Applied and Computational Harmonic Analysis最新文献

英文中文

Model agnostic signal encoding by leaky integrate-and-fire, performance and uncertainty 基于泄漏积分与发射、性能与不确定性的模型不可知信号编码

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-02-02 DOI: 10.1016/j.acha.2026.101856

Diana Carbajal, José Luis Romero

引用次数: 0

Dictionary Learning under Symmetries via Group Representations 基于群表示的对称字典学习

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-01-31 DOI: 10.1016/j.acha.2026.101855

Subhroshekhar Ghosh, Aaron Y.R. Low, Yong Sheng Soh, Zhuohang Feng, Brendan K.Y. Tan

引用次数: 0

Hierarchic flows to estimate and sample high-dimensional probabilities 估计和抽样高维概率的层次流

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-01-23 DOI: 10.1016/j.acha.2026.101854

Etienne Lempereur , Stéphane Mallat

Finding low-dimensional interpretable models of complex physical fields such as turbulence remains an open question, 80 years after the pioneer work of Kolmogorov. Estimating high-dimensional probability distributions from data samples suffers from an optimization and an approximation curse of dimensionality. It may be avoided by following a hierarchic probability flow from coarse to fine scales. This inverse renormalization group is defined by conditional probabilities across scales, renormalized in a wavelet basis. For a φ⁴ scalar potential, sampling these hierarchic models avoids the critical slowing down at the phase transition. In a well chosen wavelet basis, conditional probabilities can be captured with low dimensional parametric models, because interactions between wavelet coefficients are local in space and scales. An outstanding issue is also to approximate non-Gaussian fields having long-range interactions in space and across scales. We introduce low-dimensional models of wavelet conditional probabilities with the scattering covariance. It is calculated with a second wavelet transform, which defines interactions over two hierarchies of scales. We estimate and sample these wavelet scattering models to generate 2D vorticity fields of turbulence, and images of dark matter densities.

在Kolmogorov的开创性工作80年后，寻找复杂物理场（如湍流）的低维可解释模型仍然是一个悬而未决的问题。从数据样本中估计高维概率分布受到维度优化和近似诅咒的困扰。它可以通过遵循从粗到细的层次概率流来避免。这个逆重整化群由跨尺度的条件概率定义，在小波基中重整化。对于φ4标量势，对这些分层模型进行采样可以避免相变时的临界减速。在选择好的小波基中，条件概率可以用低维参数模型捕获，因为小波系数之间的相互作用在空间和尺度上是局部的。一个突出的问题也是近似非高斯场具有空间和跨尺度的远程相互作用。引入了具有散射协方差的小波条件概率的低维模型。它是用第二个小波变换计算的，它定义了两个层次尺度上的相互作用。我们对这些小波散射模型进行估计和采样，以生成二维湍流涡度场和暗物质密度图像。

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引用次数: 0

The spectrality of Cantor-Moran measure and Fuglede’s conjecture Cantor-Moran测度的频谱性与Fuglede猜想

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2026-01-09 DOI: 10.1016/j.acha.2026.101853

Jinsong Liu , Zheng-Yi Lu , Ting Zhou

Let

{(p_{n}, D_{n}, L_{n})}

be a sequence of Hadamard triples on

R

. Suppose that the associated Cantor-Moran measure

μ_{{p_{n}, D_{n}}} = δ_{p_{1}^{- 1} D_{1}} * δ_{{(p_{2} p_{1})}^{- 1} D_{2}} * \dots,

where

\sup_{n} {| p_{n}^{- 1} d | : d \in D_{n}} < \infty

and

\sup # D_{n} < \infty

. It has been observed that the spectrality of

μ_{{p_{n}, D_{n}}}

is determined by equi-positivity. A significant problem is what kind of Moran measures can satisfy this property. In this paper, we introduce the conception of Double Points Condition Set (DPCS) to characterize the equi-positivity equivalently. As applications of our characterization, we show that all singularly continuous Cantor-Moran measures are spectral. For the absolutely continuous case, we study Fuglede’s Conjecture on Cantor-Moran set. We show that the equi-positivity of

μ_{{p_{n}, D_{n}}}

implies the tiling of its support, and the reverse direction holds under certain conditions.

设{(pn,Dn,Ln)}是r上的Hadamard三元组序列，设相关的Cantor-Moran测度μ{pn，Dn}=δp1−1D1*δ(p2p1)−1D2*⋯，其中supn{|pn−1d|:d∈Dn}<；∞和sup#Dn<；∞。已经观察到μ{pn，Dn}的光谱是由等正性决定的。一个重要的问题是什么样的Moran测度能够满足这一性质。本文引入双点条件集（DPCS）的概念来等价地描述等正性。作为我们的表征的应用，我们证明了所有奇连续Cantor-Moran测度都是谱的。对于绝对连续的情况，我们研究了Cantor-Moran集合上的Fuglede猜想。我们证明了μ{pn，Dn}的等正性意味着它的支撑是平铺的，并且在某些条件下逆方向成立。

引用次数: 0

Ramanujan sums in signal recovery and uncertainty principle inequalities 信号恢复和不确定性原理不等式中的Ramanujan和

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-25 DOI: 10.1016/j.acha.2025.101852

Sahil Kalra, Niraj K. Shukla

This paper explores the perfect reconstruction property of filter banks based on Ramanujan sums and their applications in signal recovery. Originally introduced by Srinivasa Ramanujan, Ramanujan sums serve as powerful tools for extracting periodic components from signals and form the foundation of Ramanujan filter banks. We investigate the perfect reconstruction property of these filter banks and analyze their robustness against erasures for discrete-time signals in the finite-dimensional space

ℓ^{2} (Z_{N})

(i.e.,

C^{N}

). The study is further extended to non-uniform Ramanujan filter banks, showcasing their ability to address the limitations of uniform ones. Employing the reconstruction properties of uniform Ramanujan filter banks, we present an uncertainty principle associated with a tight frame generated by shifts of Ramanujan sums. This principle establishes representation inequalities in terms of Euler’s totient function ϕ(n), which provide sufficient conditions for the perfect recovery of signals in scenarios where signal information is lost during transmission or corrupted by noise. Finally, we illustrate that utilizing the signal’s periodicity information through Ramanujan filter banks significantly improves the efficiency of signal recovery optimization algorithms, resulting in enhanced signal-to-noise ratio (SNR) gains and more precise reconstruction.

本文探讨了基于拉马努金和的滤波器组的完美重构特性及其在信号恢复中的应用。最初由Srinivasa Ramanujan引入，Ramanujan和作为从信号中提取周期分量的强大工具，并构成了Ramanujan滤波器组的基础。我们研究了这些滤波器组的完美重构特性，并分析了它们在有限维空间l2 (ZN)（即CN）中对离散时间信号的抗擦除的鲁棒性。该研究进一步扩展到非均匀拉马努金滤波器组，展示了它们解决均匀滤波器组局限性的能力。利用均匀拉马努金滤波器组的重构性质，给出了拉马努金和移位产生的紧框架的不确定性原理。该原理建立了欧拉完备函数φ (n)的表示不等式，为信号信息在传输过程中丢失或被噪声破坏的情况下信号的完美恢复提供了充分条件。最后，我们证明了通过拉马努金滤波器组利用信号的周期性信息可以显著提高信号恢复优化算法的效率，从而提高信噪比（SNR）增益和更精确的重建。

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引用次数: 0

Integral operator approaches for scattered data fitting on spheres 球面上分散数据拟合的积分算子方法

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.acha.2025.101851

Shao-Bo Lin

This paper focuses on scattered data fitting problems on spheres. We study the approximation performance of a class of weighted spectral filter algorithms (WSFA), including Tikhonov regularization, Landweber iteration, spectral cut-off, and iterated Tikhonov, in fitting noisy data with possibly unbounded random noise. For theoretical analysis, we borrow the idea of integral operator approach from statistical learning theory to be an extension of the widely used sampling inequality approach and norming set method in the community of scattered data fitting. After providing an equivalence between the operator differences and quadrature rules, we succeed in deriving tight bounds for operator differences, explicit operator representations for WSFA and consequently optimal error estimates. Our derived error estimates do not suffer from the saturation phenomenon for Tikhonov regularization, native-space-barrier for existing error analysis and adapts to different embedding spaces. Based on the operator representations, we develop a Lepskii-type principle to determine the filter parameter of WSFA and a divide-and-conquer scheme to reduce the computational burden and provide optimal approximation rates for corresponding algorithms.

本文主要研究球面上的散射数据拟合问题。本文研究了包括Tikhonov正则化、Landweber迭代、谱截止和迭代Tikhonov算法在内的加权谱滤波算法（WSFA）在拟合可能无界随机噪声的噪声数据中的近似性能。在理论分析方面，我们借鉴统计学习理论中的积分算子方法思想，对离散数据拟合领域中广泛使用的抽样不等式方法和规范化集方法进行了扩展。在提供了算子差和正交规则之间的等价性之后，我们成功地推导出了算子差的紧界、WSFA的显式算子表示以及由此产生的最优误差估计。我们推导的误差估计不受Tikhonov正则化的饱和现象的影响，不受现有误差分析的本地空间屏障的影响，并适应不同的嵌入空间。在算子表示的基础上，提出了一种确定WSFA滤波器参数的lepskii型原理和一种分而治之的方案，以减少计算量并为相应的算法提供最优逼近率。

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引用次数: 0

Approximation and estimation capability of vision transformers for hierarchical compositional models 层次组合模型视觉变换的逼近与估计能力

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.acha.2025.101849

Zhongjie Shi , Zhiying Fang , Yuan Cao

Although the Transformer model has emerged as the preferred choice in numerous application domains, its theoretical underpinnings remain sparse. Specifically, when compared to traditional fully-connected neural networks (FNNs), there is currently no theoretical result that explains the advantages of Transformers. In this paper, we delve into the analysis of approximation and generalization errors for the Vision Transformer (ViT) model. Despite the presence of the softmax function in the self-attention mechanism, we have successfully constructed a product gate within the ViT architecture. Our analysis shows that, for target functions of the hierarchical compositional form with suitable smoothness constraints, ViTs can avoid the curse of dimensionality in the sense that the input dimension only affects the exponent of the logarithmic terms and the constant terms. Notably, our findings underscore the efficiency of ViTs in terms of parameter usage compared to FNNs. Furthermore, when the regression function is of the hierarchical compositional form with the same suitable smoothness constraints, estimators generated by the empirical risk minimization algorithm with a ViT structure can achieve near-optimal convergence rates in a regression framework. These theoretical contributions not only demonstrate the inherent strengths of the ViT model but also address a significant gap in its theoretical exploration.

尽管Transformer模型已成为许多应用程序领域的首选，但其理论基础仍然很少。具体来说，与传统的全连接神经网络（fnn）相比，目前还没有理论结果可以解释变形金刚的优势。本文对视觉变压器（Vision Transformer, ViT）模型的近似误差和泛化误差进行了深入的分析。尽管在自关注机制中存在softmax功能，但我们已经成功地在ViT架构中构建了一个产品门。我们的分析表明，对于具有适当平滑约束的分层组合形式的目标函数，vit可以避免维数诅咒，即输入维数仅影响对数项和常数项的指数。值得注意的是，我们的研究结果强调了与fnn相比，vit在参数使用方面的效率。此外，当回归函数为层次组合形式且具有相同的合适平滑约束时，具有ViT结构的经验风险最小化算法生成的估计量在回归框架中可以达到接近最优的收敛速度。这些理论贡献不仅展示了ViT模型的内在优势，而且弥补了其理论探索的重大空白。

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引用次数: 0

Quantum wave packet transforms with compact frequency support: Implementations for wavelets and Gabor atoms 具有紧凑频率支持的量子波包变换：小波和Gabor原子的实现

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-15 DOI: 10.1016/j.acha.2025.101850

Hongkang Ni, Lexing Ying

Various wave packet transforms are widely used to extract multiscale structures in signal processing. This paper introduces the quantum circuit implementation of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. Our approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers. The resulting implementation differs from existing quantum algorithms for spatially compactly supported wavelets and can be readily extended to quantum transforms of other wave packets with compact frequency support.

在信号处理中，各种波包变换被广泛用于提取多尺度结构。本文介绍了具有紧凑频率支持的一大类波包的量子电路实现，包括Gabor原子和小波。我们的方法在频率空间中运行，涉及为量子计算机操作量身定制的信号的重新分配和重组。由此产生的实现不同于现有的空间紧支持小波的量子算法，可以很容易地扩展到具有紧凑频率支持的其他波包的量子变换。

引用次数: 0

Proximal subgradient norm minimization of ISTA and FISTA ISTA和FISTA的近端亚梯度范数最小化

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-11 DOI: 10.1016/j.acha.2025.101848

Bowen Li , Bin Shi , Ya-Xiang Yuan

The study of acceleration in first-order smooth optimization has a long history. Yet, it was only recently that the mechanism behind acceleration was successfully elucidated through the introduction of the gradient correction term and its equivalent implicit-velocity formulation. Building on the high-resolution differential equation framework, augmented by phase-space representation and Lyapunov analysis, a faster convergence rate has been established for the squared gradient norm of Nesterov’s accelerated gradient descent (NAG) method. Despite this progress, such results do not directly extend to composite optimization problems widely encountered in practice, such as linear inverse problems with sparsity constraints. In this work, we refine a key descent inequality in the smooth setting and generalize it to the composite case, demonstrating that it admits a tighter bound. By incorporating this refined inequality into a carefully constructed Lyapunov function, we derive a proximal subgradient norm minimization result without relying on gradient correction or implicit-velocity scheme. Specifically, we establish that the squared proximal subgradient norm for the iterative shrinkage-thresholding algorithm (ISTA) decays at an inverse square rate, while for its accelerated variant (FISTA), it improves to an inverse cubic rate.

一阶光滑优化中加速度问题的研究由来已久。然而，直到最近，通过引入梯度校正项及其等效隐式速度公式，才成功地阐明了加速度背后的机制。在高分辨率微分方程框架的基础上，通过相空间表示和Lyapunov分析，建立了Nesterov加速梯度下降（NAG）方法的梯度范数平方的更快收敛速度。尽管取得了这些进展，但这些结果并不能直接推广到实践中广泛遇到的复合优化问题，例如具有稀疏性约束的线性逆问题。在这项工作中，我们改进了光滑情况下的关键下降不等式，并将其推广到复合情况，证明它允许更严格的约束。通过将这个精炼的不等式合并到一个精心构造的Lyapunov函数中，我们得到了一个近次梯度范数最小化结果，而不依赖于梯度校正或隐式速度方案。具体地说，我们建立了迭代收缩阈值算法（ISTA）的平方近次梯度范数以反平方速率衰减，而其加速变体（FISTA）的平方近次梯度范数以反立方速率衰减。

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引用次数: 0

Non-negative sparse recovery at minimal sampling rate 最小采样率下的非负稀疏恢复

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-11 DOI: 10.1016/j.acha.2025.101847

Hendrik Bernd Zarucha , Peter Jung

It is known that sparse recovery is possible if the number of measurements is in the order of the sparsity, but the corresponding decoders either lack polynomial decoding time or robustness to noise. Commonly, decoders that rely on a null space property are being used. These achieve polynomial time decoding and are robust to additive noise but pay the price by requiring more measurements. The non-negative least residual has been established as such a decoder for non-negative recovery. A new equivalent condition for uniform, robust recovery of non-negative sparse vectors with the non-negative least residual that is not based on null space properties is introduced. It is shown that the number of measurements for this equivalent condition only needs to be in the order of the sparsity. Further, it is explained why the robustness to additive noise is similar, but not equal, to the robustness of decoders based on null space properties.

已知，如果测量次数与稀疏度的顺序一致，则稀疏恢复是可能的，但相应的解码器要么缺乏多项式解码时间，要么缺乏对噪声的鲁棒性。通常，使用依赖于零空间属性的解码器。这些方法实现了多项式时间解码，并且对加性噪声具有鲁棒性，但需要付出更多测量的代价。建立了非负最小残差作为非负恢复的译码器。给出了非负最小残差的非负稀疏向量的一致鲁棒恢复的一个新的等价条件，该条件不基于零空间性质。结果表明，在这种等效条件下，测量次数只需在稀疏度的数量级上即可。此外，还解释了为什么对加性噪声的鲁棒性与基于零空间特性的解码器的鲁棒性相似，但不相等。

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