arXiv subjects

Sajad Daei

Publications and source records attributed to Sajad Daei.

At least 37 records · Page 2Linked to original sources

arXiv2023

Multi-User Distributed Computing Via Compressed Sensing

The multi-user linearly-separable distributed computing problem is considered here, in which $N$ servers help to compute the real-valued functions requested by $K$ users, where each function can be written as a linear combination of up to $L$ (generally non-linear) subfunctions. Each server computes a fraction $γ$ of the subfunctions, then communicates a function of its computed outputs to some of the users, and then each user collects its received data to recover its desired function. Our goal is to bound the ratio between the computation workload done by all servers over the number of datasets. To this end, we here reformulate the real-valued distributed computing problem into a matrix factorization problem and then into a basic sparse recovery problem, where sparsity implies computational savings. Building on this, we first give a simple probabilistic scheme for subfunction assignment, which allows us to upper bound the optimal normalized computation cost as $γ\leq \frac{K}{N}$ that a generally intractable $\ell_0$-minimization would give. To bypass the intractability of such optimal scheme, we show that if these optimal schemes enjoy $γ\leq - r\frac{K}{N}W^{-1}_{-1}(- \frac{2K}{e N r} )$ (where $W_{-1}(\cdot)$ is the Lambert function and $r$ calibrates the communication between servers and users), then they can actually be derived using a tractable Basis Pursuit $\ell_1$-minimization. This newly-revealed connection between distributed computation and compressed sensing opens up the possibility of designing practical distributed computing algorithms by employing tools and methods from compressed sensing.

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arXiv2022

Distribution-aware $\ell_1$ Analysis Minimization

This work is about recovering an analysis-sparse vector, i.e. sparse vector in some transform domain, from under-sampled measurements. In real-world applications, there often exist random analysis-sparse vectors whose distribution in the analysis domain are known. To exploit this information, a weighted $\ell_1$ analysis minimization is often considered. The task of choosing the weights in this case is however challenging and non-trivial. In this work, we provide an analytical method to choose the suitable weights. Specifically, we first obtain a tight upper-bound expression for the expected number of required measurements. This bound depends on two critical parameters: support distribution and expected sign of the analysis domain which are both accessible in advance. Then, we calculate the near-optimal weights by minimizing this expression with respect to the weights. Our strategy works for both noiseless and noisy settings. Numerical results demonstrate the superiority of our proposed method. Specifically, the weighted $\ell_1$ analysis minimization with our near-optimal weighting design considerably needs fewer measurements than its regular $\ell_1$ analysis counterpart.

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arXiv2022

Demixing Sines and Spikes Using Multiple Measurement Vectors

In this paper, we address the line spectral estimation problem with multiple measurement corrupted vectors. Such scenarios appear in many practical applications such as radar, optics, and seismic imaging in which the signal of interest can be modeled as the sum of a spectrally sparse and a blocksparse signal known as outlier. Our aim is to demix the two components and for that, we design a convex problem whose objective function promotes both of the structures. Using positive trigonometric polynomials (PTP) theory, we reformulate the dual problem as a semi-definite program (SDP). Our theoretical results states that for a fixed number of measurements N and constant number of outliers, up to O(N) spectral lines can be recovered using our SDP problem as long as a minimum frequency separation condition is satisfied. Our simulation results also show that increasing the number of samples per measurement vectors, reduces the minimum required frequency separation for successful recovery.

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arXiv2022

Blind Goal-Oriented Massive Access for Future Wireless Networks

Emerging communication networks are envisioned to support massive wireless connectivity of heterogeneous devices with sporadic traffic and diverse requirements in terms of latency, reliability, and bandwidth. Providing multiple access to an increasing number of uncoordinated users and sharing the limited resources become essential in this context. In this work, we revisit the random access (RA) problem and exploit the continuous angular group sparsity feature of wireless channels to propose a novel RA strategy that provides low latency, high reliability, and massive access with limited bandwidth resources in an all-in-one package. To this end, we first design a reconstruction-free goal-oriented optimization problem, which only preserves the angular information required to identify the active devices. To solve this, we propose an alternating direction method of multipliers (ADMM) and derive closed-form expressions for each ADMM step. Then, we design a clustering algorithm that assigns the users in specific groups from which we can identify active stationary devices by their angles. For mobile devices, we propose an alternating minimization algorithm to recover their data and their channel gains simultaneously, which allows us to identify active mobile users. Simulation results show significant performance gains in terms of active user detection and false alarm probabilities as compared to state-of-the-art RA schemes, even with limited number of preambles. Moreover, unlike prior work, the performance of the proposed blind goal-oriented massive access does not depend on the number of devices.

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arXiv2022

Active User Detection and Channel Estimation for Spatial-based Random Access in Crowded Massive MIMO Systems via Blind Super-resolution

This work presents a novel framework for random access in crowded scenarios of multiple-input multiple-output(MIMO) systems. A multi-antenna base station (BS) and multiple single-antenna users are considered in these systems. A huge portion of the system resources is dedicated as orthogonal pilots for accurate channel estimation which imposes a huge training overhead. This overhead can be highly mitigated by exploiting intrinsic angular domain sparsity of massive MIMO channels and the sporadic traffic of users, i.e., few number of users are active to sent or receive data in each coherence interval. In fact, the angles of arrivals (AoAs) coming from active users are continuous parameters and can take any arbitrary values. Besides, the AoAs corresponding to each active user are alongside each other forming a specific cluster. This work revolves around exploiting these features. Specifically, a blind clustering-based algorithm is proposed that not only recovers the transmitted data by users in grant free random access and primary pilots in random access blocks of coherent transmission, but also provides accurate channel estimation. Our approach is based on transforming the unknown variables into a higher dimensional space with matrix variables. An off-grid atomic norm minimization is then proposed to obtain the unknown matrix from only a few observed arrays at the BS. Then, a clustering-based approach is employed to identify which AoAs correspond to which active users. After identifying active users and their AoAs, an alternating-based approach is performed to obtain the channels and data or primary pilots of active users. Simulation results demonstrate the effectiveness of our approach in AoA detection as well as data recovery.

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arXiv2022

Multi-weight Nuclear Norm Minimization for Low-rank Matrix Recovery in Presence of Subspace Prior Information

Weighted nuclear norm minimization has been recently recognized as a technique for reconstruction of a low-rank matrix from compressively sampled measurements when some prior information about the column and row subspaces of the matrix is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted nuclear norm minimization when multiple weights are allowed. This setup might be used when one has access to prior subspaces forming multiple angles with the column and row subspaces of the ground-truth matrix. While existing works in this field use a single weight to penalize all the angles, we propose a multi-weight problem which is designed to penalize each angle independently using a distinct weight. Specifically, we prove that our proposed multi-weight problem is stable and robust under weaker conditions for the measurement operator than the analogous conditions for single-weight scenario and standard nuclear norm minimization. Moreover, it provides better reconstruction error than the state of the art methods. We illustrate our results with extensive numerical experiments that demonstrate the advantages of allowing multiple weights in the recovery procedure.

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arXiv2021

Multi-weight Matrix Completion with Arbitrary Subspace Prior Information

Matrix completion refers to completing a low-rank matrix from a few observed elements of its entries and has been known as one of the significant and widely-used problems in recent years. The required number of observations for exact completion is directly proportional to rank and the coherency parameter of the matrix. In many applications, there might exist additional information about the low-rank matrix of interest. For example, in collaborative filtering, Netflix and dynamic channel estimation in communications, extra subspace information is available. More precisely in these applications, there are prior subspaces forming multiple angles with the ground-truth subspaces. In this paper, we propose a novel strategy to incorporate this information into the completion task. To this end, we designed a multi-weight nuclear norm minimization where the weights are such chosen to penalize each angle within the matrix subspace independently. We propose a new scheme for optimally choosing the weights. Specifically, we first calculate an upper-bound expression describing the coherency of the interested matrix. Then, we obtain the optimal weights by minimizing this expression. Simulation results certify the advantages of allowing multiple weights in the completion procedure. Explicitly, they indicate that our proposed multi-weight problem needs fewer observations compared to state-of-the-art methods.

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arXiv2021

Off-the-grid Recovery of Time and Frequency Shifts with Multiple Measurement Vectors

We address the problem of estimating time and frequency shifts of a known waveform in the presence of multiple measurement vectors (MMVs). This problem naturally arises in radar imaging and wireless communications. Specifically, a signal ensemble is observed, where each signal of the ensemble is formed by a superposition of a small number of scaled, time-delayed, and frequency shifted versions of a known waveform sharing the same continuous-valued time and frequency components. The goal is to recover the continuous-valued time-frequency pairs from a small number of observations. In this work, we propose a semidefinite programming which exactly recovers $s$ pairs of time-frequency shifts from $L$ regularly spaced samples per measurement vector under a minimum separation condition between the time-frequency shifts. Moreover, we prove that the number $s$ of time-frequency shifts scales linearly with the number $L$ of samples up to a log-factor. Extensive numerical results are also provided to validate the effectiveness of the proposed method over the single measurement vectors (SMVs) problem. In particular, we find that our approach leads to a relaxed minimum separation condition and reduced number of required samples.

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arXiv2020

Blind Two-Dimensional Super Resolution in Multiple Input Single Output Linear Systems

In this paper, we consider a multiple-input single-output (MISO) linear time-varying system whose output is a superposition of scaled and time-frequency shifted versions of inputs. The goal of this paper is to determine system characteristics and input signals from the single output signal. More precisely, we want to recover the continuous time-frequency shift pairs, the corresponding (complex-valued) amplitudes and the input signals from only one output vector. This problem arises in a variety of applications such as radar imaging, microscopy, channel estimation and localization problems. While this problem is naturally ill-posed, by constraining the unknown input waveforms to lie in separate known low-dimensional subspaces, it becomes tractable. More explicitly, we propose a semidefinite program which exactly recovers time-frequency shift pairs and input signals. We prove uniqueness and optimality of the solution to this program. Moreover, we provide a grid-based approach which can significantly reduce computational complexity in exchange for adding a small gridding error. Numerical results confirm the ability of our proposed method to exactly recover the unknowns.

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arXiv2020

On the Sample Complexity of Super-Resolution Radar

We point out an issue with Lemma 8.6 of [1]. This lemma specifies the required sample complexity for recovering the delay-Doppler pairs in radar systems. In this lemma, it is claimed that the required sample complexity is linearly related to the number of delay-Doppler pairs. However, the way this result is proved seems to be wrong and leads to a quadratic relation. In this work, we provide a correction to the proof of this lemma.

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arXiv2019

Adaptive Recovery of Dictionary-sparse Signals using Binary Measurements

One-bit compressive sensing (CS) is an advanced version of sparse recovery in which the sparse signal of interest can be recovered from extremely quantized measurements. Namely, only the sign of each measurement is available to us. In many applications, the ground-truth signal is not sparse itself, but can be represented in a redundant dictionary. A strong line of research has addressed conventional CS in this signal model including its extension to one-bit measurements. However, one-bit CS suffers from the extremely large number of required measurements to achieve a predefined reconstruction error level. A common alternative to resolve this issue is to exploit adaptive schemes. Adaptive sampling acts on the acquired samples to trace the signal in an efficient way. In this work, we utilize an adaptive sampling strategy to recover dictionary-sparse signals from binary measurements. For this task, a multi-dimensional threshold is proposed to incorporate the previous signal estimates into the current sampling procedure. This strategy substantially reduces the required number of measurements for exact recovery. Our proof approach is based on the recent tools in high dimensional geometry in particular random hyperplane tessellation and Gaussian width. We show through rigorous and numerical analysis that the proposed algorithm considerably outperforms state of the art approaches. Further, our algorithm reaches an exponential error decay in terms of the number of quantized measurements.

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arXiv2019

Living near the edge: A lower-bound on the phase transition of total variation minimization

This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to success as the number of measurements increases. In fact, in large dimensions, TV minimization succeeds in recovering the gradient-sparse signal with high probability when the number of measurements exceeds a certain threshold; otherwise, it fails almost certainly. Obtaining a closed-form expression that approximates this threshold is a major challenge in this field and has not been appropriately addressed yet. In this work, we derive a tight lower-bound on this threshold in case of any random measurement matrix whose null space is distributed uniformly with respect to the Haar measure. In contrast to the conventional TV phase transition results that depend on the simple gradient-sparsity level, our bound is highly affected by generalized notions of gradient-sparsity. Our proposed bound is very close to the true phase transition of TV minimization confirmed by simulation results.

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arXiv2019

A Greedy Algorithm for Matrix Recovery with Subspace Prior Information

Matrix recovery is the problem of recovering a low-rank matrix from a few linear measurements. Recently, this problem has gained a lot of attention as it is employed in many applications such as Netflix prize problem, seismic data interpolation and collaborative filtering. In these applications, one might access to additional prior information about the column and row spaces of the matrix. These extra information can potentially enhance the matrix recovery performance. In this paper, we propose an efficient greedy algorithm that exploits prior information in the recovery procedure. The performance of the proposed algorithm is measured in terms of the rank restricted isometry property (R-RIP). Our proposed algorithm with prior subspace information converges under a more milder condition on the R-RIP in compared with the case that we do not use prior information. Additionally, our algorithm performs much better than nuclear norm minimization in terms of both computational complexity and success rate.

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arXiv2019

Exploiting Prior Information in Block Sparse Signals

We study the problem of recovering a block-sparse signal from under-sampled observations. The non-zero values of such signals appear in few blocks, and their recovery is often accomplished using a $\ell_{1,2}$ optimization problem. In applications such as DNA micro-arrays, some prior information about the block support, i.e., blocks containing non-zero elements, is available. A typical way to consider the extra information in recovery procedures is to solve a weighted $\ell_{1,2}$ problem. In this paper, we consider a block sparse model, where the block support has intersection with some given subsets of blocks with known probabilities. Our goal in this work is to minimize the number of required linear Gaussian measurements for perfect recovery of the signal by tuning the weights of a weighted $\ell_{1,2}$ problem. For this goal, we apply tools from conic integral geometry and derive closed-form expressions for the optimal weights. We show through precise analysis and simulations that the weighted $\ell_{1,2}$ problem with optimal weights significantly outperforms the regular $\ell_{1,2}$ problem. We further examine the sensitivity of the optimal weights to the mismatch of block probabilities, and conclude stability under small probability deviations.

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arXiv2019

On the Error in Phase Transition Computations for Compressed Sensing

Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and $\ell_1$ analysis minimization becomes poorly large. Next, we develop a new error bound which significantly improves the estimation gap compared to [1]. In particular, unlike the bound in [1] that is not applicable to settings with overcomplete dictionaries, our bound exhibits a decaying behavior in such cases.

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arXiv2019

Optimal Weighted Low-rank Matrix Recovery with Subspace Prior Information

Matrix sensing is the problem of reconstructing a low-rank matrix from a few linear measurements. In many applications such as collaborative filtering, the famous Netflix prize problem, and seismic data interpolation, there exists some prior information about the column and row spaces of the ground-truth low-rank matrix. In this paper, we exploit this prior information by proposing a weighted optimization problem where its objective function promotes both rank and prior subspace information. Using the recent results in conic integral geometry, we obtain the unique optimal weights that minimize the required number of measurements. As simulation results confirm, the proposed convex program with optimal weights requires substantially fewer measurements than the regular nuclear norm minimization.

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arXiv2019

Distribution-aware Block-sparse Recovery via Convex Optimization

We study the problem of reconstructing a block-sparse signal from compressively sampled measurements. In certain applications, in addition to the inherent block-sparse structure of the signal, some prior information about the block support, i.e. blocks containing non-zero elements, might be available. Although many block-sparse recovery algorithms have been investigated in Bayesian framework, it is still unclear how to incorporate the information about the probability of occurrence into regularization-based block-sparse recovery in an optimal sense. In this work, we bridge between these fields by the aid of a new concept in conic integral geometry. Specifically, we solve a weighted optimization problem when the prior distribution about the block support is available. Moreover, we obtain the unique weights that minimize the expected required number of measurements. Our simulations on both synthetic and real data confirm that these weights considerably decrease the required sample complexity.

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arXiv2018

Improved Recovery of Analysis Sparse Vectors in Presence of Prior Information

In this work, we consider the problem of recovering analysis-sparse signals from under-sampled measurements when some prior information about the support is available. We incorporate such information in the recovery stage by suitably tuning the weights in a weighted $\ell_1$ analysis optimization problem. Indeed, we try to set the weights such that the method succeeds with minimum number of measurements. For this purpose, we exploit the upper-bound on the statistical dimension of a certain cone to determine the weights. Our numerical simulations confirm that the introduced method with tuned weights outperforms the standard $\ell_1$ analysis technique.

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