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Feng Dai

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math.CA math.FA Computer Vision Machine Learning math.AP

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preprint2026arXiv

Max-Min Neural Network Operators For Approximation of Multivariate Functions

In this paper, we develop a multivariate framework for approximation by max-min neural network operators. Building on the recent advances in approximation theory by neural network operators, particularly, the univariate max-min operators, we propose and analyze new multivariate operators activated by sigmoidal functions. We establish pointwise and uniform convergence theorems and derive quantitative estimates for the order of approximation via modulus of continuity and multivariate generalized absolute moment. Our results demonstrate that multivariate max-min structure of operators, besides their algebraic elegance, provide efficient and stable approximation tools in both theoretical and applied settings.

preprint2026arXiv

TEA: Temporal Adaptive Satellite Image Semantic Segmentation

Crop mapping based on satellite images time-series (SITS) holds substantial economic value in agricultural production settings, in which parcel segmentation is an essential step. Existing approaches have achieved notable advancements in SITS segmentation with predetermined sequence lengths. However, we found that these approaches overlooked the generalization capability of models across scenarios with varying temporal length, leading to markedly poor segmentation results in such cases. To address this issue, we propose TEA, a TEmporal Adaptive SITS semantic segmentation method to enhance the model's resilience under varying sequence lengths. We introduce a teacher model that encapsulates the global sequence knowledge to guide a student model with adaptive temporal input lengths. Specifically, teacher shapes the student's feature space via intermediate embedding, prototypes and soft label perspectives to realize knowledge transfer, while dynamically aggregating student model to mitigate knowledge forgetting. Finally, we introduce full-sequence reconstruction as an auxiliary task to further enhance the quality of representations across inputs of varying temporal lengths. Through extensive experiments, we demonstrate that our method brings remarkable improvements across inputs of different temporal lengths on common benchmarks. Our code will be publicly available.

preprint2026arXiv

VIP: Visual-guided Prompt Evolution for Efficient Dense Vision-Language Inference

Pursuing training-free open-vocabulary semantic segmentation in an efficient and generalizable manner remains challenging due to the deep-seated spatial bias in CLIP. To overcome the limitations of existing solutions, this work moves beyond the CLIP-based paradigm and harnesses the recent spatially-aware dino $.$ txt framework to facilitate more efficient and high-quality dense prediction. While dino $.$ txt exhibits robust spatial awareness, we find that the semantic ambiguity of text queries gives rise to severe mismatch within its dense cross-modal interactions. To address this, we introduce Visual-guided Prompt evolution (VIP) to rectify the semantic expressiveness of text queries in dino $.$ txt, unleashing its potential for fine-grained object perception. Towards this end, VIP integrates alias expansion with a visual-guided distillation mechanism to mine valuable semantic cues, which are robustly aggregated in a saliency-aware manner to yield a high-fidelity prediction. Extensive evaluations demonstrate that VIP: 1. surpasses the top-leading methods by 1.4%-8.4% average mIoU, 2. generalizes well to diverse challenging domains, and 3. requires marginal inference time and memory overhead.

preprint2022arXiv

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 $\mathbb{R}^n.$

preprint2022arXiv

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.

preprint2016arXiv

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.

preprint2016arXiv

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.

preprint2015arXiv

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 $\ell\in{\mathbb N}$ .

preprint2015arXiv

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.

preprint2014arXiv

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 $\mathbb{R}^d$ .

preprint2014arXiv

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.

preprint2013arXiv

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 $0<p<\infty$ and $r>0$ 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.

preprint2010arXiv

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.

preprint2010arXiv

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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Feng Dai

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14 publications

Max-Min Neural Network Operators For Approximation of Multivariate Functions

TEA: Temporal Adaptive Satellite Image Semantic Segmentation

VIP: Visual-guided Prompt Evolution for Efficient Dense Vision-Language Inference

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

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

Geodesic distance Riesz energy on the sphere

Stolarsky principle and energy optimization on the sphere

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

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

Reverse Hölder's inequality for spherical harmonics

The Hardy-Rellich inequality and uncertainty principle on the sphere

Weighted Fractional Bernstein's inequalities and their applications

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

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

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