Mahdi Soltanolkotabi

Phase Retrieval via Wirtinger Flow: Theory and Algorithms

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal x ∈ ℂn about which we have phaseless samples of the form yr = |〈ar, x〉|2, r = 1, ..., m (knowledge of the phase of these samples would yield a linear system). This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.

A Geometric Analysis of Subspace Clustering with Outliers

This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [11], which signicantly broadens the range of problems where it is provably eective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods.

Robust subspace clustering

Subspace clustering refers to the task of nding a multi-subspace representation that best ts a collection of points taken from a high-dimensional space. This paper introduces an algorithm inspired by sparse subspace clustering (SSC) (25) to cluster noisy data, and develops some novel theory demonstrating its correctness. In particular, the theory uses ideas from geometric functional analysis to show that the algorithm can accurately recover the underlying subspaces under minimal requirements on their orientation, and on the number of samples per subspace. Synthetic as well as real data experiments complement our theoretical study, illustrating our approach and demonstrating its eectiveness.

Experimental robustness of Fourier ptychography phase retrieval algorithms

Fourier ptychography is a new computational microscopy technique that provides gigapixel-scale intensity and phase images with both wide field-of-view and high resolution. By capturing a stack of low-resolution images under different illumination angles, an inverse algorithm can be used to computationally reconstruct the high-resolution complex field. Here, we compare and classify multiple proposed inverse algorithms in terms of experimental robustness. We find that the main sources of error are noise, aberrations and mis-calibration (i.e. model mis-match). Using simulations and experiments, we demonstrate that the choice of cost function plays a critical role, with amplitude-based cost functions performing better than intensity-based ones. The reason for this is that Fourier ptychography datasets consist of images from both brightfield and darkfield illumination, representing a large range of measured intensities. Both noise (e.g. Poisson noise) and model mis-match errors are shown to scale with intensity. Hence, algorithms that use an appropriate cost function will be more tolerant to both noise and model mis-match. Given these insights, we propose a global Newtons method algorithm which is robust and accurate. Finally, we discuss the impact of procedures for algorithmic correction of aberrations and mis-calibration.

Sharp Time–Data Tradeoffs for Linear Inverse Problems

In this paper, we characterize sharp time–data tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a least-squares objective subject to a constraint defined as the sub-level set of a penalty function. We present a unified convergence analysis of the gradient projection algorithm applied to such problems. We sharply characterize the convergence rate associated with a wide variety of random measurement ensembles in terms of the number of measurements and structural complexity of the signal with respect to the chosen penalty function. The results apply to both convex and nonconvex constraints, demonstrating that a linear convergence rate is attainable even though the least squares objective is not strongly convex in these settings. When specialized to Gaussian measurements our results show that such linear convergence occurs when the number of measurements is merely four times the minimal number required to recover the desired signal at all (also known as the phase transition). We also achieve a slower but geometric rate of convergence precisely above the phase transition point. Extensive numerical results suggest that the derived rates exactly match the empirical performance.

Errorless Codes for Over-Loaded CDMA with Active User Detection

In this paper we introduce a new class of codes for Over-loaded synchronous wireless CDMA systems which increases the number of users for a fixed number of chips without introducing any errors. In addition these codes support active user detection. We derive an upper bound on the number of users with a fixed spreading factor. Also we propose an ML decoder for a subclass of these codes that is computationally implementable. Although for our simulations we consider a scenario that is worse than what occurs in practice, simulation results indicate that this coding/decoding scheme is robust against additive noise. As an example, for 64 chips and 88 users we propose a coding/decoding scheme that can obtain an arbitrary small probability of error which is computationally feasible and can detect active users. Furthermore, we prove that for this to be possible the number of users cannot be beyond 230.

A practical sparse channel estimation for current OFDM standards

Wireless channels especially for OFDM transmissions can be precisely approximated by a time varying filter with sparse taps (in the time domain). Sparsity of the channel is a criterion which can highly improve the channel estimation task in mobile applications. In sparse signal processing, many efficient algorithms have been developed for finding the sparsest solution to linear equations (Basis Pursuit, Matching Pursuit) in the presence of noise. In current OFDM standards, a number of the ending subcarriers at both positive and negative frequencies are left unoccupied (for ease of analog filtering at the receiver) which results in an ill-conditioned frequency to time transformation matrix. This means that the initial estimate for the impulse response of the channel (in time) easily varies as the noise vector changes. Thus in this case we cannot use most of the proposed algorithms in sparse signal processing. In this paper, we propose iteration with adaptive thresholding and MMSE methods to overcome this difficulty. Simulation results indicate that the proposed method is almost perfect for stationary channels and only minor performance degradation is observed with increase of Doppler frequency.

Fast and Reliable Parameter Estimation from Nonlinear Observations

In this paper we study the problem of recovering a structured but unknown parameter

Salt and pepper noise removal for image signals

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Phase retrieval from coded diffraction patterns

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