, and for $1\leq p<q=\infty$, the average Nikolskii factor is of order $(\ln n)^{1/2}$ as compared to the worst case bound of order $n^{d/p}$.

math.PR↗

arXiv2025

Average Nikolskii factors for random trigonometric polynomials

For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\bf a}$ is defined by $$\mathcal N_{p,q}(T_{\bf a})=\frac{\|T_{\bf a}\|_{q}}{\|T_{\bf a}\|_{p}},\ \ T_{\bf a}(x)=a_{1}+\sum\limits^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,\sigma^{2})$ coefficients and obtain that the exact order. For $1\leq p<q<\infty$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1/p-1/q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials.

math.CA↗

arXiv2024

Optimal quadrature for weighted function spaces on multivariate domains

Consider the numerical integration $${\rm Int}_{\mathbb S^d,w}(f)=\int_{\mathbb S^d}f({\bf x})w({\bf x}){\rm d}\sigma({\bf x}) $$ for weighted Sobolev classes $BW_{p,w}^r(\mathbb S^d)$ with a Dunkl weight $w$ and weighted Besov classes $BB_\gamma^\Theta(L_{p,w}(\mathbb S^d))$ with the generalized smoothness index $\Theta $ and a doubling weight $w$ on the unit sphere $\mathbb S^d$ of the Euclidean space $\mathbb R^{d+1}$ in the deterministic and randomized case settings. For $BW_{p,w}^r(\mathbb S^d)$ we obtain the optimal quadrature errors in both settings. For $BB_\gamma^\Theta(L_{p,w}(\mathbb S^d))$ we use the weighted least $\ell_p$ approximation and the standard Monte Carlo algorithm to obtain upper estimates of the quadrature errors which are optimal if $w$ is an $A_\infty$ weight in the deterministic case setting or if $w$ is a product weight in the randomized case setting. Our results show that randomized algorithms can provide a faster convergence rate than that of the deterministic ones when $p>1$. Similar results are also established on the unit ball and the standard simplex of $\mathbb R^d$.

math.NA↗

arXiv2024

Average case tractability of multivariate approximation with Gevrey type kernels

We consider multivariate approximation problems in the average case setting with a zero mean Gaussian measure whose covariance kernel is a periodic Gevrey kernel. We investigate various notions of algebraic tractability and exponential tractability, and obtain necessary and sufficient conditions in terms of the parameters of the problem.

math.NA↗

arXiv2024

Exact $L_2$ Bernstein-Markov inequalities for generalized weights

In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n^2(L_2(W_\lambda),{\rm D}):=\sup_{0\neq p\in\mathcal{P}_n}\frac{\int_I\left|{\rm D} p(x)\right|^2W_\lambda(x){\rm d}x}{\int_I| p(x)|^2W_\lambda(x){\rm d}x},\ \lambda>0,$$ where $\mathcal{P}_n$ denotes the set of all algebraic polynomials of degree at most $n$, ${\rm D}$ is the differential operator given by $${\rm D}=\Bigg\{\begin{aligned}&\frac {\rm d}{{\rm d}x}\ {\rm or}\ \mathcal{D}_\lambda, &&{\rm if}\ W_\lambda(x)=|x|^{2\lambda}e^{-x^2}\ {\rm and}\ I=\mathbb R, \\&(1-x^2)^{\frac12}\,\frac {\rm d}{{\rm d}x}\ {\rm or}\ (1-x^2)^{\frac12}\,\mathcal{D}_\lambda, &&{\rm if}\ W_\lambda(x):=|x|^{2\lambda}(1-x^2)^{\mu-\frac 12},\mu>-\frac12\ {\rm and}\ I=[-1,1],\end{aligned} $$ and $\mathcal{D}_\lambda$ is the univariate Dunkl operator, i.e., $\mathcal{D}_\lambda f(x)=f'(x)+\lambda{(f(x)-f(-x))}/{x}$. Furthermore, the corresponding extremal polynomials are also obtained.

math.CA↗

arXiv2024

Weighted least $\ell_p$ approximation on compact Riemannian manifolds

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gröchenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in $L_q,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces $B_{p,τ}^r(\Bbb M),\, 0<τ\le \infty, r>d/p $ defined by best polynomial approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.

math.NA↗

arXiv2024

Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{α,β}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>0$, which gives a clue to Open problem 3 as posed by Krieg and Vybíral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $α>d/2$, and apply Hilbert space structure and Vybíral's theorem about Schur product theory to obtain lower bounds in the case $α=d/2,\,β>1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>d/2$ or $α=d/2,\,β>1/2$, which are order optimal.

math.NA↗

arXiv2023

On the power of standard information for tractability for $L_\infty$ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic function in the worst case setting with the error measured in the $L_\infty$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ under the absolute or normalized error criterion, and show that the power of $Λ^{\rm std}$ is the same as the one of $Λ^{\rm all}$ for some notions of algebraic and exponential tractability. Our result can be applied to weighted Korobov spaces and Korobov spaces with exponential weight. This gives a special solution to Open problem 145 as posed by Novak and Woźniakowski (2012).

math.NA↗

arXiv2022

Outlier-Detection Based Robust Information Fusion for Networked Systems

We consider state estimation for networked systems where measurements from sensor nodes are contaminated by outliers. A new hierarchical measurement model is formulated for outlier detection by integrating the outlier-free measurement model with a binary indicator variable. The binary indicator variable, which is assigned a beta-Bernoulli prior, is utilized to characterize if the sensor's measurement is nominal or an outlier. Based on the proposed outlier-detection measurement model, both centralized and decentralized information fusion filters are developed. Specifically, in the centralized approach, all measurements are sent to a fusion center where the state and outlier indicators are jointly estimated by employing the mean-field variational Bayesian inference in an iterative manner. In the decentralized approach, however, every node shares its information, including the prior and likelihood, only with its neighbors based on a hybrid consensus strategy. Then each node independently performs the estimation task based on its own and shared information. In addition, an approximation distributed solution is proposed to reduce the local computational complexity and communication overhead. Simulation results reveal that the proposed algorithms are effective in dealing with outliers compared with several recent robust solutions.

stat.AP↗

arXiv2022

Weighted $\ell_q$ approximation problems on the ball and on the sphere

Let $L_{q,μ},\, 1\le q<\infty, \ μ\ge0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_μ$ on the ball $\Bbb B^d$. We consider the weighted least $\ell_q$ approximation problem for a given $L_{q,μ}$-Marcinkiewicz-Zygmund family on $\Bbb B^d$. We obtain the weighted least $\ell_q$ approximation errors for the weighted Sobolev space $W_{q,μ}^r$, $r>(d+2μ)/q$, which are order optimal. We also discuss the least squares quadrature induced by an $L_{2,μ}$-Marcinkiewicz-Zygmund family, and get the quadrature errors for $W_{2,μ}^r$, $r>(d+2μ)/2$, which are also order optimal. Meanwhile, we give the corresponding the weighted least $\ell_q$ approximation theorem and the least squares quadrature errors on the sphere.

math.NA↗

arXiv2022

Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

We consider the numerical integration $${\rm INT}_d(f)=\int_{\mathbb{B}^{d}}f(x)w_μ(x)dx $$ for the weighted Sobolev classes $BW^{r}_{p,μ}$ and the weighted Besov classes $BB_τ^r(L_{p,μ})$ in the randomized case setting, where $w_μ, \,μ\ge0,$ is the classical Jacobi weight on the ball $\Bbb B^d$, $1\le p\le \infty$, $r>(d+2μ)/p$, and $0<τ\le\infty$. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are $n^{-r/d-1/2+(1/p-1/2)_+}$. Compared to the orders $n^{-r/d}$ of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when $p>1$.

math.NA↗

arXiv2022

Weighted $L_p$ Markov factors with doubling weights on the ball

Let $L_{p,w},\ 1 \le p<\infty,$ denote the weighted $L_p$ space of functions on the unit ball $\Bbb B^d$ with a doubling weight $w$ on $\Bbb B^d$. The Markov factor for $L_{p,w}$ on a polynomial $P$ is defined by $\frac{\|\, |\nabla P|\,\|_{p,w}}{\|P\|_{p,w}}$, where $\nabla P$ is the gradient of $P$. We investigate the worst case Markov factors for $L_{p,w}\ (1\le p<\infty)$ and obtain that the degree of these factors are at most $2$. In particular, for the Jacobi weight $w_μ(x)=(1-|x|^2)^{μ-1/2}, \ μ\ge0$, the exponent $2$ is sharp. We also study the average case Markov factor for $L_{2,w}$ on random polynomials with independent $N(0, σ^2)$ coefficients and obtain that the upper bound of the average (expected) Markov factor is order degree to the $3/2$, as compared to the degree squared worst case upper bound.

math.CA↗

arXiv2021

Entropy numbers of weighted Sobolev classes on the unit sphere

We obtain the asymptotic orders of entropy numbers of Sobolev classes on the unit sphere with Dunkl weight which associates with the finite reflection group. Moreover, the asymptotic order of entropy numbers of weighted Sobolev classes on the unit ball and on the standard simplex are discussed.

math.CA↗

arXiv2021

On the power of standard information for tractability for $L_2$-approximation in the average case setting

We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ for the absolute error criterion, and show that the power of $Λ^{\rm std}$ is the same as that of $Λ^{\rm all}$ for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by E.Novak and H.Woźniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

math.NA↗

arXiv2021

Approximation and quadrature by weighted least squares polynomials on the sphere

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a usual compact space $\mathcal M$, Gröchenig introduced the weighted least squares polynomials and the least squares quadrature from pointwise samples of a function, and obtained approximation theorems and quadrature errors. In this paper we obtain approximation theorems and quadrature errors on the sphere which are optimal. We also give upper bounds of the operator norms of the weighted least squares operators.

math.NA↗