In graph-based data analysis, k-nearest neighbor (kNN) graphs are widely used due to their adaptivity to local data densities. Allowing weighted edges in the graph, the kernelized graph affinity provides a more general type of kNN graph where the kNN distance is used to set the kernel bandwidth adaptively. In this work, we consider a general class of kNN graph where the graph affinity is Wij=ϵ−d/2k0(∥xi−xj∥2/ϵϕ(ρ^(xi),ρ^(xj))2), with ρ^(x) being the (rescaled) kNN distance at the point x, ϕ a symmetric bi-variate function, and k0 a non-negative function on [0, ∞). Under the manifold data setting, where N i.i.d. samples xi are drawn from a density p on a d-dimensional unknown manifold embedded in a high dimensional Euclidean space, we prove the operator pointwise convergence of the kNN graph Laplacian to the limiting manifold operator (depending on p) at the rate of O(N−2/(d+6)), up to a log factor, when k0 and ϕ have C3 regularity and satisfy other technical conditions. This is obtained when ϵ∼N−2/(d+6) and k∼N6/(d+6), both at the optimal order to balance the theoretical bias and variance errors. Our improved convergence rate is based on a refined analysis of the kNN estimator, which can be of independent interest. We validate our theory by numerical experiments on simulated data.
In this article we introduce the linear canonical Riesz potential (for short, LCRP) and give its symbol in terms of linear canonical transforms. Driven by image processing, we establish the convergence/divergence of these LCRPs for different kinds of functions. Concretely, for grating functions, we prove that their classical Riesz potentials diverge, whereas their LCRP converge due to the key role of chirp functions. For the characteristic function 1P of a bounded polygonal domain P, we show that the limit of its Riesz potential at any non-boundary point x equals 1P(x), but its limit at the boundaries differ from 1P, while it is known that, for any Schwartz function f, the limit of its Riesz potential at any point x always equals f(x). Based on these and the inverse operator of the LCRP (namely the linear canonical Laplacian operator), we propose an asymmetric cascaded LCRP method for the multi-image encryption and construct an efficient numerical cryptosystem. Systematic security evaluations, including sensitivity, statistical, noise attack, and occlusion attack analyses, demonstrate its robustness and its security. Even for a single image, the proposed method is more efficient than the known encryption approach based on the fractional Riesz potential. The novelty of these results lies in that the convergence and the divergence of LCRPs at the critical indices, respectively, for “good” Schwartz functions and for “bad” discrete image functions essentially affect the security of image encryption and decryption.
We study the recovery of an unknown three-dimensional band-limited signal from multiple noisy observations that are randomly rotated by latent elements of SO(3), where the rotations are drawn from an unknown, non-uniform distribution. Because the rotations are unobserved, only the signal orbit under the rotation group can be recovered. We show that the signal orbit and the rotation distribution are jointly identifiable from the first and second moments. This yields an improved high-noise sample complexity that scales quadratically with the noise variance, rather than cubically as in the uniform-rotation case. We further develop a provable, computationally efficient reconstruction algorithm that recovers the 3-D signal by successively solving a sequence of well-conditioned linear systems. The algorithm is validated through extensive numerical experiments. Our results provide a principled and tractable framework for high-noise 3-D orbit recovery, with potential relevance to cryo-electron microscopy and cryo-electron tomography modeling, where molecules are observed in unknown orientations.
Let g∈L2(R) be a rational function of degree M, i.e., there exist polynomials P, Q such that g=PQ and degP 0 and any M∈N, there exists a universal set Λ⊂R of uniform Beurling density less than 1+ϵ such that the system {e2πiλtg(t−n):(λ,n)∈Λ×Z} forms a frame in L2(R) for any well-behaved rational function g of degree at most M. In particular, this result stands for the class of Herglotz function of degree at most M.
In this paper, it is shown that a necessary condition for unique identifiability of K chirps from N regularly spaced samples of their mixture is N ≥ 2K when K ≥ 2. A necessary and sufficient condition is that a rank-constrained matrix optimization problem has a unique solution; this is the first result of such kind. An algorithm is proposed to solve the optimization problem and to identify the parameters numerically. The lower bound of N=2K is shown to be tight by providing diverse problem instances for which the proposed algorithm succeeds to identify the parameters. The advantageous performance of the proposed algorithm is also demonstrated compared with the state of the art.
The higher-order autocorrelations of integer-valued or rational-valued functions on finite Abelian groups appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a rational-valued function on finite Abelian groups from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order 3r+3 are always sufficient to determine the data up to translation, where r is the rank of the group. We also provide examples of rational-valued functions on finite Abelian group which are not determined by their autocorrelations up to order 3r+2. In particular, we provide a sharp upper bound on the separating degree of the regular representation of a finite Abelian group in terms of its rank.
We prove the group version of the Matrix Spencer conjecture. For every finite group G, there exist signs ε ∈ { ± 1}G such that∥∑g∈Gεgρ(g)∥≤C|G|,where ρ is the left regular representation of G and C is a universal constant. This conjecture was posed in [1], which settled it for simple groups; we establish it for all finite groups, combining the Peter–Weyl decomposition with the intrinsic-freeness inequalities of [2] in an iterated partial-coloring argument.
In this paper we consider a Tikhonov regularized robust learning algorithm with Huber loss in a reproducing kernel Hilbert space (RKHS). By imposing a pth moment (1 < p <= 2) condition on the conditional distribution and a source condition on the regression function, we conduct a rigorous leave one out analysis of regularized Huber regression, and utilize the results to derive capacity independent generalization error bounds and learning rates of the algorithm. Under the strong moment condition (p = 2), we achieve the capacity independent optimal learning rates by tuning the regularization parameter and scale parameter of Huber loss in according with the sample size. We also show that for data with infinite variance (1 < p < 2), the generalization performance of Huber regression estimators is still effective regardless the regression function lies in the associated RKHS or not. In this sense, our study theoretically justifies the super robustness of the algorithm with respect to heavy-tailed distributions.
We investigate the Slepian spatiospectral localization problem within subdomains of the d-dimensional ball. Opposed to the more classical setups of the Euclidean space or the sphere, the ball lacks a standard or universally accepted definition of bandwidth. Here, we consider a Fourier-Jacobi function system, decoupling the spherical and radial contributions via spherical harmonics and Jacobi polynomials. Special cases of this setup are of interest for various inverse problems in geophysics and medical imaging, since they relate to the underlying non-uniqueness, as well as in optics, where they represent the widely used Zernike polynomials. Bandwidth can be prescribed separately for the spherical and the radial contributions, where the particular choice of coupling between the two contributions determines the spectral shape, i.e., the overall notion of bandlimit. Understanding the effects of the spectral shape on the eigenvalue distribution of the Slepian spatiospectral localization problem can provide hints on particularly suitable notions of bandwidth for different applications. We provide rigorous asymptotic results for the spectral shape being defined via the overall polynomial degree as well as for being defined via sequential limits for the spherical and radial contributions. For various other spectral shapes, we provide numerical illustrations of the asymptotic eigenvalue distribution. Furthermore, we demonstrate a direct connection of the spectral shape to common indexing schemes for Zernike polynomials.
Neural operators, built on neural networks, have emerged as a crucial tool in deep learning for approximating nonlinear operators. The present work develops an approximation and generalization theory for neural operators with prespecified encoder-decoder structures, extending previous work by considering target operators that are Fr & eacute;chet differentiable. To utilize the Fr & eacute;chet differentiability, we expand the target operator by the Taylor formula and apply a re-discretizing technique. This enables us to derive novel upper bounds on approximation and generalization errors. We further apply these bounds to two commonly used network architectures, functional neural network (FNN) and PCA-Net, and derive improved convergence rates compared with those for Lipschitz continuous operators. These results also quantitatively demonstrate how the reconstruction errors of infinite dimensional spaces and the order of Fr & eacute;chet differentiability of target operators influence learning performances. Notably, for PCA-Net, we conduct error analysis over the entire input space endowed with a sub-Gaussian measure, broadening the applicability of the theory.
This article presents a systematic analysis for the generalized tail-atomic norm method in gridless spectrum estimations. The technique is to impose a tail-penalty on the complement of an estimated frequency set F. In gridless formulations, the complementary frequency set F-c equivalent to T\F is a continuous spectrum, where T is the frequency torus. Hence, to match the discrete measure of r frequencies in the measurement model, tail-penalty is preferably imposed over sufficiently "dense" samples of this continuum F-c. Such "density" requirement coincides with the beneficial sparsity enhancement to the amplitude variable. This naturally extends the original m x r (r < m) steering matrix to an over-complete m x n system, often with n >> m. A key contribution of this paper is the derivation of an equivalent semi-definite programming (SDP) solution for this generalized m x n-tail-atomic norm formulation. We further extend the analysis to the multiple measurement vector (MMV) model, showing its superior frequency recovery capabilities. Extensive numerical experiments are conducted, comparing the proposed framework with several state-of-the-art gridless techniques. The results demonstrate that the generalized tail-atomic norm method achieves significant improvements in both frequency estimation capacities and robustness against noise across a range of signal-to-noise ratios.
We analyze domain adaptation within the framework of reproducing kernel Hilbert spaces under the covariate shift assumption. In this setting, previously known results concerning the least squares excess risk bounds have mainly been derived for importance-weighted kernel ridge regression and either in terms of the smoothness of the target function or in terms of the capacity of the underlying space. The primary novelty of the current research lies in the analysis of general importance-weighted spectral algorithms in both above terms, which enables a substantial improvement of the excess risk bounds. Furthermore, we delve into the covariate shift adaptation utilizing estimated density ratios and explore its application in the context of imbalanced learning from real clinical data.
Unitary invariants-polynomials in both the variables and their conjugates-often separate orbits in far lower degree than ordinary polynomial invariants, yet their separating power is little studied. We illustrate this for the finite Heisenberg group H-N: although C[V](HN) has no nonconstant invariants below degree N, degree-six unitary invariants (a Heisenberg analogue of the bispectrum) already separate generic H-N-orbits up to a global phase, and a single degree-N polynomial invariant resolves the phase, giving full generic orbit separation. The construction rests on results from phase retrieval. This gives a concrete case where the minimal separating degree for unitary invariants is dramatically lower than for polynomial invariants.
We analyze the Double Fourier Sphere (DFS) method on the rotation group SO(3) in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on SO(3) are commonly formulated as a Wigner transform-mapping harmonic to Fourier coefficients-followed by a Fourier transform. We revisit this formulation and interpret the Wigner transform as an explicit realization of the DFS method, lifting functions from SO(3) to T3. In this context, we analyze the Sobolev regularity loss induced by this lifting. Furthermore, we compare different Wigner transform implementations, examine additional symmetry enhancements, and observe that the direct method is often faster and more stable than the fast polynomial transform approaches.
We introduce a generalized Fourier ratio, the l(1)/l(2) norm ratio of coefficients in a arbitrary orthonormal system, as a single, basis-invariant measure of effective dimension that governs fundamental limits across signal recovery, localization, and learning. First, we prove that functions with small Fourier ratio can be stably recovered from random missing samples via l(1) minimization, extending and clarifying compressed sensing guarantees for general bounded orthonormal systems. Second, we establish a sharp localization obstruction: any attempt to localize recovery to subslices of a product space necessarily inflates the Fourier ratio by a factor scaling with the square root of the slice count, demonstrating that global complexity cannot be distributed locally. Finally, we show that the same parameter controls key complexity-theoretic measures: it provides explicit upper bounds on Kolmogorov rate-distortion description length and on the statistical query (SQ) dimension of the associated function class. These results unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter, revealing the Fourier ratio as a fundamental invariant in information-theoretic signal processing.
Spectral interference, commonly referred to as the beating phenomenon, can severely distort time-frequency representations (TFRs) in physical applications. We study this phenomenon for the short-time Fourier transform (STFT) with a Gaussian window and for nonlinear refinements based on the reassignment method, with an emphasis on the synchrosqueezing transform (SST). Working with a two-component harmonic model, we quantify when STFT can (and cannot) resolve two nearby frequencies: a sharp transition occurs at a critical gap that scales inversely to kernel bandwidth and depends explicitly on the amplitude ratio. Below this threshold, the spectrogram ridges undergo bifurcation and form repeating time-frequency bubbles, which we describe asymptotically and, in the balanced-amplitude case, approximate closely by ellipses. We then analyze the STFT phase, showing a canonical winding behavior, and relate the complex-valued SST reassignment map to a holomorphic structure via the Bargmann transform. In the two-component setting the reassignment rule admits an explicit M & ouml;bius-geometry description, sending frequency lines to circular arcs in the complex plane. Finally, viewing SST and reassignment through a measure mapping perspective, we derive small-kernel asymptotics that explain when reassignment sharpens energy and when it produces distorted or misleading TFRs; we also introduce a generalized synchrosqueezing framework that isolates the role of STFT weighting and clarifies how alternative choices can mitigate interference in certain regimes.
Various wave packet transforms are widely used to extract multiscale structures in signal processing. This paper introduces the quantum circuit implementation of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. Our approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers. The resulting implementation differs from existing quantum algorithms for spatially compactly supported wavelets and can be readily extended to quantum transforms of other wave packets with compact frequency support.
We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces M-m(p,q) (R-d), we prove dimension-independent approximation rates in Sobolev norms W-n,W-r (Omega) for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for f is an element of M-m(p,q) (R-d) one can achieve & Vert;f - f (N)& Vert;(n,r (Omega))(W) less than or similar to N-1/2 & Vert; f & Vert; (p,q)(M)m (R-d), on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on R-d using weighted modulation dictionaries, and derive consequences for Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions confirm that modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks, consistent with the theoretical estimates.
This paper explores the perfect reconstruction property of filter banks based on Ramanujan sums and their applications in signal recovery. Originally introduced by Srinivasa Ramanujan, Ramanujan sums serve as powerful tools for extracting periodic components from signals and form the foundation of Ramanujan filter banks. We investigate the perfect reconstruction property of these filter banks and analyze their robustness against erasures for discrete-time signals in the finite-dimensional space l2(& Zopf;N) (i.e., CN). The study is further extended to non-uniform Ramanujan filter banks, showcasing their ability to address the limitations of uniform ones. Employing the reconstruction properties of uniform Ramanujan filter banks, we present an uncertainty principle associated with a tight frame generated by shifts of Ramanujan sums. This principle establishes representation inequalities in terms of Euler's totient function phi(n), which provide sufficient conditions for the perfect recovery of signals in scenarios where signal information is lost during transmission or corrupted by noise. Finally, we illustrate that utilizing the signal's periodicity information through Ramanujan filter banks significantly improves the efficiency of signal recovery optimization algorithms, resulting in enhanced signal-to-noise ratio (SNR) gains and more precise reconstruction.
