Applied and Computational Harmonic Analysis最新文献

Sparse binary matrices are of great interest in the field of sparse recovery, nonnegative compressed sensing, statistics in networks, and theoretical computer science. This class of matrices makes it possible to perform signal recovery with lower storage costs and faster decoding algorithms. In particular, Bernoulli (p) matrices formed by independent identically distributed (i.i.d.) Bernoulli (p) random variables are of practical relevance in the context of noise-blind recovery in nonnegative compressed sensing.

In this work, we investigate the robust nullspace property of Bernoulli (p) matrices. Previous results in the literature establish that such matrices can accurately recover n-dimensional s-sparse vectors with $m=O(\frac{s}{c\left(p\right)}\log \frac{en}{s})$ measurements, where $c\left(p\right)\le p$ is a constant dependent only on the parameter p. These results suggest that in the sparse regime, as p approaches zero, the (sparse) Bernoulli (p) matrix requires significantly more measurements than the minimal necessary, as achieved by standard isotropic subgaussian designs. However, we show that this is not the case.

Our main result characterizes, for a wide range of sparsity levels s, the smallest p for which sparse recovery can be achieved with the minimal number of measurements. We also provide matching lower bounds to establish the optimality of our results and explore connections with the theory of invertibility of discrete random matrices and integer compressed sensing.

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

Given a finite group, we study the Gaussian series of the matrices in the image of its left regular representation. We propose such random matrices as a benchmark for improvements to the noncommutative Khintchine inequality, and we highlight an application to the matrix Spencer conjecture.

We prove the existence of a positive semidefinite matrix $A\in R^{n\times n}$ such that any decomposition into rank-1 matrices has to have factors with a large ${\ell }^1-$norm, more precisely$\sum _k{x}_k{x}_k^{⁎}=A\Rightarrow \sum _k{\Vert x_k\Vert }_1^2\ge c\sqrt{n}{\Vert A\Vert }_1,$ where c is independent of n. This provides a lower bound for the Balan–Jiang matrix problem. The construction is probabilistic.

Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces $H^s$, but are easier to obtain numerically. With such rules we construct “generalised needlets”. The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper.

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.

The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points. The proposed tools include a total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator and an improvement of the Berezin-Lieb inequality using the projection functional of the data augmentation operator. The framework for our approach is provided by quantum harmonic analysis.

The approximation properties of infinitely wide shallow neural networks heavily depend on the choice of the activation function. To understand this influence, we study embeddings between Barron spaces with different activation functions. These embeddings are proven by providing push-forward maps on the measures μ used to represent functions f. An activation function of particular interest is the rectified power unit (RePU) given by ${\mathrm{RePU}}_s\left(x\right)=\max {(0,x)}^s$. For many commonly used activation functions, the well-known Taylor remainder theorem can be used to construct a push-forward map, which allows us to prove the embedding of the associated Barron space into a Barron space with a RePU as activation function. Moreover, the Barron spaces associated with the ${\mathrm{RePU}}_s$ have a hierarchical structure similar to the Sobolev spaces $H^s$.

We show that spectral data of the Koopman operator arising from an analytic expanding circle map τ can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if $m\ge \delta n$, where δ is an explicitly given positive number quantifying by how much τ expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in n. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle.

In data science, a number of applications have been emerging involving data recovery from permutations. Here, we study this problem theoretically for data organized in a rank-deficient matrix. Specifically, we give unique recovery guarantees for matrices of bounded rank that have undergone arbitrary permutations of their entries. We use methods and results of commutative algebra and algebraic geometry, for which we include a preparation for a general audience.