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Feng Dai

Publications and source records attributed to Feng Dai.

At least 37 records · Page 2Linked to original sources
arXiv2025

Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces

Let $Λ_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*} \label{0-1}Δ_r f(x,y): =\sup_{|h|\leq y} |Δ_h^r f(x)|\leq L y^s, \ x\in\mathbb{R}^n, \ y \in(0, 1]. \end{equation*} Here (and throughout the article) $\mathfrak{C}$ refers to continuous functions, and $Δ_h^r$ is the usual $r$-th order difference operator with step $h\in\mathbb{R}^n$. For each $f\in Λ_s$ and $\varepsilon\in(0,L)$, let $ S(f,\varepsilon):= \{ (x,y)\in\mathbb{R}^n\times [0,1]: \frac {Δ_r f(x,y)}{y^s}>\varepsilon\}$, and let $μ: \mathcal{B}(\mathbb{R}_+^{n+1})\to [0,\infty]$ be a suitably defined nonnegative extended real-valued function on the Borel $σ$-algebra of subsets of $\mathbb{R}_+^{n+1}$. Let $\varepsilon(f)$ be the infimum of all $\varepsilon\in(0,\infty)$ such that $μ(S(f,\varepsilon))<\infty$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap Λ_s$ of $Λ_s$ for various function spaces $V$ (including Sobolev, Besov--Triebel--Lizorkin, and Besov--Triebel--Lizorkin-type spaces) in terms of $\varepsilon(f)$, showing that \begin{equation*} \varepsilon(f)\sim \mathrm{dist} (f, V\cap Λ_s)_{Λ_s}: = \inf_{g\in Λ_s\cap V} \|f-g\|_{Λ_s}.\end{equation*} Moreover, we present our results in a general framework based on quasi-normed lattices of function sequences $X$ and Daubechies $s$-Lipschitz $X$-based spaces.

math.FA↗
arXiv2024

Exact: Exploring Space-Time Perceptive Clues for Weakly Supervised Satellite Image Time Series Semantic Segmentation

Automated crop mapping through Satellite Image Time Series (SITS) has emerged as a crucial avenue for agricultural monitoring and management. However, due to the low resolution and unclear parcel boundaries, annotating pixel-level masks is exceptionally complex and time-consuming in SITS. This paper embraces the weakly supervised paradigm (i.e., only image-level categories available) to liberate the crop mapping task from the exhaustive annotation burden. The unique characteristics of SITS give rise to several challenges in weakly supervised learning: (1) noise perturbation from spatially neighboring regions, and (2) erroneous semantic bias from anomalous temporal periods. To address the above difficulties, we propose a novel method, termed exploring space-time perceptive clues (Exact). First, we introduce a set of spatial clues to explicitly capture the representative patterns of different crops from the most class-relative regions. Besides, we leverage the temporal-to-class interaction of the model to emphasize the contributions of pivotal clips, thereby enhancing the model perception for crop regions. Build upon the space-time perceptive clues, we derive the clue-based CAMs to effectively supervise the SITS segmentation network. Our method demonstrates impressive performance on various SITS benchmarks. Remarkably, the segmentation network trained on Exact-generated masks achieves 95% of its fully supervised performance, showing the bright promise of weakly supervised paradigm in crop mapping scenario. Our code will be publicly available.

cs.CV↗
arXiv2024

On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^α$-domains

We prove new Bernstein and Markov type inequalities in $L^p$ spaces associated with the normal and the tangential derivatives on the boundary of a general compact $C^α$-domain with $1\leq α\leq 2$. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of $L^p$ norms of algebraic polynomials on $C^α$-domains with asymptotically optimal number of function samples used.

math.NA↗
arXiv2024

Compression using Quasi-Interpolation

We consider quasi-interpolation with a main application in radial basis function approximations and compression in this article. Constructing and using these quasi-interpolants, we consider wavelet and compression-type approximations from their linear spaces and provide convergence estimates. The results include an error estimate for nonlinear approximation by quasi-interpolation, results about compression in the space of continuous functions and a pointwise convergence estimate for approximands of low smoothness.

math.NA↗
arXiv2024

TopoLogic: An Interpretable Pipeline for Lane Topology Reasoning on Driving Scenes

As an emerging task that integrates perception and reasoning, topology reasoning in autonomous driving scenes has recently garnered widespread attention. However, existing work often emphasizes "perception over reasoning": they typically boost reasoning performance by enhancing the perception of lanes and directly adopt MLP to learn lane topology from lane query. This paradigm overlooks the geometric features intrinsic to the lanes themselves and are prone to being influenced by inherent endpoint shifts in lane detection. To tackle this issue, we propose an interpretable method for lane topology reasoning based on lane geometric distance and lane query similarity, named TopoLogic. This method mitigates the impact of endpoint shifts in geometric space, and introduces explicit similarity calculation in semantic space as a complement. By integrating results from both spaces, our methods provides more comprehensive information for lane topology. Ultimately, our approach significantly outperforms the existing state-of-the-art methods on the mainstream benchmark OpenLane-V2 (23.9 v.s. 10.9 in TOP$_{ll}$ and 44.1 v.s. 39.8 in OLS on subset_A. Additionally, our proposed geometric distance topology reasoning method can be incorporated into well-trained models without re-training, significantly boost the performance of lane topology reasoning. The code is released at https://github.com/Franpin/TopoLogic.

cs.CV↗
arXiv2024

Rethinking Boundary Discontinuity Problem for Oriented Object Detection

Oriented object detection has been developed rapidly in the past few years, where rotation equivariance is crucial for detectors to predict rotated boxes. It is expected that the prediction can maintain the corresponding rotation when objects rotate, but severe mutation in angular prediction is sometimes observed when objects rotate near the boundary angle, which is well-known boundary discontinuity problem. The problem has been long believed to be caused by the sharp loss increase at the angular boundary, and widely used joint-optim IoU-like methods deal with this problem by loss-smoothing. However, we experimentally find that even state-of-the-art IoU-like methods actually fail to solve the problem. On further analysis, we find that the key to solution lies in encoding mode of the smoothing function rather than in joint or independent optimization. In existing IoU-like methods, the model essentially attempts to fit the angular relationship between box and object, where the break point at angular boundary makes the predictions highly unstable.To deal with this issue, we propose a dual-optimization paradigm for angles. We decouple reversibility and joint-optim from single smoothing function into two distinct entities, which for the first time achieves the objectives of both correcting angular boundary and blending angle with other parameters.Extensive experiments on multiple datasets show that boundary discontinuity problem is well-addressed. Moreover, typical IoU-like methods are improved to the same level without obvious performance gap. The code is available at https://github.com/hangxu-cv/cvpr24acm.

cs.CV↗
arXiv2023

Polynomial approximation on $C^2$-domains

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $Ω\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for the best polynomial approximation in $L_p(Ω)$. The Jackson inequality is established for the full range of $0<p\leq \infty$, while its proof relies on a recently established Whitney type estimates with constants depending only on certain parameters; and on a highly localized polynomial partitions of unity on a $C^2$-domain which is of independent interest. The inverse inequality is established for $1\leq p\leq \infty$, and its proof relies on a recently proved Bernstein type inequality associated with the tangential derivatives on the boundary of $Ω$. Such an inequality also allows us to establish the inverse theorem for Ivanov's average moduli of smoothness on general compact $C^2$-domains.

math.CA↗
arXiv2022

Brezis--Van Schaftingen--Yung Formulae in Ball Banach Function Spaces with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

Let $X$ be a ball Banach function space on ${\mathbb R}^n$. In this article, under some mild assumptions about both $X$ and the boundedness of the Hardy--Littlewood maximal operator on the associate space of the convexification of $X$, the authors prove that, for any locally integrable function $f$ with $\|\,|\nabla f|\,\|_{X}<\infty$, $$\sup_{λ\in(0,\infty)}λ\left \|\left|\left\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\right\}\right|^{\frac{1}{q}} \right\|_X\sim \|\,|\nabla f|\,\|_X$$ with the positive equivalence constants independent of $f$, where the index $q\in(0,\infty)$ is related to $X$ and $|\{y\in{\mathbb R}^n:\ |f(\cdot)-f(y)| >λ|\cdot-y|^{\frac{n}{q}+1}\}|$ is the Lebesgue measure of the set under consideration. In particular, when $X:=L^p({\mathbb R}^n)$ with $p\in [1,\infty)$, the above formulae hold true for any given $q\in (0,\infty)$ with $n(\frac{1}{p}-\frac{1}{q})<1$, which when $q=p$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo--Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on $X'$ of the Hardy--Littlewood maximal operator, and the exquisite geometry of $\mathbb{R}^n.$

math.CA↗
arXiv2022

Cycle Self-Training for Semi-Supervised Object Detection with Distribution Consistency Reweighting

Recently, many semi-supervised object detection (SSOD) methods adopt teacher-student framework and have achieved state-of-the-art results. However, the teacher network is tightly coupled with the student network since the teacher is an exponential moving average (EMA) of the student, which causes a performance bottleneck. To address the coupling problem, we propose a Cycle Self-Training (CST) framework for SSOD, which consists of two teachers T1 and T2, two students S1 and S2. Based on these networks, a cycle self-training mechanism is built, i.e., S1${\rightarrow}$T1${\rightarrow}$S2${\rightarrow}$T2${\rightarrow}$S1. For S${\rightarrow}$T, we also utilize the EMA weights of the students to update the teachers. For T${\rightarrow}$S, instead of providing supervision for its own student S1(S2) directly, the teacher T1(T2) generates pseudo-labels for the student S2(S1), which looses the coupling effect. Moreover, owing to the property of EMA, the teacher is most likely to accumulate the biases from the student and make the mistakes irreversible. To mitigate the problem, we also propose a distribution consistency reweighting strategy, where pseudo-labels are reweighted based on distribution consistency across the teachers T1 and T2. With the strategy, the two students S2 and S1 can be trained robustly with noisy pseudo labels to avoid confirmation biases. Extensive experiments prove the superiority of CST by consistently improving the AP over the baseline and outperforming state-of-the-art methods by 2.1% absolute AP improvements with scarce labeled data.

cs.CV↗
arXiv2021

Poincaré Inequality Meets Brezis--Van Schaftingen--Yung Formula on Metric Measure Spaces

Let $(\mathcal{X}, ρ, μ)$ be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with compact support satisfying that $\operatorname{Lip} f:=\limsup _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ is also a continuous function with compact support and $\operatorname{Lip} f=\lim _{r \rightarrow 0} \sup_{y\in B(\cdot, r)} |f(\cdot)-f(y)|/r$ converges uniformly. Let $p \in[1,\infty)$. In this article, the authors prove that, for any $f\in\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$, \begin{align*} &\sup_{λ\in(0,\infty)}λ^p\int_{\mathcal{X}} μ\left(\left\{y\in \mathcal{X}:\ |f(x)-f(y)|>λρ(x,y) [V(x,y)]^{\frac 1p}\right\}\right)\, dμ(x)\\ &\quad\sim \int_{\mathcal{X}} [\operatorname{Lip}f(x)]^p \,dμ(x) \end{align*} with the positive equivalence constants independent of $f$, where $V(x,y):=μ(B(x,ρ(x,y)))$. This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the $n$-dimensional Euclidean space ${\mathbb R}^n$ to $\mathcal{X}$. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo--Nirenberg inequalities in $\mathcal{X}$. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, ${\mathbb R}^n$ with weighted Lebesgue measure and the complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincaré inequality.

math.FA↗
arXiv2021

$L^p$-Bernstein inequalities on $C^2$-domains and applications to discretization

We prove a new Bernstein type inequality in $L^p$ spaces associated with the tangential derivatives on the boundary of a general compact $C^2$-domain. We give two applications: Marcinkiewicz type inequality for discretization of $L^p$ norm and positive cubature formula. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.

math.CA↗
arXiv2021

Unbiased IoU for Spherical Image Object Detection

As one of the most fundamental and challenging problems in computer vision, object detection tries to locate object instances and find their categories in natural images. The most important step in the evaluation of object detection algorithm is calculating the intersection-over-union (IoU) between the predicted bounding box and the ground truth one. Although this procedure is well-defined and solved for planar images, it is not easy for spherical image object detection. Existing methods either compute the IoUs based on biased bounding box representations or make excessive approximations, thus would give incorrect results. In this paper, we first identify that spherical rectangles are unbiased bounding boxes for objects in spherical images, and then propose an analytical method for IoU calculation without any approximations. Based on the unbiased representation and calculation, we also present an anchor free object detection algorithm for spherical images. The experiments on two spherical object detection datasets show that the proposed method can achieve better performance than existing methods.

cs.CV↗
arXiv2021

Universal sampling discretization

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(Ω, μ)$ spanned by a uniformly bounded Riesz basis $Φ_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal discretization for integral norms $L_q(Ω,μ)$ of functions from the collection of all subspaces of $X_N$ spanned by $v$ elements of $Φ_N$ with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$. This last bound on $m$ is much better than previously known bounds which are quadratic in $v$. Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.

math.FA↗
arXiv2021

On directional Whitney inequality

This paper studies a new Whitney type inequality on a compact domain $Ω\subset {\mathbb{R}}^d$ that takes the form $$\inf_{Q\in Π_{r-1}^d({\mathcal{E}})} \|f-Q\|_p \leq C(p,r,Ω) ω_{\mathcal{E}}^r(f,{\rm diam}(Ω))_p,\ \ r\in {\mathbb{N}},\ \ 0 0$. It is proved that ${\mathcal{N}}_d(Ω)=d$ for every connected $C^2$-domain $Ω\subset {\mathbb{R}}^d$, for $d=2$ and every planar convex body $Ω\subset {\mathbb{R}}^2$, and for $d\ge 3$ and every almost smooth convex body $Ω\subset {\mathbb{R}}^d$. [See the pre-print for the complete abstract - not included here due to arXiv limitations.]

math.NA↗
arXiv2020

Discretization on high-dimensional domains

Let $μ$ be a Borel probability measure on a compact path-connected metric space $(X, ρ)$ for which there exist constants $c,β>1$ such that $μ(B) \geq c r^β$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $Φ:[0,\infty)\to R$ that piecewisely lie in a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $ρ$ and the measure $μ$ that for each positive integer $N\geq 2$, and each $g\in L^\infty(X, dμ)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N}\in X$ and real numbers $λ_1, \ldots, λ_{ N}$ such that for any $x\in X$, \begin{align*} & \left| \int_X Φ(ρ(x, y)) g(y) \,d μ(y) - \sum_{j = 1}^{ N} λ_j Φ(ρ(x, y_j)) \right| \leq C N^{- \frac{1}{2} - \frac{3}{2β}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $S^d$ of $R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac12}\sqrt{\log N}$.

math.NA↗
arXiv2019

Polynomial approximation on $C^2$-domains

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $Ω\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for best polynomial approximation in $L_p(Ω)$. The Jackson inequality is established for the full range of $0<p\leq \infty$, while its proof relies on (i) Whitney type estimates with constants depending only on certain parameters; and (ii) highly localized polynomial partitions of unity on a $C^2$-domain. Both (i) and (ii) are of independent interest. In particular, our Whitney type estimate (i) is established for directional moduli of smoothness rather than the ordinary moduli of smoothness, and is applicable to functions on a very wide class of domains (not necessarily convex). It generalizes an earlier result of Dekel and Leviatan on Whitney type estimates on convex domains. The inverse inequality is established for $1\leq p\leq \infty$, and its proof relies on a new Bernstein type inequality associated with the tangential derivatives on the boundary of $Ω$. Such an inequality also allows us to establish the Marcinkiewicz-Zygmund type inequalities, positive cubature formula, as well as the inverse theorem for Ivanov's average moduli of smoothness on general compact $C^2$-domains.

math.CA↗
arXiv2019

Estimates of the asymptotic Nikolskii constants for spherical polynomials

Let $Π_n^d$ denote the space of spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$ that is equipped with the surface Lebesgue measure $dσ$ normalized by $\int_{\mathbb{S}^d} \, dσ(x)=1$. This paper establishes a close connection between the asymptotic Nikolskii constant, $$ \mathcal{L}^\ast(d):=\lim_{n\to \infty} \frac 1 {\dim Π_n^d} \sup_{f\in Π_n^d} \frac { \|f\|_{L^\infty(\mathbb{S}^d)}}{\|f\|_{L^1(\mathbb{S}^d)}},$$ and the following extremal problem: $$ \mathcal{I}_α:=\inf_{a_k} \Bigl\| j_{α+1} (t)- \sum_{k=1}^\infty a_k j_α \bigl( q_{α+1,k}t/q_{α+1,1}\bigr)\Bigr\|_{L^\infty(\mathbb{R}_+)} $$ with the infimum being taken over all sequences $\{a_k\}_{k=1}^\infty\subset \mathbb{R}$ such that the infinite series converges absolutely a.e. on $\mathbb{R}_+$. Here $j_α$ denotes the Bessel function of the first kind normalized so that $j_α(0)=1$, and $\{q_{α+1,k}\}_{k=1}^\infty$ denotes the strict increasing sequence of all positive zeros of $j_{α+1}$. We prove that for $α\ge -0.272$, $$\mathcal{I}_α= \frac{\int_{0}^{q_{α+1,1}}j_{α+1}(t)t^{2α+1}\,dt}{\int_{0}^{q_{α+1,1}}t^{2α+1}\,dt}= {}_{1}F_{2}\Bigl(α+1;α+2,α+2;-\frac{q_{α+1,1}^{2}}{4}\Bigr). $$ As a result, we deduce that the constant $\mathcal{L}^\ast(d)$ goes to zero exponentially fast as $d\to\infty$: \[ 0.5^d\le \mathcal{L}^{*}(d)\le (0.857\cdots)^{d\,(1+\varepsilon_d)} \ \ \ \ \ \text{with $\varepsilon_d =O(d^{-2/3})$.} \]

math.CA↗