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 $(g, \Lambda)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $\Lambda\subset \mathbb{Z}_M\times\mathbb{Z}_M$ 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 $\Lambda$, and to investigate their dependence on the structure of $\Lambda$, 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.Comment: 21 pages, 3 figure

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

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

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

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. As only the outer weights of such architectures need to be learned, 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 provided its hidden layer is exponentially wide in the input dimension. Although it has been established before that such approximation can be achieved in $L_2$ sense, we prove it for $L_\infty$ approximation error and Gaussian inner weights. To the best of our knowledge, our result is the first of this kind. We give a nonasymptotic lower bound for the number of hidden layer nodes, 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

Stability of Phase Retrieval Problem

Phase retrieval is a non-convex inverse problem of signal reconstruction from intensity measurements with respect to a measurement frame. One of the main problems in phase retrieval is to determine for which frames the associated phaseless measurement map is injective and stable. In this paper we address the question of stability of phase retrieval for two classes of random measurement maps, namely, frames with independent frame vectors satisfying bounded fourth moment assumption and frames with no independence assumptions. We propose a new method based on the frame order statistics, which can be used to establish stability of the measurement maps for other classes of frames

Frame Bounds for Gabor Frames in Finite Dimensions

One of the key advantages of a frame compared to a basis is its redundancy. Provided we have a control on the frame bounds, this redundancy allows, among other things, to achieve robust reconstruction of a signal from its frame coefficients that are corrupted by noise, rounding error, or erasures. In this paper, we discuss frame bounds for Gabor frames (g, Λ) with generic frame set Λ and random window g. We show that, with high probability, such frames have frame bounds similar to the frame bounds of randomly generated frames with independent frame vectors

Frame Bounds for Gabor Frames in Finite Dimensions

One of the key advantages of a frame compared to a basis is its redundancy. Provided we have a control on the frame bounds, this redundancy allows, among other things, to achieve robust reconstruction of a signal from its frame coefficients that are corrupted by noise, rounding error, or erasures. In this paper, we discuss frame bounds for Gabor frames (g, Λ) with generic frame set Λ and random window g. We show that, with high probability, such frames have frame bounds similar to the frame bounds of randomly generated frames with independent frame vectors

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 $(g, \Lambda)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $\Lambda\subset \mathbb{Z}_M\times\mathbb{Z}_M$ 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 $\Lambda$, and to investigate their dependence on the structure of $\Lambda$, 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