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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 $\mathbb R^d$., Comment: 48 pages

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., Comment: 23 pages

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig 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,\tau}^r(\Bbb M),\, 0<\tau\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., Comment: 23 pages

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ 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 ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>0$, which gives a clue to Open problem 3 as posed by Krieg and Vyb\'iral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{\alpha,\beta}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $\alpha>d/2$, and apply Hilbert space structure and Vyb\'iral's theorem about Schur product theory to obtain lower bounds in the case $\alpha=d/2,\,\beta>1/2$ 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 ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$, which are order optimal., Comment: 24 pages

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 $w_\mu(x)=(1-|x|^2)^{\mu-1/2}, \ \mu\ge0$, the exponent $2$ is sharp. We also study the average case Markov factor for $L_{2,w}$ on random polynomials with independent $N(0, \sigma^2)$ 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., Comment: 17 pages

We consider the numerical integration $${\rm INT}_d(f)=\int_{\mathbb{B}^{d}}f(x)w_\mu(x)dx $$ for the weighted Sobolev classes $BW^{r}_{p,\mu}$ and the weighted Besov classes $BB_\tau^r(L_{p,\mu})$ in the randomized case setting, where $w_\mu, \,\mu\ge0,$ is the classical Jacobi weight on the ball $\Bbb B^d$, $1\le p\le \infty$, $r>(d+2\mu)/p$, and $0<\tau\le\infty$. 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 $p>1$., Comment: 19 pages

Let $L_{q,\mu},\, 1\le q<\infty, \ \mu\ge0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_\mu$ on the ball $\Bbb B^d$. We consider the weighted least $\ell_q$ approximation problem for a given $L_{q,\mu}$-Marcinkiewicz-Zygmund family on $\Bbb B^d$. We obtain the weighted least $\ell_q$ approximation errors for the weighted Sobolev space $W_{q,\mu}^r$, $r>(d+2\mu)/q$, which are order optimal. We also discuss the least squares quadrature induced by an $L_{2,\mu}$-Marcinkiewicz-Zygmund family, and get the quadrature errors for $W_{2,\mu}^r$, $r>(d+2\mu)/2$, 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., Comment: 17 pages

The training of deep neural networks (DNNs) is usually memory-hungry due to the limited device memory capacity of DNN accelerators. Characterizing the memory behaviors of DNN training is critical to optimize the device memory pressures. In this work, we pinpoint the memory behaviors of each device memory block of GPU during training by instrumenting the memory allocators of the runtime system. Our results show that the memory access patterns of device memory blocks are stable and follow an iterative fashion. These observations are useful for the future optimization of memory-efficient training from the perspective of raw memory access patterns., Comment: Submitted to ISPASS'21 poster

Accelerating the deep learning inference is very important for real-time applications. In this paper, we propose a novel method to fuse the layers of convolutional neural networks (CNNs) on Graphics Processing Units (GPUs), which applies data reuse analysis and access optimization in different levels of the memory hierarchy. To achieve the balance between computation and memory access, we explore the fusion opportunities in the CNN computation graph and propose three fusion modes of convolutional neural networks: straight, merge and split. Then, an approach for generating efficient fused code is designed, which goes deeper in multi-level memory usage for cross-layer data reuse. The effectiveness of our method is evaluated with the network layers from state-of-the-art CNNs on two different GPU platforms, NVIDIA TITAN Xp and Tesla P4. The experiments show that the average speedup is 2.02x on representative structures of CNNs, and 1.57x on end-to-end inference of SqueezeNet., Comment: 15 pages, 8 figures, to be published in Euro-Par 2020

In recent years, domain-specific hardware has brought significant performance improvements in deep learning (DL). Both industry and academia only focus on throughput when evaluating these AI accelerators, which usually are custom ASICs deployed in datacenter to speed up the inference phase of DL workloads. Pursuing higher hardware throughput such as OPS (Operation Per Second) using various optimizations seems to be their main design target. However, they ignore the importance of accuracy in the DL nature. Motivated by this, this paper argue that a single throughput metric can not comprehensively reflect the real-world performance of AI accelerators. To reveal this pitfall, we evaluates several frequently-used optimizations on a typical AI accelerator and quantifies their impact on accuracy and throughout under representative DL inference workloads. Based on our experimental results, we find that some optimizations cause significant loss on accuracy in some workloads, although it can improves the throughout. Furthermore, our results show the importance of end-to-end evaluation in DL.