arXiv subjects

Feng Dai

Publications and source records attributed to Feng Dai.

At least 55 records · Page 3Linked to original sources

arXiv2019

Dense Scale Network for Crowd Counting

Crowd counting has been widely studied by computer vision community in recent years. Due to the large scale variation, it remains to be a challenging task. Previous methods adopt either multi-column CNN or single-column CNN with multiple branches to deal with this problem. However, restricted by the number of columns or branches, these methods can only capture a few different scales and have limited capability. In this paper, we propose a simple but effective network called DSNet for crowd counting, which can be easily trained in an end-to-end fashion. The key component of our network is the dense dilated convolution block, in which each dilation layer is densely connected with the others to preserve information from continuously varied scales. The dilation rates in dilation layers are carefully selected to prevent the block from gridding artifacts. To further enlarge the range of scales covered by the network, we cascade three blocks and link them with dense residual connections. We also introduce a novel multi-scale density level consistency loss for performance improvement. To evaluate our method, we compare it with state-of-the-art algorithms on four crowd counting datasets (ShanghaiTech, UCF-QNRF, UCF_CC_50 and UCSD). Experimental results demonstrate that DSNet can achieve the best performance and make significant improvements on all the four datasets (30% on the UCF-QNRF and UCF_CC_50, and 20% on the others).

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arXiv2019

Nikolskii inequality for lacunary spherical polynomials

We prove that for $d\ge 2$, the asymptotic order of the usual Nikolskii inequality on $\mathbb{S}^d$ (also known as the reverse Hölder's inequality) can be significantly improved in many cases, for lacunary spherical polynomials of the form $f=\sum_{j=0}^m f_{n_j}$ with $f_{n_j}$ being a spherical harmonic of degree $n_j$ and $n_{j+1}-n_j\ge 3$. As is well known, for $d=1$, the Nikolskii inequality for trigonometric polynomials on the unit circle does not have such a phenomenon.

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arXiv2018

General and Refined Montgomery Lemmata

Montgomery's Lemma on the torus $\mathbb{T}^d$ states that a sum of $N$ Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: let $(M,g)$ be a smooth compact $d-$dimensional manifold without boundary, let $(ϕ_k)_{k=0}^{\infty}$ denote the Laplacian eigenfunctions, let $\left\{ x_1, \dots, x_N\right\} \subset M$ be a set of points and $\left\{a_1, \dots, a_N\right\} \subset \mathbb{R}_{\geq 0}$ be a sequence of nonnegative weights. Then $$\sum_{k=0}^{X}{ \left| \sum_{n=1}^{N}{ a_n ϕ_k(x_n)} \right|^2} \gtrsim_{(M,g)} \left(\sum_{i=1}^{N}{a_i^2} \right) \frac{ X}{(\log{X})^{\frac{d}{2}}}.$$ This result is sharp up to the logarithmic factor. Furthermore, we prove a refined spherical version of Montgomery's Lemma, and provide applications to estimates of discrepancy and discrete energies of $N$ points on the sphere $\mathbb{S}^{d}$.

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arXiv2017

DR2-Net: Deep Residual Reconstruction Network for Image Compressive Sensing

Most traditional algorithms for compressive sensing image reconstruction suffer from the intensive computation. Recently, deep learning-based reconstruction algorithms have been reported, which dramatically reduce the time complexity than iterative reconstruction algorithms. In this paper, we propose a novel \textbf{D}eep \textbf{R}esidual \textbf{R}econstruction Network (DR$^{2}$-Net) to reconstruct the image from its Compressively Sensed (CS) measurement. The DR$^{2}$-Net is proposed based on two observations: 1) linear mapping could reconstruct a high-quality preliminary image, and 2) residual learning could further improve the reconstruction quality. Accordingly, DR$^{2}$-Net consists of two components, \emph{i.e.,} linear mapping network and residual network, respectively. Specifically, the fully-connected layer in neural network implements the linear mapping network. We then expand the linear mapping network to DR$^{2}$-Net by adding several residual learning blocks to enhance the preliminary image. Extensive experiments demonstrate that the DR$^{2}$-Net outperforms traditional iterative methods and recent deep learning-based methods by large margins at measurement rates 0.01, 0.04, 0.1, and 0.25, respectively. The code of DR$^{2}$-Net has been released on: https://github.com/coldrainyht/caffe\_dr2

cs.CV↗

arXiv2017

Nikolskii constants for polynomials on the unit sphere

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space $Π_n^d$ of spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$ as $n\to\infty$. It is shown that for $0<p<\infty$, \[ \lim_{n\to \infty} \sup\Bigl\{\frac{\|P\|_{L^\infty(\mathbb{S}^d)}}{n^{\frac dp}\|P\|_{L^p(\mathbb{S}^d)}}:\ \ P\inΠ_n^d\Bigr\} =\sup\Bigl\{ \frac{\|f\|_{L^\infty(\mathbb{R}^{d})}}{\|f\|_{L^p(\mathbb{R}^d)}}:\ \ f\in\mathcal{E}_p^d \Bigr\}, \] where $\mathcal{E}_p^d$ denotes the space of all entire functions of spherical exponential type at most $1$ whose restrictions to $\mathbb{R}^d$ belong to the space $L^p(\mathbb{R}^d)$, and it is agreed that $0/0=0$. It is further proved that for $0<p<q<\infty$, \[ \liminf_{n\to \infty} \sup\Bigl\{\frac{\|P\|_{L^q(\mathbb{S}^d)}}{n^{d(1/p-1/q)}\|P\|_{L^p(\mathbb{S}^d)}}:\ \ P\inΠ_n^d\Bigr\} \ge \sup\Bigl\{ \frac{\|f\|_{L^q(\mathbb{R}^{d})}}{\|f\|_{L^p(\mathbb{R}^d)}}:\ \ f\in\mathcal{E}_p^d\Bigr\}. \] These results extend the recent results of Levin and Lubinsky for trigonometric polynomials on the unit circle. The paper also determines the exact value of the Nikolskii constant for nonnegative functions with $p=1$ and $q=\infty$: $$\lim_{n\to \infty} \sup_{0\leq P\inΠ_n^d}\frac{\|P\|_{L^\infty(\mathbb{S}^d)}}{\|P\|_{L^1(\mathbb{S}^d)}} =\sup_{0\leq f\in\mathcal{E}_1^d}\frac{\|f\|_{L^\infty(\mathbb{R}^{d})}}{\|f\|_{L^1(\mathbb{R}^d)}} =\frac1{4^d π^{d/2}Γ(d/2+1)}.$$

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arXiv2017

Chebyshev-type cubature formulas for doubling weights on spheres, balls and simplexes

This paper proves that given a doubling weight $w$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$, there exists a positive constant $K_w$ such that for each positive integer $n$ and each integer $N\geq \max_{x\in \mathbb{S}^{d-1}} \frac {K_w} {w(B(x, n^{-1}))}$, there exists a set of $N$ distinct nodes $z_1,\cdots, z_N$ on $\mathbb{S}^{d-1}$ which admits a strict Chebyshev-type cubature formula (CF) of degree $n$ for the measure $w(x) dσ_d(x)$, $$ \frac 1{w(\mathbb{S}^{d-1})} \int_{\mathbb{S}^{d-1}} f(x) w(x)\, dσ_d(x)=\frac 1N \sum_{j=1}^N f(z_j),\ \ \forall f\inΠ_n^d, $$ and which, if in addition $w\in L^\infty(\mathbb{S}^{d-1})$, satisfies $$\min_{1\leq i\neq j\leq N}\mathtt{d}(z_i,z_j)\geq c_{w,d} N^{-\frac1{d-1}}$$ for some positive constant $c_{w,d}$. Here, $dσ_d$ and $\mathtt{d}(\cdot, \cdot)$ denote the surface Lebesgue measure and the geodesic distance on $\mathbb{S}^{d-1}$ respectively, $B(x,r)$ denotes the spherical cap with center $x\in\mathbb{S}^{d-1}$ and radius $r>0$, $w(E)=\int_E w(x) \, dσ_d(x)$ for $E\subset\mathbb{S}^{d-1}$, and $Π_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\mathbb{S}^{d-1}$. It is also shown that the minimal number of nodes $\mathcal{N}_{n} (wdσ_d)$ in a strict Chebyshev-type CF of degree $n$ for a doubling weight $w$ on $\mathbb{S}^{d-1}$ satisfies $$\mathcal{N}_n (wdσ_d) \sim \max_{x\in \mathbb{S}^{d-1}} \frac 1 {w(B(x, n^{-1}))},\ \ n=1,2,\cdots.$$ Proofs of these results rely on new convex partitions of $\mathbb{S}^{d-1}$ that are regular with respect to a given weight $w$ and integer $N$. Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs ({\it Ann. of Math. (2)} {\bf 178}(2013), no. 2, 443--452,{\it Constr. Approx.} {\bf 41}(2015), no. 1, 93--112).

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arXiv2016

Geodesic distance Riesz energy on the sphere

We study energy integrals and discrete energies on the sphere, in particular, analogs of the Riesz energy with the geodesic distance in place of Euclidean, and observe that the range of exponents for which the uniform distribution optimizes such energies is different from the classical case. We also obtain a general form of the Stolarsky principle, which relates discrete energies to certain $L^2$ discrepancies. This leads to new proofs of discrepancy estimates, as well as the sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case.

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arXiv2016

Stolarsky principle and energy optimization on the sphere

The classical Stolarsky invariance principle connects the spherical cap $L^2$ discrepancy of a finite point set on the sphere to the pairwise sum of Euclidean distances between the points. In this paper we further explore and extend this phenomenon. In addition to a new elementary proof of this fact, we establish several new analogs, which relate various notions of discrepancy to different discrete energies. In particular, we find that the hemisphere discrepancy is related to the sum of geodesic distances. We also extend these results to arbitrary measures on the sphere and arbitrary notions of discrepancy and apply them to problems of energy optimization and combinatorial geometry and find that, surprisingly, the geodesic distance energy behaves differently than its Euclidean counterpart.

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arXiv2015

Littlewood-Paley Characterizations of Fractional Sobolev Spaces via Averages on Balls

In this paper, the authors characterize Sobolev spaces $W^{α,p}({\mathbb R}^n)$ with the smoothness order $α\in(0,2]$ and $p\in(\max\{1, \frac{2n}{2α+n}\},\infty)$, via the Lusin area function and the Littlewood-Paley $g_λ^\ast$-function in terms of centered ball averages. The authors also show that the condition $p\in(\max\{1, \frac{2n}{2α+n}\},\infty)$ is nearly sharp in the sense that these characterizations are no longer true when $p\in (1,\max\{1, \frac{2n}{2α+n}\})$. These characterizations provide a new possible way to introduce fractional Sobolev spaces with smoothness order in $(1,2]$ on metric measure spaces.

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arXiv2015

Characterizations of Besov and Triebel-Lizorkin Spaces via Averages on Balls

Let $\ell\in\mathbb{N}$ and $p\in(1,\infty]$. In this article, the authors prove that the sequence $\{f-B_{\ell,2^{-k}}f\}_{k\in\mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\ell,2^{-k}}f$ characterizes the Besov space $\dot B^α_{p,q}(\rn)$ with $q\in (0, \infty]$ and the Triebel-Lizorkin space $\dot F^α_{p,q}(\rn)$ with $q\in (1,\infty]$ when the smoothness order $α\in(0,2\ell)$. More precisely, it is proved that $f-B_{\ell,2^{-k}}f$ plays the same role as the approximation to the identity $φ_{2^{-k}}\ast f$ appearing in the definitions of $\dot B^α_{p,q}(\rn)$ and $\dot F^α_{p,q}(\rn)$. The corresponding results for inhomogeneous Besov and Triebel-Lizorkin spaces are also obtained. These results, for the first time, give a way to introduce Besov and Triebel-Lizorkin spaces with any smoothness order in $(0, 2\ell)$ on spaces of homogeneous type, where $\ell\in{\mathbb N}$.

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arXiv2014

The Hardy-Rellich inequality and uncertainty principle on the sphere

Let $Δ_0$ be the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. We show that the Hardy-Rellich inequality of the form $$ \int_{\mathbb{S}^{d-1}} \left | f (x)\right|^2 dσ(x) \leq c_d \min_{e\in \mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) \left |(-Δ_0)^{\frac{1}{2}}f(x) \right |^2 dσ(x) $$ holds for $d =2$ and $d \ge 4$ but does not hold for $d=3$ with any finite constant, and the optimal constant for the inequality is $c_d = 8/(d-3)^2$ for $d =2, 4, 5$ and, under additional restrictions on the function space, for $d\ge 6$. This inequality yields an uncertainty principle of the form $$ \min_{e\in\mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) |f(x)|^2 dσ(x) \int_{\mathbb{S}^{d-1}}\left |\nabla_0 f(x)\right |^2 dσ(x) \ge c'_d $$ on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new. This paper is published in Constructive Approximation, 40(2014): 141-171. An erratum is now appended.

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arXiv2014

Reverse Hölder's inequality for spherical harmonics

This paper determines the sharp asymptotic order of the following reverse Hölder inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$ as $n\to \infty$: \[\|Y_n\|_{L^q(\mathbb{S}^{d-1})}\leq C n^{α(p,q)}\|Y_n\|_{L^p(\mathbb{S}^{d-1})},\quad 0<p<q\leq \infty.\] In many cases, these sharp estimates turn out to be significantly better than the corresponding estimates in the Nilkolskii inequality for spherical polynomials. Furthermore, they allow us to improve two recent results on the restriction conjecture and the sharp Pitt inequalities for the Fourier transform on $\mathbb{R}^d$.

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arXiv2013

Weighted Fractional Bernstein's inequalities and their applications

This paper studies the following weighted, fractional Bernstein inequality for spherical polynomials on $\sph$: \begin{equation}\label{4-1-TD-ab} \|(-Δ_0)^{r/2} f\|_{p,w}\leq C_w n^{r} \|f\|_{p,w}, \ \ \forall f\in Π_n^d, \end{equation} where $Π_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\sph$, and $(-Δ_0)^{r/2}$ is the fractional Laplacian-Beltrami operator on $\sph$. A new class of doubling weights with conditions weaker than the $A_p$ is introduced, and used to fully characterize those doubling weights $w$ on $\sph$ for which the weighted Bernstein inequality \eqref{4-1-TD-ab} holds for some $1\leq p\leq \infty$ and all $r>τ$. In the unweighted case, it is shown that if $0 0$ is not an even integer, then \eqref{4-1-TD-ab} with $w\equiv 1$ holds if and only if $r>(d-1)(\f 1p-1)$. As applications, we show that any function $f\in L_p(\sph)$ with $0<p<1$ can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series, and establish a sharp Sobolev type Embedding theorem for the weighted Besov spaces with respect to general doubling weights.

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arXiv2010

Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball

This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these spaces, including simultaneous approximation by polynomials and relation between best approximation to a function and to its derivatives.

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arXiv2010

Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a $K$-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding $K$-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.

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arXiv2007

Boundedness of projection operators and Cesàro means in weighted $L^p$ space on the unit sphere

For the weight function $\prod_{i=1}^{d+1}|x_i|^{2\k_i}$ on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Cesàro $(C,δ)$ means in the weighted $L^p$ space for $δ$ below the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex.

math.CA↗

arXiv2007

Cesàro means of orthogonal expansions in several variables

Cesàro $(C,δ)$ means are studied for orthogonal expansions with respect to the weight function $\prod_{i=1}^{d}|x_i|^{2\k_i}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the $(C,\d)$ kernel with $\d > -1$ and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Cesàro means and the projection operator on these domains.

math.CA↗

arXiv2007

Maximal function and Multiplier Theorem for Weighted Space on the Unit Sphere

For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.

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