Applied and Computational Harmonic Analysis最新文献

It is usually difficult to study the structure of the spectra for the measures in $R^2$ and higher dimensions. In this paper, by employing the projective techniques and our previous results on the line we prove that the Beurling dimension of spectra for a class of random convolutions in $R^2$ satisfies an intermediate value property.

In this paper, we study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements in a bounded domain. We aim to provide a mathematical foundation for sparsity-based super-resolution in such spectral estimation problems in both one- and multi-dimensional spaces. In particular, we estimate the resolution and stability of the location recovery of a cluster of closely spaced point sources when considering the sparsest solution under the measurement constraint, and characterize their dependence on the cut-off frequency, the noise level, the sparsity of point sources, and the incoherence of the amplitude vectors of point sources. Our estimate emphasizes the importance of the high incoherence of amplitude vectors in enhancing the resolution of multi-snapshot spectral estimation. Moreover, to the best of our knowledge, it also provides the first stability result in the super-resolution regime for the well-known sparse MMV problem in DOA estimation.

This paper focuses on parameter selection issues of kernel ridge regression (KRR). Due to special spectral properties of KRR, we find that delicate subdivision of the parameter interval shrinks the difference between two successive KRR estimates. Based on this observation, we develop an early-stopping type parameter selection strategy for KRR according to the so-called Lepskii-type principle. Theoretical verifications are presented in the framework of learning theory to show that KRR equipped with the proposed parameter selection strategy succeeds in achieving optimal learning rates and adapts to different norms, providing a new record of parameter selection for kernel methods.

In this paper, we prove the existence of a spherical t-design formed by adding extra points to an arbitrarily given point set on the sphere and, subsequently, deduce the existence of nested spherical designs. Estimates on the number of required points are also given. For the case that the given point set is a spherical $t_1$-design such that $t_1<t$ and the number of points is of optimal order $t_1^d$, we show that the upper bound of the total number of extra points and given points for forming nested spherical t-design is of order $t^{2d+1}$. A brief discussion concerning the optimal order in nested spherical designs is also given.

We provide a surface density threshold to guarantee mobile sampling in terms of the surface density of the set. This threshold is sharp if the Fourier transform is supported in either a ball or a cube, and further examples in the two-dimensional case where the result is sharp are given.

Consider the quotient of a Hilbert space by a subgroup of its automorphisms. We study whether this orbit space can be embedded into a Hilbert space by a bilipschitz map, and we identify constraints on such embeddings.

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.