Applied and Computational Harmonic Analysis最新文献

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

In this short note, we propose a new method for quantizing the weights of a fully trained neural network. A simple deterministic pre-processing step allows us to quantize network layers via memoryless scalar quantization while preserving the network performance on given training data. On one hand, the computational complexity of this pre-processing slightly exceeds that of state-of-the-art algorithms in the literature. On the other hand, our approach does not require any hyper-parameter tuning and, in contrast to previous methods, allows a plain analysis. We provide rigorous theoretical guarantees in the case of quantizing single network layers and show that the relative error decays with the number of parameters in the network if the training data behave well, e.g., if it is sampled from suitable random distributions. The developed method also readily allows the quantization of deep networks by consecutive application to single layers.

The problem of synchronization over a group $G$ aims to estimate a collection of group elements $G_1^{⁎},\dots ,G_n^{⁎}\in G$ based on noisy observations of a subset of all pairwise ratios of the form $G_i^{⁎}{G_j^{⁎}}^{-1}$. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup — an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.

Classical wavelet, wavelet packets and time-frequency dictionaries have been generalized to the graph setting, the main goal being to obtain atoms which are jointly localized both in the vertex and the graph spectral domain. We present a new method to generate a whole dictionary of frames of wavelet packets defined in the graph spectral domain to represent signals on weighted graphs.

We will give some concrete examples on how the spectral graph wavelet packets can be used for compressing, denoising and reconstruction by considering a signal, given by the fRMI (functional magnetic resonance imaging) data, on the nodes of voxel-wise brain graph with 900760 nodes, representing the brain voxels.

This paper provides algebraic and analytic, as well as numerical arguments why and how double preconditioning of the Gabor frame operator yields an efficient method to compute approximate dual (respectively tight) Gabor atoms for a given time-frequency lattice. We extend the definition of the approach to the continuous setting, making use of the so-called Banach Gelfand Triple, based on the Segal algebra $\left(S_0\right(R^d),{\Vert \text{ ⋅ }\Vert }_{S_0})$ and show the continuous dependency of the double preconditioning operators on their parameters. The generalization allows to investigate the influence of the order of the two main single preconditioners (diagonal and convolutional). In the applied section we demonstrate the quality of double preconditioning over all possible lattices and adapt the method to approximate the canonical tight Gabor window, which yields a significant generalization of the FAB-method used in OFDM-applications. Finally, we demonstrate that our approach provides a way to efficiently compute approximate dual families for Gabor families which arise from a slowly varying pattern instead of a regular lattice.

Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators representing the geometry of the variables, (ii) Riemannian geometry of symmetric positive-definite matrices for multiscale composition of operators corresponding to different time samples, and (iii) spectral analysis of the composite operators for extracting different dynamic modes. We propose a method that is analogous to the classical wavelet analysis, which we term Riemannian multi-resolution analysis (RMRA). We provide some theoretical results on the spectral analysis of the composite operators, and we demonstrate the proposed method on simulations and on real data.

It is frequently observed that overparameterized neural networks generalize well. Regarding such phenomena, existing theoretical work mainly devotes to linear settings or fully-connected neural networks. This paper studies the learning ability of an important family of deep neural networks, deep convolutional neural networks (DCNNs), under both underparameterized and overparameterized settings. We establish the first learning rates of underparameterized DCNNs without parameter or function variable structure restrictions presented in the literature. We also show that by adding well-defined layers to a non-interpolating DCNN, we can obtain some interpolating DCNNs that maintain the good learning rates of the non-interpolating DCNN. This result is achieved by a novel network deepening scheme designed for DCNNs. Our work provides theoretical verification of how overfitted DCNNs generalize well.

Rectified-linear-unit (ReLU) neural networks, which play a prominent role in deep learning, generate continuous and piecewise-linear (CPWL) functions. While they provide a powerful parametric representation, the mapping between the parameter and function spaces lacks stability. In this paper, we investigate an alternative representation of CPWL functions that relies on local hat basis functions and that is applicable to low-dimensional regression problems. It is predicated on the fact that any CPWL function can be specified by a triangulation and its values at the grid points. We give the necessary and sufficient condition on the triangulation (in any number of dimensions and with any number of vertices) for the hat functions to form a Riesz basis, which ensures that the link between the parameters and the corresponding CPWL function is stable and unique. In addition, we provide an estimate of the ${\ell }_2\to L_2$ condition number of this local representation. As a special case of our framework, we focus on a systematic parameterization of $R^d$ with control points placed on a uniform grid. In particular, we choose hat basis functions that are shifted replicas of a single linear box spline. In this setting, we prove that our general estimate of the condition number is exact. We also relate the local representation to a nonlocal one based on shifts of a causal ReLU-like function. Finally, we indicate how to efficiently estimate the Lipschitz constant of the CPWL mapping.

Is it possible to detect if the sample paths of a stochastic process almost surely admit a finite expansion with respect to some/any basis? The determination is to be made on the basis of a finite collection of discretely/noisily observed sample paths. We show that it is indeed possible to construct a hypothesis testing scheme that is almost surely guaranteed to make only finite many incorrect decisions as more data are collected. Said differently, our scheme almost certainly detects whether the process has a finite or infinite basis expansion for all sufficiently large sample sizes. Our approach relies on Cover's classical test for the irrationality of a mean, combined with tools for the non-parametric estimation of covariance operators.