Applied and Computational Harmonic Analysis最新文献

An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.

In this paper, we consider networks with topologies described by some connected undirected graph $G=(V,E)$ and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem ${\min }_x\left\{F\right(x)={\sum }_{i\in V}{f}_i(x\left)\right\}$ with local objective functions $f_i$ depending only on neighboring variables of the vertex $i\in V$. We introduce a divide-and-conquer algorithm to solve the above optimization problem in a distributed and decentralized manner. The proposed divide-and-conquer algorithm has exponential convergence, its computational cost is almost linear with respect to the size of the network, and it can be fully implemented at fusion centers of the network. In addition, our numerical demonstrations indicate that the proposed divide-and-conquer algorithm has superior performance than popular decentralized optimization methods in solving the least squares problem, both with and without the ${\ell }^1$ penalty, and exhibits great performance on networks equipped with asynchronous local peer-to-peer communication.

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on $R$ that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval $[0,1]$. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues.

In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on $L^2\left(R^d\right)$, which is an alternating product of projection operators associated to given spatial and frequency domains in $R^d$. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval $[0,1]$.

To prove our results, we design an orthonormal system of wave packets in $L^2\left(R^d\right)$ that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for $L^2[0,1]$, and we lift the basis to higher dimensions using a tensor product.

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori $T_N^2=R^2/(Z\times NZ)=[0,1]\times [0,N]$ act as phase spaces. We work on an N-dimensional subspace $S_N$ of distributions periodic in time and frequency in the dual $S_0^{\prime }\left(R\right)$ of the Feichtinger algebra $S_0\left(R\right)$ and equip it with an inner product. To construct the Hilbert space $S_N$ we apply a suitable double periodization operator to $S_0\left(R\right)$. On $S_N$, the STFT is applied as the usual STFT defined on $S_0^{\prime }\left(R\right)$. This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallstén for Gabor frames with Gaussian windows and which, for N odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.

Using the embedded gradient vector field method (see P. Birtea, D. Comănescu (2015) [7]), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of $n\times n$ matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal group using the ambient Euclidean coordinates. We apply this new formula for some relevant functions.

This paper is concerned with functional learning by utilizing two-stage sampled distribution regression. We study a multi-penalty regularization algorithm for distribution regression in the framework of learning theory. The algorithm aims at regressing to real-valued outputs from probability measures. The theoretical analysis of distribution regression is far from maturity and quite challenging since only second-stage samples are observable in practical settings. In our algorithm, to transform information of distribution samples, we embed the distributions to a reproducing kernel Hilbert space $H_K$ associated with Mercer kernel K via mean embedding technique. One of the primary contributions of this work is the introduction of a novel multi-penalty regularization algorithm, which is able to capture more potential features of distribution regression. Optimal learning rates of the algorithm are obtained under mild conditions. The work also derives learning rates for distribution regression in the hard learning scenario $f_{\rho }\notin H_K$, which has not been explored in the existing literature. Moreover, we propose a new distribution-regression-based distributed learning algorithm to face large-scale data or information challenges arising from distribution data. The optimal learning rates are derived for the distributed learning algorithm. By providing new algorithms and showing their learning rates, the work improves the existing literature in various aspects.

We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings.

We then apply the general results to a particular class of dilation groups, the so-called shearlet dilation groups. We present a general, algebraic characterization of matrices that are coorbit compatible with a given shearlet dilation group. We explicitly determine the groups of compatible dilations, for a variety of concrete examples.

Image denoising is always considered an important area of image processing. In this work, we address a new PDE-based model for image denoising that have been contaminated by multiplicative noise, specially the Speckle one. We propose a new class of PDEs whose nonlinear structure depends on a spatially tensor depending quantity attached to the desired solution, which takes into account the gray level information by introducing a gray level indicator function in the diffusion coefficient. We give some theoretical results, discretization and also stability condition for the suggested model. Finally, we carry out some numerical results to approve the effectiveness of our model by comparing the results obtained with some competitive models.

We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are in 0 and their upper entropy dimensions. Moreover, for such a Moran spectral measure μ, we show that the Beurling dimension for the spectra of μ has the intermediate value property: let t be any value in 0 and the upper entropy dimension of μ, then there exists a spectrum whose Beurling dimension is t. In particular, this result settles affirmatively a conjecture involving spectral Bernoulli convolution in Fu et al. (2018) [20]. Furthermore, we prove that the set of the spectra whose Beurling dimensions are equal to any fixed value in 0 and ${\bar{\dim }}_e\mu$ has the cardinality of the continuum.