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版本更新 · v2arXiv:2611.06998

Resolving Tusnády's Problem

Nicola Wengiel

原文摘要

We prove a lower bound of $Ω_d((\log N)^{d-1/2})$ for the worst-case combinatorial discrepancy of $N$-point sets with respect to anchored boxes in every fixed dimension $d\ge3$. Together with Nikolov's upper bound of the same order and Wei's planar lower bound, this settles Tusnády's problem: the discrepancy of axis-parallel boxes is $Θ_d((\log N)^{d-1/2})$ in every fixed dimension $d\ge2$. The proof combines smoothed product test functions, sampling discretization, and an entropy argument for conditioned Gaussian vectors. The matching upper and lower bounds are also formalized in Lean 4.

math.CA · cs.CG · cs.DS · math.CO

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.29211

On the gradient conjecture at infinity: Finite spherical length for gradient trajectories of definable potentials

Thang Do, Arnulf Jentzen, Benno Kuckuck

原文摘要

We prove that for every open $U \subseteq \mathbb{R}^n$ the spherical projection of every gradient trajectory $θ(t)$, $t \geq 0$, of a definable $C^1$-potential $F \colon U \to \mathbb{R}$ has finite length and, in particular, converges (directional convergence). In other words, every gradient trajectory of a definable $C^1$-potential has finite total angular motion. Here, definable means definable in an arbitrary o-minimal structure expanding the real field. Notably, the gradient trajectory in this result is an arbitrary solution of the gradient flow differential equation (whose solutions may not be unique) with no further assumptions: The gradient trajectory may be unbounded and diverge to infinity or it may be convergent. The established theorem therefore provides affirmative answers to both the gradient conjecture at infinity for definable potentials and the definable Thom gradient conjecture. We extend our conclusions from the Euclidean metric to a general class of ambient metrics including Riemannian metrics and we establish uniform local upper bounds for the lengths of the spherical projections of gradient flow trajectories. We finally apply our findings to the training of deep neural networks (DNNs) to conclude that, in the case of definable and continuously differentiable activation and loss functions, the spherical projection of the DNN training flow must always converge (directional convergence).

math.DS · cs.LG · math.AG · math.CA · math.LO

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.11461

Sharp Rates for Optimal Recovery of Bandlimited Functions from Localized Oversampling

Wentao Huang, Haizhang Zhang

原文摘要

It is well-known that one can recover a function $f$ bandlimited to $[-δ,δ]$ ( $0<δ<π$) from its finite sample data $f(j),-n\le j\le n$, with an approximation error exponentially decaying to zero as $n$ tends to infinity. Various regularized Shannon sampling formula have been proposed to achieve such exponential approximation. So far the best rate has been $$ O\biggl(\frac1{\sqrt{n}}\exp(-(π-δ)n)\biggr). $$ We establish that the optimal recovery algorithm in the worst-case scenario achieves an approximation error $ρ_{n,δ}$ satisfying $$ c_* \frac1{\sqrt{n}}(\tan\fracδ4)^{2n+1/2}\le ρ_{n,δ}\le C^* \frac1{\sqrt{n}}(\tan\fracδ4)^{2n+1/2}, \quad 0<δ\leqπ,\quad n\geq1, $$ where $c_*,C^*$ are positive constants independent of $n$ and $δ$. This result identifies the best possible exponential rate to reconstruct bandlimited functions from two-sided finite oversampling data. In other words, $$ ρ_{n,δ} \asymp \frac{\tan\fracδ4}{\sqrt{n}}\exp\bigl(-c_δ(π-δ)n\bigr), $$ where $c_δ$ is strictly decreasing on $(0,π]$ and $$ \lim_{δ\toπ^-}c_δ=1,\ \lim_{δ\to 0^+}c_δ=+\infty. $$ We also make the new finding that one can also reconstruct $f$ bandlimited to $[-δ,δ]$ with exponentially decaying error from one-sided sample data $f(j)$, $1\le j\le n$, and the sharp rate of approximation error $ρ_{n,δ}'$ of the optimal algorithm is $$ ρ_{n,δ}' \asymp \sqrt{δ/π}\sin^n(δ/2). $$

cs.IT · math.CA

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.09887

关于 ReLU 网络集合的（非）闭性

On the (non-)closedness of the set of ReLU networks

Eloi Martinet, Leon Bungert

AI 速读 基于摘要

作者证明单隐层 ReLU 网络集合在 Sobolev 空间中是弱闭的，而宽度有界的双隐层 ReLU 网络集合在 Sobolev 空间与连续函数空间中均不闭。

cs.LG · math.CA · math.FA · math.NA · math.OC

arXiv ↗PDF ↗
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We prove that the set $\mathcal{N}_N$ of ReLU networks with one hidden layer and at most $N\in\mathbb N$ neurons is weakly closed in $W^{1,p}(Ω)$, for a bounded connected open set $Ω\subset\mathbb R^d$. Consequently, every coercive, weakly lower semicontinuous energy on this space attains its minimum over $\mathcal{N}_N$. This covers the Deep Ritz method with convex integrands, Sobolev training and Lipschitz learning. Moreover, as the weight decay parameter tends to zero, minimizers of the regularized problem converge, up to subsequences, to minimizers with minimal parameter norm. We also prove that the set of ReLU networks with two hidden layers of bounded widths is not closed in $W^{1,p}(Ω)$ or in $C(\overlineΩ)$. This disproves a conjecture of Petersen, Raslan and Voigtlaender.

版本更新 · v4arXiv:2603.09722

A Novel Two-Parameter Penalty: Relaxation Degree Analysis and Sparse Signal Recovery

Ziwei Li, Wengu Chen, Huanmin Ge, Dachun Yang

原文摘要

In this article, we introduce a nonconvex two-parameter penalty function $P_{a,p}$, parameterized by $a\in(0,\infty)$ and $p\in(0,1]$, and the relaxation degree RD$_P$ for a separable nonconvex penalty function $P$. Based on $P_{a,p}$, we further propose the $P_{a,p}$ minimization framework for sparse signal recovery. This framework generalizes the TL1 minimization model established by S. Zhang and J. Xin (corresponding to the special case $p=1$) and provides a unified family of nonconvex penalty functions for sparse signal recovery. Using the sparse convex-combination technique, we establish both exact and stable sparse signal recovery under the restricted isometry property (RIP). To efficiently solve the resulting nonconvex optimization problem, we apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to develop the IRLSTLp algorithm for unconstrained $P_{a,p}$ minimization and prove some convergence results. Finally, numerical experiments compare recovery and runtime against single- and double-loop baselines, examine sensitivity to $(a,p)$, and assess the empirical correlation between RD$_{P_{a,p}}$ and recovery within the proposed family.

math.FA · cs.IT · math.CA

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.39145

Phase Retrieval of Bandlimited Signals Using a Morlet Wavelet and an Admissible Companion

Bei Liu, Meng Ning, Liping Wu, Qingyue Zhang

原文摘要

In this paper, we demonstrate that complex bandlimited signals can be uniquely recovered (up to a global phase factor) from two families of wavelet transform magnitudes, sampled at twice the Nyquist rate in two frequency bins. These families come from translations of the fixed dilated Morlet wavelet and a fixed companion wavelet. The companion wavelet admits an augmented Morlet--Gaussian representation. We rewrite the two families as two modulated Gaussian convolution measurements of an auxiliary bandlimited signal. Using a Gaussian convolution uniqueness result, we prove that samples on \((4B)^{-1}\mathbb{Z}\) determine every signal in \(PW_B^2\) (up to a global phase factor). Compared with earlier wavelet phase retrieval results, our approach requires significantly fewer sampling points.

math.CA · cs.IT

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2604.08095

The estimates for Boolean surface area of polynomial threshold functions

Fan Chang, Joseph Slote, Chun-Kai Tseng, Alexander Volberg, Haonan Zhang

原文摘要

Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture recently solved by OpenAI, \cite{GLsol}. In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the second bound \[ \mathbf{BSA}[f]\le \exp(C(d)\sqrt{\log n}). \] This bound was obtained in \cite{FSVZ} without clarifying the magnitude of $C(d)$, which we do below. For lower moments $\bE |\nabla f|^η \le C(d, η)$ there is no dependence on $n$ for $0<η<1$.

cs.CC · math.AP · math.CA · math.PR

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arXiv ↗PDF ↗