Applied and Computational Harmonic Analysis最新文献

Algorithmic stability is a fundamental concept in statistical learning theory to understand the generalization behavior of optimization algorithms. Existing high-probability bounds are developed for the generalization gap as measured by function values and require the algorithm to be uniformly stable. In this paper, we introduce a novel stability measure called pointwise uniform stability by considering the sensitivity of the algorithm with respect to the perturbation of each training example. We show this weaker pointwise uniform stability guarantees almost optimal bounds, and gives the first high-probability bound for the generalization gap as measured by gradients. Sharper bounds are given for strongly convex and smooth problems. We further apply our general result to derive improved generalization bounds for stochastic gradient descent. As a byproduct, we develop concentration inequalities for a summation of weakly-dependent vector-valued random variables.

The analysis of the time–frequency content of a signal is a classical problem in signal processing, with a broad number of applications in real life. Many different approaches have been developed over the decades, which provide alternative time–frequency representations of a signal each with its advantages and limitations. In this work, following the success of nonlinear methods for the decomposition of signals into intrinsic mode functions (IMFs), we first provide more theoretical insights into the so–called Iterative Filtering decomposition algorithm, proving an energy conservation result for the derived decompositions. Furthermore, we present a new time–frequency representation method based on the IMF decomposition of a signal, which is called IMFogram. We prove theoretical results regarding this method, including its convergence to the spectrogram representation for a certain class of signals, and we present a few examples of applications, comparing results with some of the most well-known approaches available in the literature.

We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number k. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in $O(k^d\log k)$ operations. The non-oscillatory component has a multiresolution representation via a linear combination of Gaussians and is applied efficiently in space.

Since the Helmholtz Green's function can be viewed as a point source, this partitioning can be interpreted as a splitting into propagating and evanescent components. We show that the non-oscillatory component is significant only in the vicinity of the source at distances $O(c_1{k}^{-1}+c_2{k}^{-1}{\log }_{10}k)$, for some constants $c_1$, $c_2$, whereas the propagating component can be observed at large distances.

This paper discusses the generation of multivariate $C^{\infty }$ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports $[0,1+ϵ]$ in comparison to the support $[0,2]$ of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of $C^{\infty }$ compactly supported wavelets of small support in any dimension.

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.