arXiv subjects

Rayan Saab

Publications and source records attributed to Rayan Saab.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the Number of Faces and Radii of Cells Induced by Gaussian Spherical Tessellations

We study a geometric property related to spherical hyperplane tessellations in $\mathbb{R}^{d}$. We first consider a fixed $x$ on the Euclidean sphere and tessellations with $M \gg d$ hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of $d\log(d)\log(M)$ such that the radius of the cell containing $x$ induced by these hyperplanes is bounded above by, up to constants, $d\log(d)\log(M)/M$. We extend this result to hold for all cells in the tessellation with high probability. Up to logarithmic terms, this upper bound matches the previously established lower bound of Goyal et al. (IEEE T. Inform. Theory 44(1):16-31, 1998).

math.PR↗

arXiv2021

A Greedy Algorithm for Quantizing Neural Networks

We propose a new computationally efficient method for quantizing the weights of pre- trained neural networks that is general enough to handle both multi-layer perceptrons and convolutional neural networks. Our method deterministically quantizes layers in an iterative fashion with no complicated re-training required. Specifically, we quantize each neuron, or hidden unit, using a greedy path-following algorithm. This simple algorithm is equivalent to running a dynamical system, which we prove is stable for quantizing a single-layer neural network (or, alternatively, for quantizing the first layer of a multi-layer network) when the training data are Gaussian. We show that under these assumptions, the quantization error decays with the width of the layer, i.e., its level of over-parametrization. We provide numerical experiments, on multi-layer networks, to illustrate the performance of our methods on MNIST and CIFAR10 data, as well as for quantizing the VGG16 network using ImageNet data.

cs.LG↗

arXiv2021

On the $\ell^\infty$-norms of the Singular Vectors of Arbitrary Powers of a Difference Matrix with Applications to Sigma-Delta Quantization

Let $\| A \|_{\max} := \max_{i,j} |A_{i,j}|$ denote the maximum magnitude of entries of a given matrix $A$. In this paper we show that $$\max \left\{ \|U_r \|_{\max},\|V_r\|_{\max} \right\} \le \frac{(Cr)^{6r}}{\sqrt{N}},$$ where $U_r$ and $V_r$ are the matrices whose columns are, respectively, the left and right singular vectors of the $r$-th order finite difference matrix $D^{r}$ with $r \geq 2$, and where $D$ is the $N\times N$ finite difference matrix with $1$ on the diagonal, $-1$ on the sub-diagonal, and $0$ elsewhere. Here $C$ is a universal constant that is independent of both $N$ and $r$. Among other things, this establishes that both the right and left singular vectors of such finite difference matrices are Bounded Orthonormal Systems (BOSs) with known upper bounds on their BOS constants, objects of general interest in classical compressive sensing theory. Such finite difference matrices are also fundamental to standard $r^{\rm th}$ order Sigma-Delta quantization schemes more specifically, and as a result the new bounds provided herein on the maximum $\ell^{\infty}$-norms of their $\ell^2$-normalized singular vectors allow for several previous Sigma-Delta quantization results to be generalized and improved.

math.NA↗

arXiv2021

Faster Binary Embeddings for Preserving Euclidean Distances

We propose a fast, distance-preserving, binary embedding algorithm to transform a high-dimensional dataset $\mathcal{T}\subseteq\mathbb{R}^n$ into binary sequences in the cube $\{\pm 1\}^m$. When $\mathcal{T}$ consists of well-spread (i.e., non-sparse) vectors, our embedding method applies a stable noise-shaping quantization scheme to $A x$ where $A\in\mathbb{R}^{m\times n}$ is a sparse Gaussian random matrix. This contrasts with most binary embedding methods, which usually use $x\mapsto \mathrm{sign}(Ax)$ for the embedding. Moreover, we show that Euclidean distances among the elements of $\mathcal{T}$ are approximated by the $\ell_1$ norm on the images of $\{\pm 1\}^m$ under a fast linear transformation. This again contrasts with standard methods, where the Hamming distance is used instead. Our method is both fast and memory efficient, with time complexity $O(m)$ and space complexity $O(m)$. Further, we prove that the method is accurate and its associated error is comparable to that of a continuous valued Johnson-Lindenstrauss embedding plus a quantization error that admits a polynomial decay as the embedding dimension $m$ increases. Thus the length of the binary codes required to achieve a desired accuracy is quite small, and we show it can even be compressed further without compromising the accuracy. To illustrate our results, we test the proposed method on natural images and show that it achieves strong performance.

cs.IT↗

arXiv2019

Admissible Measurements and Robust Algorithms for Ptychography

We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say $x_0\in \mathbb{C}^d$) are illuminated one at a time, often with a physical mask between the sample and light source. The corresponding measurements are the noisy magnitudes of the Fourier transform coefficients resulting from the pointwise product of the mask and the sample. The goal is to recover the original signal from such measurements. The algorithmic framework we study herein relies on first inverting a linear system of equations to recover a fraction of the entries in $x_0 x_0^*$ and then using non-linear techniques to recover the magnitudes and phases of the entries of $x_0$. Thus, this paper's contributions are three-fold. First, focusing on the linear part, it expands the theory studying which measurement schemes (i.e., masks, shifts of the sample) yield invertible linear systems, including an analysis of the conditioning of the resulting systems. Second, it analyzes a class of improved magnitude recovery algorithms and, third, it proposes and analyzes algorithms for phase recovery in the ptychographic setting where large shifts --- up to $50\%$ the size of the mask --- are permitted.

math.NA↗

arXiv2019

New Algorithms and Improved Guarantees for One-Bit Compressed Sensing on Manifolds

We study the problem of approximately recovering signals on a manifold from one-bit linear measurements drawn from either a Gaussian ensemble, partial circulant ensemble, or bounded orthonormal ensemble and quantized using Sigma-Delta or distributed noise shaping schemes. We assume we are given a Geometric Multi-Resolution Analysis, which approximates the manifold, and we propose a convex optimization algorithm for signal recovery. We prove an upper bound on the recovery error which outperforms prior works that use memoryless scalar quantization, requires a simpler analysis, and extends the class of measurements beyond Gaussians. Finally, we illustrate our results with numerical experiments.

cs.IT↗

arXiv2019

A direct solver for the phase retrieval problem in ptychographic imaging

Measurements achieved with ptychographic imaging are a special case of diffraction measurements. They are generated by illuminating small parts of a sample with, e.g., a focused X-ray beam. By shifting the sample, a set of far-field diffraction patterns of the whole sample are then obtained. From a mathematical point of view those measurements are the squared modulus of the windowed Fourier transform of the sample. Thus, we have a phase retrieval problem for local Fourier measurements. A direct solver for this problem was introduced by Iwen, Viswanathan and Wang in 2016 and improved by Iwen, Preskitt, Saab and Viswanathan in 2018. Motivated by the applied perspective of ptychographic imaging, we present a generalization of this method and compare the different versions in numerical experiments. The new method proposed herein turns out to be more stable, particularly in the case of missing data.

math.NA↗

arXiv2018

Fast binary embeddings, and quantized compressed sensing with structured matrices

This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in $\mathcal{X}$. Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise-shaping, and include Sigma-Delta schemes and distributed noise-shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in $m$, depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson-Lindenstrauss maps while preserving $\ell_2$ norms. Second, we again consider noise-shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for Sigma-Delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise-shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems.

cs.IT↗

arXiv2018

Quantization for Low-Rank Matrix Recovery

We study Sigma-Delta quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the oversampling factor, and we leverage our results to obtain root-exponential accuracy by optimizing over the choice of quantization scheme. Additionally, we show that a random encoding scheme, applied to the quantized measurements, yields a near-optimal exponential bit-rate. As an added benefit, our schemes are robust both to noise and to deviations from the low-rank assumption. In short, we provide a full generalization of analogous results, obtained in the classical setup of bandlimited function acquisition, and more recently, in the finite frame and compressed sensing setups to the case of low-rank matrices sampled with sub-Gaussian linear operators. Finally, we believe our techniques for generalizing results from the compressed sensing setup to the analogous low-rank matrix setup is applicable to other quantization schemes.

cs.IT↗

arXiv2017

Simple Classification using Binary Data

Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this work, we study the problem of data classification from binary data and propose a framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, and provide a theoretical analysis for a simple case. We hope that our framework and analysis will serve as a foundation for studying similar types of approaches.

cs.LG↗

arXiv2017

Quantized Compressed Sensing for Partial Random Circulant Matrices

We provide the first analysis of a non-trivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in line with analogous results on Sigma-Delta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of non-quantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality.

cs.IT↗

arXiv2016

Weighted $\ell_1$-Minimization for Sparse Recovery under Arbitrary Prior Information

Weighted $\ell_1$-minimization has been studied as a technique for the reconstruction of a sparse signal from compressively sampled measurements when prior information about the signal, in the form of a support estimate, is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted $\ell_1$-minimization when arbitrarily many distinct weights are permitted. For example, such a setup might be used when one has multiple estimates for the support of a signal, and these estimates have varying degrees of accuracy. Our analysis yields an extension to existing works that assume only a single support estimate set upon which a constant weight is applied. We include numerical experiments, with both synthetic signals and real video data, that demonstrate the benefits of allowing non-uniform weights in the reconstruction procedure.

cs.IT↗

arXiv2016

Phase Retrieval from Local Measurements: Improved Robustness via Eigenvector-Based Angular Synchronization

We improve a phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks [27]. The improved algorithm admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees from [27] are improved as a result for the new and more robust reconstruction approach proposed herein. Numerical experiments demonstrate robustness and computational efficiency that outperforms competing approaches on large problems. Finally, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.

math.NA↗

arXiv2016

From compressed sensing to compressed bit-streams: practical encoders, tractable decoders

Compressed sensing is now established as an effective method for dimension reduction when the underlying signals are sparse or compressible with respect to some suitable basis or frame. One important, yet under-addressed problem regarding the compressive acquisition of analog signals is how to perform quantization. This is directly related to the important issues of how "compressed" compressed sensing is (in terms of the total number of bits one ends up using after acquiring the signal) and ultimately whether compressed sensing can be used to obtain compressed representations of suitable signals. Building on our recent work, we propose a concrete and practicable method for performing "analog-to-information conversion". Following a compressive signal acquisition stage, the proposed method consists of a quantization stage, based on $ΣΔ$ (sigma-delta) quantization, and a subsequent encoding (compression) stage that fits within the framework of compressed sensing seamlessly. We prove that, using this method, we can convert analog compressive samples to compressed digital bitstreams and decode using tractable algorithms based on convex optimization. We prove that the proposed AIC provides a nearly optimal encoding of sparse and compressible signals. Finally, we present numerical experiments illustrating the effectiveness of the proposed analog-to-information converter.

cs.IT↗

arXiv2016

One-bit compressive sensing with norm estimation

Consider the recovery of an unknown signal ${x}$ from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that ${x}$ is sparse, and that the measurements are of the form $\operatorname{sign}(\langle {a}_i, {x} \rangle) \in \{\pm1\}$. Since such measurements give no information on the norm of ${x}$, recovery methods from such measurements typically assume that $\| {x} \|_2=1$. We show that if one allows more generally for quantized affine measurements of the form $\operatorname{sign}(\langle {a}_i, {x} \rangle + b_i)$, and if the vectors ${a}_i$ are random, an appropriate choice of the affine shifts $b_i$ allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed ${x}$ in the annulus $r \leq \| {x} \|_2 \leq R$, one may estimate the norm $\| {x} \|_2$ up to additive error $δ$ from $m \gtrsim R^4 r^{-2} δ^{-2}$ such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors ${x}$ within a Euclidean ball of known radius.

stat.ML↗

arXiv2015

A Deterministic Analysis of Decimation for Sigma-Delta Quantization of Bandlimited Functions

We study Sigma-Delta ($ΣΔ$) quantization of oversampled bandlimited functions. We prove that digitally integrating blocks of bits and then down-sampling, a process known as decimation, can efficiently encode the associated $ΣΔ$ bit-stream. It allows a large reduction in the bit-rate while still permitting good approximation of the underlying bandlimited function via an appropriate reconstruction kernel. Specifically, in the case of stable $r$th order $ΣΔ$ schemes we show that the reconstruction error decays exponentially in the bit-rate. For example, this result applies to the 1-bit, greedy, first-order $ΣΔ$ scheme.

cs.IT↗

arXiv2015

Quantization of compressive samples with stable and robust recovery

In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma-Delta quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.

cs.IT↗

arXiv2015

Noise-shaping Quantization Methods for Frame-based and Compressive Sampling Systems

Noise shaping refers to an analog-to-digital conversion methodology in which quantization error is arranged to lie mostly outside the signal spectrum by means of oversampling and feedback. Recently it has been successfully applied to more general redundant linear sampling and reconstruction systems associated with frames as well as non-linear systems associated with compressive sampling. This chapter reviews some of the recent progress in this subject.

cs.IT↗