Pub Date : 2025-01-20DOI: 10.1109/TSP.2025.3531356
Yiming Jiang;Helei Kang;Jinlan Liu;Dongpo Xu
Stochastic optimization algorithms are widely used to solve large-scale machine learning problems. However, their theoretical analysis necessitates access to unbiased estimates of the true gradients. To address this issue, we perform a comprehensive convergence rate analysis of stochastic gradient descent (SGD) with biased gradients for decentralized optimization. In non-convex settings, we show that for decentralized SGD utilizing biased gradients, the gradient in expectation is bounded asymptotically at a rate of $mathcal{O}(1/sqrt{nT}+n/T)$, and the bound is linearly correlated to the biased gradient gap. In particular, we can recover the convergence results in the unbiased stochastic gradient setting when the biased gradient gap is zero. Lastly, we provide empirical support for our theoretical findings through extensive numerical experiments.
{"title":"On the Convergence of Decentralized Stochastic Gradient Descent With Biased Gradients","authors":"Yiming Jiang;Helei Kang;Jinlan Liu;Dongpo Xu","doi":"10.1109/TSP.2025.3531356","DOIUrl":"10.1109/TSP.2025.3531356","url":null,"abstract":"Stochastic optimization algorithms are widely used to solve large-scale machine learning problems. However, their theoretical analysis necessitates access to unbiased estimates of the true gradients. To address this issue, we perform a comprehensive convergence rate analysis of stochastic gradient descent (SGD) with biased gradients for decentralized optimization. In non-convex settings, we show that for decentralized SGD utilizing biased gradients, the gradient in expectation is bounded asymptotically at a rate of <inline-formula><tex-math>$mathcal{O}(1/sqrt{nT}+n/T)$</tex-math></inline-formula>, and the bound is linearly correlated to the biased gradient gap. In particular, we can recover the convergence results in the unbiased stochastic gradient setting when the biased gradient gap is zero. Lastly, we provide empirical support for our theoretical findings through extensive numerical experiments.","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"549-558"},"PeriodicalIF":4.6,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-20DOI: 10.1109/tsp.2025.3531292
Aleksandar Armacki, Dragana Bajović, Dušan Jakovetić, Soummya Kar
{"title":"Distributed Center-based Clustering: A Unified Framework","authors":"Aleksandar Armacki, Dragana Bajović, Dušan Jakovetić, Soummya Kar","doi":"10.1109/tsp.2025.3531292","DOIUrl":"https://doi.org/10.1109/tsp.2025.3531292","url":null,"abstract":"","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"13 1","pages":""},"PeriodicalIF":5.4,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142991539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-16DOI: 10.1109/TSP.2025.3530149
Nilesh Kumar Jha;Huayan Guo;Vincent K. N. Lau
In-memory computation (IMC) is a promising technology for enabling low-latency and energy-efficient deep learning and artificial intelligence (AI) applications at edge devices. However, the IMC crossbar array, typically implemented using resistive random access memory (RRAM), faces hardware defects that pose a significant challenge to reliable computation. This paper presents a robust IMC scheme utilizing Bayesian neural network-accelerated analog codes. Our approach includes a new datapath design comprising a parity matrix generator and a low-complexity decoder module to facilitate analog codes for IMC. Moreover, we introduce a Gaussian mixture model-based error prior to capture impulsive error statistics and leverage variational Bayesian inference (VBI) techniques for training neural network weights. Extensive simulations confirm the effectiveness of our proposed solution compared to various state-of-the-art baseline schemes.
{"title":"Robust In-Memory Computation With Bayesian Analog Error Mitigating Codes","authors":"Nilesh Kumar Jha;Huayan Guo;Vincent K. N. Lau","doi":"10.1109/TSP.2025.3530149","DOIUrl":"10.1109/TSP.2025.3530149","url":null,"abstract":"In-memory computation (IMC) is a promising technology for enabling low-latency and energy-efficient deep learning and artificial intelligence (AI) applications at edge devices. However, the IMC crossbar array, typically implemented using resistive random access memory (RRAM), faces hardware defects that pose a significant challenge to reliable computation. This paper presents a robust IMC scheme utilizing Bayesian neural network-accelerated analog codes. Our approach includes a new datapath design comprising a parity matrix generator and a low-complexity decoder module to facilitate analog codes for IMC. Moreover, we introduce a Gaussian mixture model-based error prior to capture impulsive error statistics and leverage variational Bayesian inference (VBI) techniques for training neural network weights. Extensive simulations confirm the effectiveness of our proposed solution compared to various state-of-the-art baseline schemes.","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"534-548"},"PeriodicalIF":4.6,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142987669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-15DOI: 10.1109/TSP.2025.3529722
Chenhao Yang;Xin Wang;Wei Ni;Yi Jiang
This paper optimizes the beamforming design of a downlink multiple-input multiple-output (MIMO) dual-function radar-communication (DFRC) system to maximize the weighted communication sum-rate under a prescribed transmit covariance constraint for radar performance guarantee. In the single-user case, we show that the transmit covariance constraint implies the existence of inherent orthogonality among the transmit beamforming vectors in use. Then, leveraging Cauchy's interlace theorem, we derive the globally optimal transmit and receive beamforming solution in closed form. In the multi-user case, we exploit the connection between the weighted sum-rate and weighted minimum mean squared error (MMSE) to reformulate the intended problem, and develop a block-coordinate-descent (BCD) algorithm to iteratively compute the transmit beamforming and receive beamforming solutions. Under this approach, we reveal that the optimal receive beamforming is given by the classic MMSE one and the optimal transmit beamforming design amounts to solving an orthogonal Procrustes problem, thereby allowing for closed-form solutions to subproblems in each BCD step and fast convergence of the proposed algorithm to a high-quality (near-optimal) overall beamforming design. Numerical results demonstrate the superiority of our approach to the existing methods, with at least 40% higher sum-rate under a multi-user MIMO setting in the high signal-to-noise regime.
{"title":"Optimal Beamforming for MIMO DFRC Systems With Transmit Covariance Constraints","authors":"Chenhao Yang;Xin Wang;Wei Ni;Yi Jiang","doi":"10.1109/TSP.2025.3529722","DOIUrl":"10.1109/TSP.2025.3529722","url":null,"abstract":"This paper optimizes the beamforming design of a downlink multiple-input multiple-output (MIMO) dual-function radar-communication (DFRC) system to maximize the weighted communication sum-rate under a prescribed transmit covariance constraint for radar performance guarantee. In the single-user case, we show that the transmit covariance constraint implies the existence of inherent orthogonality among the transmit beamforming vectors in use. Then, leveraging Cauchy's interlace theorem, we derive the globally optimal transmit and receive beamforming solution in closed form. In the multi-user case, we exploit the connection between the weighted sum-rate and weighted minimum mean squared error (MMSE) to reformulate the intended problem, and develop a block-coordinate-descent (BCD) algorithm to iteratively compute the transmit beamforming and receive beamforming solutions. Under this approach, we reveal that the optimal receive beamforming is given by the classic MMSE one and the optimal transmit beamforming design amounts to solving an orthogonal Procrustes problem, thereby allowing for closed-form solutions to subproblems in each BCD step and fast convergence of the proposed algorithm to a high-quality (near-optimal) overall beamforming design. Numerical results demonstrate the superiority of our approach to the existing methods, with at least 40% higher sum-rate under a multi-user MIMO setting in the high signal-to-noise regime.","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"601-616"},"PeriodicalIF":4.6,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-14DOI: 10.1109/TSP.2025.3529950
Bo Tang;Da Li;Wenjun Wu;Astha Saini;Prabhu Babu;Petre Stoica
This paper investigates the transmit beamforming design for multiple-input multiple-output systems to support both multi-target localization and multi-user communications. To enhance the target localization performance, we derive the asymptotic Cramér-Rao bound (CRB) for target angle estimation by assuming that the receive array is linear and uniform. Then we formulate a beamforming design problem based on minimizing an upper bound on the asymptotic CRB (which is shown to be equivalent to maximizing the harmonic mean of the weighted beampattern responses at the target directions). Moreover, we impose a constraint on the SINR of each received communication signal to guarantee reliable communication performance. Two iterative algorithms are derived to tackle the non-convex design problem: one is based on the alternating direction method of multipliers, and the other uses the majorization-minimization technique to solve an equivalent minimax problem. Numerical results show that, through elaborate dual-function beamforming matrix design, the proposed algorithms can simultaneously achieve superior angle estimation performance as well as high-quality multi-user communications.
{"title":"Dual-Function Beamforming Design for Multi-Target Localization and Reliable Communications","authors":"Bo Tang;Da Li;Wenjun Wu;Astha Saini;Prabhu Babu;Petre Stoica","doi":"10.1109/TSP.2025.3529950","DOIUrl":"10.1109/TSP.2025.3529950","url":null,"abstract":"This paper investigates the transmit beamforming design for multiple-input multiple-output systems to support both multi-target localization and multi-user communications. To enhance the target localization performance, we derive the asymptotic Cramér-Rao bound (CRB) for target angle estimation by assuming that the receive array is linear and uniform. Then we formulate a beamforming design problem based on minimizing an upper bound on the asymptotic CRB (which is shown to be equivalent to maximizing the harmonic mean of the weighted beampattern responses at the target directions). Moreover, we impose a constraint on the SINR of each received communication signal to guarantee reliable communication performance. Two iterative algorithms are derived to tackle the non-convex design problem: one is based on the alternating direction method of multipliers, and the other uses the majorization-minimization technique to solve an equivalent minimax problem. Numerical results show that, through elaborate dual-function beamforming matrix design, the proposed algorithms can simultaneously achieve superior angle estimation performance as well as high-quality multi-user communications.","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"559-573"},"PeriodicalIF":4.6,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142981500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-13DOI: 10.1109/TSP.2025.3529468
Shuning Sun;Dongpo Xu;Qiankun Diao;Danilo P. Mandic
The Cramér-Rao lower bound (CRLB) is a fundamental result in statistical signal processing, however, the CRLB for quaternion parameters is not yet established. To this end, we develop the theory of quaternion Cramér-Rao lower bound (QCRLB), based on the generalized Hamilton-real (GHR) calculus. For generality, this is achieved in a way that conforms with the real and complex CRLB. We first provide the properties of the quaternion covariance matrix and the quaternion Fisher information matrix (FIM), paving the way for the derivation of the QCRLB. This serves as a basis for the formulation of the QCRLB without constraints and a criterion for determining whether the QCRLB is attained. We also establish the QCRLB for constrained quaternion parameters, including both nonsingular and singular cases of the quaternion FIM. These broaden the theoretical framework and enhance its applicability to diverse practical scenarios. The practical efficacy of the QCRLB is demonstrated through two illustrative examples. Numerical validations confirm that the maximum-likelihood estimator (MLE) attains the QCRLB for the linear model, and the quaternion gradient ascent (QGA) algorithm achieves the QCRLB at each iteration with theoretical guarantees. We also propose the quaternion constrained scoring (QCS) algorithm, which converges in one step in the linear constrained MLE case, for the linear model. These results significantly contribute to both the theory and practical application of quaternion signal processing, bringing valuable insights into the quaternion parameter estimation.
{"title":"Cramér-Rao Lower Bounds for Unconstrained and Constrained Quaternion Parameters","authors":"Shuning Sun;Dongpo Xu;Qiankun Diao;Danilo P. Mandic","doi":"10.1109/TSP.2025.3529468","DOIUrl":"10.1109/TSP.2025.3529468","url":null,"abstract":"The Cramér-Rao lower bound (CRLB) is a fundamental result in statistical signal processing, however, the CRLB for quaternion parameters is not yet established. To this end, we develop the theory of quaternion Cramér-Rao lower bound (QCRLB), based on the generalized Hamilton-real (GHR) calculus. For generality, this is achieved in a way that conforms with the real and complex CRLB. We first provide the properties of the quaternion covariance matrix and the quaternion Fisher information matrix (FIM), paving the way for the derivation of the QCRLB. This serves as a basis for the formulation of the QCRLB without constraints and a criterion for determining whether the QCRLB is attained. We also establish the QCRLB for constrained quaternion parameters, including both nonsingular and singular cases of the quaternion FIM. These broaden the theoretical framework and enhance its applicability to diverse practical scenarios. The practical efficacy of the QCRLB is demonstrated through two illustrative examples. Numerical validations confirm that the maximum-likelihood estimator (MLE) attains the QCRLB for the linear model, and the quaternion gradient ascent (QGA) algorithm achieves the QCRLB at each iteration with theoretical guarantees. We also propose the quaternion constrained scoring (QCS) algorithm, which converges in one step in the linear constrained MLE case, for the linear model. These results significantly contribute to both the theory and practical application of quaternion signal processing, bringing valuable insights into the quaternion parameter estimation.","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"508-518"},"PeriodicalIF":4.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142974993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-13DOI: 10.1109/tsp.2025.3529657
Lin Chen, Xue Jiang, Pei Xiao, Xingzhao Liu, Martin Haardt
{"title":"Structured Tensor Decomposition for FDD Massive MIMO Downlink Channel Reconstruction","authors":"Lin Chen, Xue Jiang, Pei Xiao, Xingzhao Liu, Martin Haardt","doi":"10.1109/tsp.2025.3529657","DOIUrl":"https://doi.org/10.1109/tsp.2025.3529657","url":null,"abstract":"","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"42 1","pages":""},"PeriodicalIF":5.4,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142974994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-13DOI: 10.1109/TSP.2025.3527755
Pei-Duo Yu;Chee Wei Tan
Identifying the source of epidemic-like spread in networks is crucial for removing internet viruses or finding the source of rumors in online social networks. The challenge lies in tracing the source from a snapshot observation of infected nodes. How do we accurately pinpoint the source? Utilizing snapshot data, we apply a probabilistic approach, focusing on the graph boundary and the observed time, to detect sources via an effective maximum likelihood algorithm. A novel starlike tree approximation extends applicability to general graphs, demonstrating versatility. Unlike previous works that rely heavily on structural properties alone, our method also incorporates temporal data for more precise source detection. We highlight the utility of the Gamma function for analyzing the ratio of the likelihood being the source between nodes asymptotically. Comprehensive evaluations confirm algorithmic effectiveness in diverse network scenarios, advancing source detection in large-scale network analysis and information dissemination strategies.
{"title":"Unraveling the Viral Spread of Misinformation: Maximum-Likelihood Estimation and Starlike Tree Approximation in Markovian Spreading Models","authors":"Pei-Duo Yu;Chee Wei Tan","doi":"10.1109/TSP.2025.3527755","DOIUrl":"10.1109/TSP.2025.3527755","url":null,"abstract":"Identifying the source of epidemic-like spread in networks is crucial for removing internet viruses or finding the source of rumors in online social networks. The challenge lies in tracing the source from a snapshot observation of infected nodes. How do we accurately pinpoint the source? Utilizing snapshot data, we apply a probabilistic approach, focusing on the graph boundary and the observed time, to detect sources via an effective maximum likelihood algorithm. A novel starlike tree approximation extends applicability to general graphs, demonstrating versatility. Unlike previous works that rely heavily on structural properties alone, our method also incorporates temporal data for more precise source detection. We highlight the utility of the Gamma function for analyzing the ratio of the likelihood being the source between nodes asymptotically. Comprehensive evaluations confirm algorithmic effectiveness in diverse network scenarios, advancing source detection in large-scale network analysis and information dissemination strategies.","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"446-461"},"PeriodicalIF":4.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142974841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-10DOI: 10.1109/TSP.2024.3522179
Junren Chen;Michael K. Ng;Zhaoqiang Liu
The problem of recovering a signal <inline-formula><tex-math>$boldsymbol{x}inmathbb{R}^{n}$</tex-math></inline-formula> from a quadratic system <inline-formula><tex-math>${y_{i}=boldsymbol{x}^{top}boldsymbol{A}_{i}boldsymbol{x}, i=1,ldots,m}$</tex-math></inline-formula> with full-rank matrices <inline-formula><tex-math>$boldsymbol{A}_{i}$</tex-math></inline-formula> frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. With i.i.d. standard Gaussian matrices <inline-formula><tex-math>$boldsymbol{A}_{i}$</tex-math></inline-formula>, this paper addresses the high-dimensional case where <inline-formula><tex-math>$mll n$</tex-math></inline-formula> by incorporating prior knowledge of <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula>. First, we consider a <inline-formula><tex-math>$k$</tex-math></inline-formula>-sparse <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> and introduce the thresholded Wirtinger flow (TWF) algorithm that does not require the sparsity level <inline-formula><tex-math>$k$</tex-math></inline-formula>. TWF comprises two steps: the spectral initialization that identifies a point sufficiently close to <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> (up to a sign flip) when <inline-formula><tex-math>$m=O(k^{2}log n)$</tex-math></inline-formula>, and the thresholded gradient descent which, when provided a good initialization, produces a sequence linearly converging to <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> with <inline-formula><tex-math>$m=O(klog n)$</tex-math></inline-formula> measurements. Second, we explore the generative prior, assuming that <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> lies in the range of an <inline-formula><tex-math>$L$</tex-math></inline-formula>-Lipschitz continuous generative model with <inline-formula><tex-math>$k$</tex-math></inline-formula>-dimensional inputs in an <inline-formula><tex-math>$ell_{2}$</tex-math></inline-formula>-ball of radius <inline-formula><tex-math>$r$</tex-math></inline-formula>. With an estimate correlated with the signal, we develop the projected gradient descent (PGD) algorithm that also comprises two steps: the projected power method that provides an initial vector with <inline-formula><tex-math>$Obig{(}sqrt{klog(L)/m}big{)}$</tex-math></inline-formula> <inline-formula><tex-math>$ell_{2}$</tex-math></inline-formula>-error given <inline-formula><tex-math>$m=O(klog(Lnr))$</tex-math></inline-formula> measurements, and the projected gradient descent that refines the <inline-formula><tex-math>$ell_{2}$</tex-math></inline-formula>-error to <inline-formula><tex-math>$O(delta)$</tex-math></inline-formula> at a geometric rate when <inline-formula><tex-math>$m=O(klogfrac{Lrn}{delta^{2}})$</tex-math></inline-formula>. Experimental results corroborate our theoretical findings and show that: (i) our approach for the sparse case nota
{"title":"Solving Quadratic Systems With Full-Rank Matrices Using Sparse or Generative Priors","authors":"Junren Chen;Michael K. Ng;Zhaoqiang Liu","doi":"10.1109/TSP.2024.3522179","DOIUrl":"10.1109/TSP.2024.3522179","url":null,"abstract":"The problem of recovering a signal <inline-formula><tex-math>$boldsymbol{x}inmathbb{R}^{n}$</tex-math></inline-formula> from a quadratic system <inline-formula><tex-math>${y_{i}=boldsymbol{x}^{top}boldsymbol{A}_{i}boldsymbol{x}, i=1,ldots,m}$</tex-math></inline-formula> with full-rank matrices <inline-formula><tex-math>$boldsymbol{A}_{i}$</tex-math></inline-formula> frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. With i.i.d. standard Gaussian matrices <inline-formula><tex-math>$boldsymbol{A}_{i}$</tex-math></inline-formula>, this paper addresses the high-dimensional case where <inline-formula><tex-math>$mll n$</tex-math></inline-formula> by incorporating prior knowledge of <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula>. First, we consider a <inline-formula><tex-math>$k$</tex-math></inline-formula>-sparse <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> and introduce the thresholded Wirtinger flow (TWF) algorithm that does not require the sparsity level <inline-formula><tex-math>$k$</tex-math></inline-formula>. TWF comprises two steps: the spectral initialization that identifies a point sufficiently close to <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> (up to a sign flip) when <inline-formula><tex-math>$m=O(k^{2}log n)$</tex-math></inline-formula>, and the thresholded gradient descent which, when provided a good initialization, produces a sequence linearly converging to <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> with <inline-formula><tex-math>$m=O(klog n)$</tex-math></inline-formula> measurements. Second, we explore the generative prior, assuming that <inline-formula><tex-math>$boldsymbol{x}$</tex-math></inline-formula> lies in the range of an <inline-formula><tex-math>$L$</tex-math></inline-formula>-Lipschitz continuous generative model with <inline-formula><tex-math>$k$</tex-math></inline-formula>-dimensional inputs in an <inline-formula><tex-math>$ell_{2}$</tex-math></inline-formula>-ball of radius <inline-formula><tex-math>$r$</tex-math></inline-formula>. With an estimate correlated with the signal, we develop the projected gradient descent (PGD) algorithm that also comprises two steps: the projected power method that provides an initial vector with <inline-formula><tex-math>$Obig{(}sqrt{klog(L)/m}big{)}$</tex-math></inline-formula> <inline-formula><tex-math>$ell_{2}$</tex-math></inline-formula>-error given <inline-formula><tex-math>$m=O(klog(Lnr))$</tex-math></inline-formula> measurements, and the projected gradient descent that refines the <inline-formula><tex-math>$ell_{2}$</tex-math></inline-formula>-error to <inline-formula><tex-math>$O(delta)$</tex-math></inline-formula> at a geometric rate when <inline-formula><tex-math>$m=O(klogfrac{Lrn}{delta^{2}})$</tex-math></inline-formula>. Experimental results corroborate our theoretical findings and show that: (i) our approach for the sparse case nota","PeriodicalId":13330,"journal":{"name":"IEEE Transactions on Signal Processing","volume":"73 ","pages":"477-492"},"PeriodicalIF":4.6,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142961228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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