Applied and Computational Harmonic Analysis最新文献

We derive upper bounds on the Wasserstein distance ($W_1$), with respect to sup-norm, between any continuous $R^d$ valued random field indexed by the n-sphere and the Gaussian, based on Stein's method. We develop a novel Gaussian smoothing technique that allows us to transfer a bound in a smoother metric to the $W_1$ distance. The smoothing is based on covariance functions constructed using powers of Laplacian operators, designed so that the associated Gaussian process has a tractable Cameron-Martin or Reproducing Kernel Hilbert Space. This feature enables us to move beyond one dimensional interval-based index sets that were previously considered in the literature. Specializing our general result, we obtain the first bounds on the Gaussian random field approximation of wide random neural networks of any depth and Lipschitz activation functions at the random field level. Our bounds are explicitly expressed in terms of the widths of the network and moments of the random weights. We also obtain tighter bounds when the activation function has three bounded derivatives.

Federated learning aims to protect data privacy by collaboratively learning a model without sharing private data among users. However, an adversary may still be able to infer the private training data by attacking the released model. Differential privacy provides a statistical protection against such attacks at the price of significantly degrading the accuracy or utility of the trained models. In this paper, we investigate a utility enhancement scheme based on Laplacian smoothing for differentially private federated learning (DP-Fed-LS), to improve the statistical precision of parameter aggregation with injected Gaussian noise without losing privacy budget. Our key observation is that the aggregated gradients in federated learning often enjoy a type of smoothness, i.e. sparsity in a graph Fourier basis with polynomial decays of Fourier coefficients as frequency grows, which can be exploited by the Laplacian smoothing efficiently. Under a prescribed differential privacy budget, convergence error bounds with tight rates are provided for DP-Fed-LS with uniform subsampling of heterogeneous non-iid data, revealing possible utility improvement of Laplacian smoothing in effective dimensionality and variance reduction, among others. Experiments over MNIST, SVHN, and Shakespeare datasets show that the proposed method can improve model accuracy with DP-guarantee and membership privacy under both uniform and Poisson subsampling mechanisms.

In 2016 Aldroubi et al. constructed the first class of frames having the form ${\left\{T^k\phi \right\}}_{k=0}^{\infty }$ for a bounded linear operator on the underlying Hilbert space. In this paper we show that a subclass of these frames has a number of additional remarkable features that have not been identified for any other frames in the literature. Most importantly, the subfamily obtained by selecting each Nth element from the frame is itself a frame, regardless of the choice of $N\in N$. Furthermore, the frame property is kept upon removal of an arbitrarily finite number of elements.

An effective tail-atomic norm methodology and algorithms for gridless spectral estimations are developed with a tail-minimization mechanism. We prove that the tail-atomic norm can be equivalently reformulated as a positive semi-definite programming (PSD) problem as well. Some delicate and critical weighting constraints are derived. Iterative tail-minimization algorithms based on PSD programming are also derived and implemented. Extensive simulation results demonstrate that the tail-atomic norm mechanism substantially outperforms state-of-the-art gridless spectral estimation techniques. Numerical studies also show that the tail-atomic norm approach is more robust to noisy measurements than other known related atomic norm methodologies.

In this paper, we perform the joint study of scale-invariant operators and self-similar processes of complex order. More precisely, we introduce general families of scale-invariant complex-order fractional-derivation and integration operators by constructing them in the Fourier domain. We analyze these operators in detail, with special emphasis on the decay properties of their output. We further use them to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst exponents. These random processes are expressed via their characteristic functionals over the Schwartz space of functions. They are therefore defined as generalized random processes in the sense of Gel'fand. Beside their self-similarity and stationarity, we study the Sobolev regularity of the proposed random processes. Our work illustrates the strong connection between scale-invariant operators and self-similar processes, with the construction of adequate complex-order scale-invariant integration operators being preparatory to the construction of the random processes.

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.

We describe the full structure of the frame set for the Gabor system $G(g;\alpha ,\beta ):=\left\{e^{-2\pi im\beta \cdot }g\right(\cdot -n\alpha ):m,n\in Z\}$ with the window being the Haar function $g=-{\chi }_{[-1/2,0)}+{\chi }_{[0,1/2)}$. This is the first compactly supported window function for which the frame set is represented explicitly.

The strategy of this paper is to introduce the piecewise linear transformation $M$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system $G(g;\alpha ,\beta )$ to be a frame, i.e., the symmetric invariant set of the transformation $M$ is empty.

We prove that frame set $F_g$ for imaginary shift of sinc-function$g\left(t\right)=\frac{\sin \pi b(t-iw)}{t-iw},b,w\in R\setminus \left\{0\right\}$ can be described as $F_g=\left\{\right(\alpha ,\beta ):\alpha \beta ⩽1,\beta ⩽|b\left|\right\}$.

In addition, we prove that $F_g=\left\{\right(\alpha ,\beta ):\alpha \beta ⩽1\}$ for window functions g of the form$\frac{1}{t-iw}(1-\sum _{k=1}^{\infty }{a}_k{e}^{2\pi i{b}_{k}t}),$ such that ${\sum }_{k⩾1}\left|a_k\right|e^{2\pi \left|w{b}_k\right|}<1$, $w{b}_k<0$.

This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.

This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nyström-based interpolation method.