arXiv subjects

Palina Salanevich

Publications and source records attributed to Palina Salanevich.

12 recordsLinked to original sources

arXiv2025

Robustness to noise and erasures of Gabor frames in finite dimensions

In this paper, we investigate the robustness of structured frames to measurement noise and erasures, with the focus on Gabor frames $(g, Λ)$ with arbitrary sets of time-frequency shifts $Λ$. This property of frames is important in many signal processing applications, from wireless communication to phase retrieval and quantization. We aim to analyze the dependence of the frame bounds of such frames on their structure and cardinality and provide constructions of nearly tight Gabor frames with small $\vert Λ\vert$. We show that the frame bounds of Gabor frames with a random window $g$ and frame set $Λ$ show similar behavior to the frame bounds of random frames with independent entries. Moreover, we study uniform estimates for the frame bounds in the case of measurement erasures. We prove numerical robustness to erasures for mutually unbiased bases frames with erasure rate up to 50\% and show that the Gabor frame generated by the Alltop window can provide results similar to the best previously known deterministic constructions.

math.FA↗

arXiv2025

On best approximation by multivariate ridge functions with applications to generalized translation networks

In this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., for approximation by functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n \varrho_k(A_k x) \in \mathbb{R}$ with $\varrho_k : \mathbb{R}^\ell \to \mathbb{R}$ and $A_k \in \mathbb{R}^{\ell \times d}$. We show that the order of approximation asymptotically behaves as $n^{-r/(d-\ell)}$, where $r$ is the regularity (order of differentiability) of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\infty$-Sobolev functions of regularity $r$ with error measured in $L^1$, while our upper bound applies to the approximation of $L^p$-Sobolev functions in $L^p$ for any $1 \leq p \leq \infty$. These bounds generalize well-known results regarding the approximation properties of univariate ridge functions to the multivariate case. We use our results to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks.

math.FA↗

arXiv2025

Efficient uniform approximation using Random Vector Functional Link networks

A Random Vector Functional Link (RVFL) network is a depth-2 neural network with random inner weights and biases. Only the outer weights of such an architecture are to be learned, so the learning process boils down to a linear optimization task, allowing one to sidestep the pitfalls of nonconvex optimization problems. In this paper, we prove that an RVFL with ReLU activation functions can approximate Lipschitz continuous functions in $L_\infty$ norm. To the best of our knowledge, our result is the first approximation result in $L_\infty$ norm using nice inner weights; namely, Gaussians. We give a nonasymptotic lower bound for the number of hidden-layer nodes to achieve a given accuracy with high probability, depending on, among other things, the Lipschitz constant of the target function, the desired accuracy, and the input dimension. Our method of proof is rooted in probability theory and harmonic analysis.

stat.ML↗

arXiv2025

Phasebook: A Survey of Selected Open Problems in Phase Retrieval

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

cs.IT↗

arXiv2025

Spectral method for low-dose Poisson and Bernoulli phase retrieval

We consider the problem of phaseless reconstruction from measurements with Poisson or Bernoulli distributed noise. This is of particular interest in biological imaging experiments where a low dose of radiation has to be used to mitigate potential damage of the specimen, resulting in low observed particle counts. We derive recovery guarantees for the spectral method for these noise models in the case of Gaussian measurements. Our results give a quantitative insight in the trade-off between the employed radiation dose per measurement and the overall sampling complexity.

cs.IT↗

arXiv2025

PtyGenography: using generative models for regularization of the phase retrieval problem

In phase retrieval and similar inverse problems, the stability of solutions across different noise levels is crucial for applications. One approach to promote it is using signal priors in a form of a generative model as a regularization, at the expense of introducing a bias in the reconstruction. In this paper, we explore and compare the reconstruction properties of classical and generative inverse problem formulations. We propose a new unified reconstruction approach that mitigates overfitting to the generative model for varying noise levels.

stat.ML↗

arXiv2024

Random Vector Functional Link Networks for Function Approximation on Manifolds

The learning speed of feed-forward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradient-based learning algorithms, which are used extensively to train neural networks, tend to work slowly when all of the network parameters must be iteratively tuned. To counter this, both researchers and practitioners have tried introducing randomness to reduce the learning requirement. Based on the original construction of Igelnik and Pao, single layer neural-networks with random input-to-hidden layer weights and biases have seen success in practice, but the necessary theoretical justification is lacking. In this paper, we begin to fill this theoretical gap. We provide a (corrected) rigorous proof that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with approximation error decaying asymptotically like $O(1/\sqrt{n})$ for the number $n$ of network nodes. We then extend this result to the non-asymptotic setting, proving that one can achieve any desired approximation error with high probability provided $n$ is sufficiently large. We further adapt this randomized neural network architecture to approximate functions on smooth, compact submanifolds of Euclidean space, providing theoretical guarantees in both the asymptotic and non-asymptotic forms. Finally, we illustrate our results on manifolds with numerical experiments.

stat.ML↗

arXiv2023

Injectivity of Multi-window Gabor Phase Retrieval

In many signal processing problems arising in practical applications, we wish to reconstruct an unknown signal from its phaseless measurements with respect to a frame. This inverse problem is known as the phase retrieval problem. For each particular application, the set of relevant measurement frames is determined by the problem at hand, which motivates the study of phase retrieval for structured, application-relevant frames. In this paper, we focus on one class of such frames that appear naturally in diffraction imaging, ptychography, and audio processing, namely, multi-window Gabor frames. We study the question of injectivity of the phase retrieval problem with these measurement frames in the finite-dimensional setup and propose an explicit construction of an infinite family of phase retrievable multi-window Gabor frames. We show that phase retrievability for the constructed frames can be achieved with a much smaller number of phaseless measurements compared to the previous results for this type of measurement frames. Additionally, we show that the sufficient for reconstruction number of phaseless measurements depends on the dimension of the signal space, and not on the ambient dimension of the problem.

cs.IT↗

arXiv2023

On Graph Uncertainty Principle and Eigenvector Delocalization

Uncertainty principles present an important theoretical tool in signal processing, as they provide limits on the time-frequency concentration of a signal. In many real-world applications the signal domain has a complicated irregular structure that can be described by a graph. In this paper, we focus on the global uncertainty principle on graphs and propose new connections between the uncertainty bound for graph signals and graph eigenvectors delocalization. We also derive uncertainty bounds for random $d$-regular graphs and provide numerically efficient upper and lower approximations for the uncertainty bound on an arbitrary graph.

cs.IT↗

arXiv2021

On audio enhancement via online non-negative matrix factorization

We propose a method for noise reduction, the task of producing a clean audio signal from a recording corrupted by additive noise. Many common approaches to this problem are based upon applying non-negative matrix factorization to spectrogram measurements. These methods use a noiseless recording, which is believed to be similar in structure to the signal of interest, and a pure-noise recording to learn dictionaries for the true signal and the noise. One may then construct an approximation of the true signal by projecting the corrupted recording on to the clean dictionary. In this work, we build upon these methods by proposing the use of \emph{online} non-negative matrix factorization for this problem. This method is more memory efficient than traditional non-negative matrix factorization and also has potential applications to real-time denoising.

eess.AS↗

arXiv2019

Robust Phase Retrieval Algorithm for Time-Frequency Structured Measurements

We address the problem of signal reconstruction from intensity measurements with respect to a measurement frame. This non-convex inverse problem is known as phase retrieval. The case considered in this paper concerns phaseless measurements taken with respect to a Gabor frame. It arises naturally in many practical applications, such as diffraction imaging and speech recognition. We present a reconstruction algorithm that uses a nearly optimal number of phaseless time-frequency structured measurements and discuss its robustness in the case when the measurements are corrupted by noise. We show how geometric properties of the measurement frame are related to the robustness of the phaseless reconstruction. The presented algorithm is based on the idea of polarization as proposed by Alexeev, Bandeira, Fickus, and Mixon.

math.FA↗

arXiv2019

Extreme Singular Values of Random Time-Frequency Structured Matrices

In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame $(g, Λ)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $Λ\subset \mathbb{Z}_M\times\mathbb{Z}_M$ is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set $Λ$, and to investigate their dependence on the structure of $Λ$, as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the respective results for matrices with independent identically distributed entries.

math.PR↗