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Matrix completion focuses on recovering missing or incomplete information in matrices. This problem arises in various applications, including image processing and network analysis. Previous research proposed Poisson matrix completion for count data with noise that follows a Poisson distribution, which assumes that the mean and variance are equal. Since overdispersed count data, whose variance is greater than the mean, is more likely to occur in realistic settings, we assume that the noise follows the negative binomial (NB) distribution, which can be more general than the Poisson distribution. In this paper, we introduce NB matrix completion by proposing a nuclear-norm regularized model that can be solved by proximal gradient descent. In our experiments, we demonstrate that the NB model outperforms Poisson matrix completion in various noise and missing data settings on real data., Comment: 6 pages, Accepted by the IEEE International Workshop on Machine Learning for Signal Processing (MLSP)

Distributed learning and adaptation have received significant interest and found wide-ranging applications in machine learning and signal processing. While various approaches, such as shared-memory optimization, multi-task learning, and consensus-based learning (e.g., federated learning and learning over graphs), focus on optimizing either local rewards or a global reward, there remains a need for further exploration of their interconnections. This paper specifically focuses on a scenario where agents collaborate towards a common task (i.e., optimizing a global reward equal to aggregated local rewards) while effectively having distinct individual tasks (i.e., optimizing individual local parameters in a local reward). Each agent's actions can potentially impact other agents' performance through interactions. Notably, each agent has access to only its local zeroth-order oracle (i.e., reward function value) and shares scalar values, rather than gradient vectors, with other agents, leading to communication bandwidth efficiency and agent privacy. Agents employ zeroth-order optimization to update their parameters, and the asynchronous message-passing between them is subject to bounded but possibly random communication delays. This paper presents theoretical convergence analyses and establishes a convergence rate for nonconvex problems. Furthermore, it addresses the relevant use-case of deep learning-based resource allocation in communication networks and conducts numerical experiments in which agents, acting as transmitters, collaboratively train their individual policies to maximize a global reward, e.g., a sum of data rates.

This paper studies the robust Hankel recovery problem, which simultaneously removes the sparse outliers and fulfills missing entries from the partial observation. We propose a novel non-convex algorithm, coined Hankel Structured Newton-Like Descent (HSNLD), to tackle the robust Hankel recovery problem. HSNLD is highly efficient with linear convergence, and its convergence rate is independent of the condition number of the underlying Hankel matrix. The recovery guarantee has been established under some mild conditions. Numerical experiments on both synthetic and real datasets show the superior performance of HSNLD against state-of-the-art algorithms.

Recently, a tensor-on-tensor (ToT) regression model has been proposed to generalize tensor recovery, encompassing scenarios like scalar-on-tensor regression and tensor-on-vector regression. However, the exponential growth in tensor complexity poses challenges for storage and computation in ToT regression. To overcome this hurdle, tensor decompositions have been introduced, with the tensor train (TT)-based ToT model proving efficient in practice due to reduced memory requirements, enhanced computational efficiency, and decreased sampling complexity. Despite these practical benefits, a disparity exists between theoretical analysis and real-world performance. In this paper, we delve into the theoretical and algorithmic aspects of the TT-based ToT regression model. Assuming the regression operator satisfies the restricted isometry property (RIP), we conduct an error analysis for the solution to a constrained least-squares optimization problem. This analysis includes upper error bound and minimax lower bound, revealing that such error bounds polynomially depend on the order $N+M$. To efficiently find solutions meeting such error bounds, we propose two optimization algorithms: the iterative hard thresholding (IHT) algorithm (employing gradient descent with TT-singular value decomposition (TT-SVD)) and the factorization approach using the Riemannian gradient descent (RGD) algorithm. When RIP is satisfied, spectral initialization facilitates proper initialization, and we establish the linear convergence rate of both IHT and RGD., Comment: arXiv admin note: text overlap with arXiv:2401.02592

Directly parameterizing and learning gradients of functions has widespread significance, with specific applications in optimization, generative modeling, and optimal transport. This paper introduces gradient networks (GradNets): novel neural network architectures that parameterize gradients of various function classes. GradNets exhibit specialized architectural constraints that ensure correspondence to gradient functions. We provide a comprehensive GradNet design framework that includes methods for transforming GradNets into monotone gradient networks (mGradNets), which are guaranteed to represent gradients of convex functions. We establish the approximation capabilities of the proposed GradNet and mGradNet. Our results demonstrate that these networks universally approximate the gradients of (convex) functions. Furthermore, these networks can be customized to correspond to specific spaces of (monotone) gradient functions, including gradients of transformed sums of (convex) ridge functions. Our analysis leads to two distinct GradNet architectures, GradNet-C and GradNet-M, and we describe the corresponding monotone versions, mGradNet-C and mGradNet-M. Our empirical results show that these architectures offer efficient parameterizations and outperform popular methods in gradient field learning tasks.

The sufficiently scattered condition (SSC) is a key condition in the study of identifiability of various matrix factorization problems, including nonnegative, minimum-volume, symmetric, simplex-structured, and polytopic matrix factorizations. The SSC allows one to guarantee that the computed matrix factorization is unique/identifiable, up to trivial ambiguities. However, this condition is NP-hard to check in general. In this paper, we show that it can however be checked in a reasonable amount of time in realistic scenarios, when the factorization rank is not too large. This is achieved by formulating the problem as a non-convex quadratic optimization problem over a bounded set. We use the global non-convex optimization software Gurobi, and showcase the usefulness of this code on synthetic data sets and on real-world hyperspectral images., Comment: 14 pages, code available from https://gitlab.com/ngillis/check-ssc

In this paper, we provide the first convergence guarantee for the factorization approach. Specifically, to avoid the scaling ambiguity and to facilitate theoretical analysis, we optimize over the so-called left-orthogonal TT format which enforces orthonormality among most of the factors. To ensure the orthonormal structure, we utilize the Riemannian gradient descent (RGD) for optimizing those factors over the Stiefel manifold. We first delve into the TT factorization problem and establish the local linear convergence of RGD. Notably, the rate of convergence only experiences a linear decline as the tensor order increases. We then study the sensing problem that aims to recover a TT format tensor from linear measurements. Assuming the sensing operator satisfies the restricted isometry property (RIP), we show that with a proper initialization, which could be obtained through spectral initialization, RGD also converges to the ground-truth tensor at a linear rate. Furthermore, we expand our analysis to encompass scenarios involving Gaussian noise in the measurements. We prove that RGD can reliably recover the ground truth at a linear rate, with the recovery error exhibiting only polynomial growth in relation to the tensor order. We conduct various experiments to validate our theoretical findings.

The generalized minimax concave (GMC) penalty is a nonconvex sparse regularizer which can preserve the overall-convexity of the regularized least-squares problem. In this paper, we focus on a significant instance of the GMC model termed scaled GMC (sGMC), and present various notable findings on its solution-set geometry and regularization path. Our investigation indicates that while the sGMC penalty is a nonconvex extension of the LASSO penalty (i.e., the $\ell_1$-norm), the sGMC model preserves many celebrated properties of the LASSO model, hence can serve as a less biased surrogate of LASSO without losing its advantages. Specifically, for a fixed regularization parameter $\lambda$, we show that the solution-set geometry, solution uniqueness and sparseness of the sGMC model can be characterized in a similar elegant way to the LASSO model (see, e.g., Osborne et al. 2000, R. J. Tibshirani 2013). For a varying $\lambda$, we prove that the sGMC solution set is a continuous polytope-valued mapping of $\lambda$. Most noticeably, our study indicates that similar to LASSO, the minimum $\ell_2$-norm regularization path of the sGMC model is continuous and piecewise linear in $\lambda$. Based on these theoretical results, an efficient regularization path algorithm is proposed for the sGMC model, extending the well-known least angle regression (LARS) algorithm for LASSO. We prove the correctness and finite termination of the proposed algorithm under a mild assumption, and confirm its correctness-in-general-situation, efficiency, and practical utility through numerical experiments. Many results in this study also contribute to the theoretical research of LASSO., Comment: 53 pages, 10 figures. Submitted to journal

We consider a discrete-time model of continuous-time distributed optimization over dynamic directed-graphs (digraphs) with applications to distributed learning. Our optimization algorithm works over general strongly connected dynamic networks under switching topologies, e.g., in mobile multi-agent systems and volatile networks due to link failures. Compared to many existing lines of work, there is no need for bi-stochastic weight designs on the links. The existing literature mostly needs the link weights to be stochastic using specific weight-design algorithms needed both at the initialization and at all times when the topology of the network changes. This paper eliminates the need for such algorithms and paves the way for distributed optimization over time-varying digraphs. We derive the bound on the gradient-tracking step-size and discrete time-step for convergence and prove dynamic stability using arguments from consensus algorithms, matrix perturbation theory, and Lyapunov theory. This work, particularly, is an improvement over existing stochastic-weight undirected networks in case of link removal or packet drops. This is because the existing literature may need to rerun time-consuming and computationally complex algorithms for stochastic design, while the proposed strategy works as long as the underlying network is weight-symmetric and balanced. The proposed optimization framework finds applications to distributed classification and learning.

Deep learning (DL) based resource allocation (RA) has recently gained significant attention due to its performance efficiency. However, most related studies assume an ideal case where the number of users and their utility demands, e.g., data rate constraints, are fixed, and the designed DL-based RA scheme exploits a policy trained only for these fixed parameters. Consequently, computationally complex policy retraining is required whenever these parameters change. In this paper, we introduce a DL-based resource allocator (ALCOR) that allows users to adjust their utility demands freely, such as based on their application layer requirements. ALCOR employs deep neural networks (DNNs) as the policy in a time-sharing problem. The underlying optimization algorithm iteratively optimizes the on-off status of users to satisfy their utility demands in expectation. The policy performs unconstrained RA (URA)--RA without considering user utility demands--among active users to maximize the sum utility (SU) at each time instant. Depending on the chosen URA scheme, ALCOR can perform RA in either a centralized or distributed scenario. Derived convergence analyses provide guarantees for ALCOR's convergence, and numerical experiments corroborate its effectiveness.

We consider the problem of reconstructing rank-one matrices from random linear measurements, a task that appears in a variety of problems in signal processing, statistics, and machine learning. In this paper, we focus on the Alternating Least Squares (ALS) method. While this algorithm has been studied in a number of previous works, most of them only show convergence from an initialization close to the true solution and thus require a carefully designed initialization scheme. However, random initialization has often been preferred by practitioners as it is model-agnostic. In this paper, we show that ALS with random initialization converges to the true solution with $\varepsilon$-accuracy in $O(\log n + \log (1/\varepsilon)) $ iterations using only a near-optimal amount of samples, where we assume the measurement matrices to be i.i.d. Gaussian and where by $n$ we denote the ambient dimension. Key to our proof is the observation that the trajectory of the ALS iterates only depends very mildly on certain entries of the random measurement matrices. Numerical experiments corroborate our theoretical predictions., Comment: 51 pages, 3 figures

Gaussian processes (GPs) stand as crucial tools in machine learning and signal processing, with their effectiveness hinging on kernel design and hyper-parameter optimization. This paper presents a novel GP linear multiple kernel (LMK) and a generic sparsity-aware distributed learning framework to optimize the hyper-parameters. The newly proposed grid spectral mixture (GSM) kernel is tailored for multi-dimensional data, effectively reducing the number of hyper-parameters while maintaining good approximation capabilities. We further demonstrate that the associated hyper-parameter optimization of this kernel yields sparse solutions. To exploit the inherent sparsity property of the solutions, we introduce the Sparse LInear Multiple Kernel Learning (SLIM-KL) framework. The framework incorporates a quantized alternating direction method of multipliers (ADMM) scheme for collaborative learning among multiple agents, where the local optimization problem is solved using a distributed successive convex approximation (DSCA) algorithm. SLIM-KL effectively manages large-scale hyper-parameter optimization for the proposed kernel, simultaneously ensuring data privacy and minimizing communication costs. Theoretical analysis establishes convergence guarantees for the learning framework, while experiments on diverse datasets demonstrate the superior prediction performance and efficiency of our proposed methods.

We present ConvexECG, an explainable and resource-efficient method for reconstructing six-lead electrocardiograms (ECG) from single-lead data, aimed at advancing personalized and continuous cardiac monitoring. ConvexECG leverages a convex reformulation of a two-layer ReLU neural network, enabling the potential for efficient training and deployment in resource constrained environments, while also having deterministic and explainable behavior. Using data from 25 patients, we demonstrate that ConvexECG achieves accuracy comparable to larger neural networks while significantly reducing computational overhead, highlighting its potential for real-time, low-resource monitoring applications.

This paper is about a deep learning approach for linear and nonlinear filtering. The idea is to train a neural network with Monte Carlo samples generated from a nominal dynamic model. Then the network weights are applied to Monte Carlo samples from an actual dynamic model. A main focus of this paper is on the deep filters with three major neural network architectures (DNN, CNN, RNN). Our deep filter compares favorably to the traditional Kalman filter in linear cases and outperform the extended Kalman filter in nonlinear cases. Then a switching model with jumps is studied to show the adaptiveness and power of our deep filtering. Among the three major NNs, the CNN outperform the others on average. while the RNN does not seem to be suitable for the filtering problem. One advantage of the deep filter is its robustness when the nominal model and actual model differ. The other advantage of deep filtering is real data can be used directly to train the deep neutral network. Therefore, model calibration can be by-passed all together., Comment: arXiv admin note: text overlap with arXiv:2008.03878

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature.

Sparse signal recovery is one of the most fundamental problems in various applications, including medical imaging and remote sensing. Many greedy algorithms based on the family of hard thresholding operators have been developed to solve the sparse signal recovery problem. More recently, Natural Thresholding (NT) has been proposed with improved computational efficiency. This paper proposes and discusses convergence guarantees for stochastic natural thresholding algorithms by extending the NT from the deterministic version with linear measurements to the stochastic version with a general objective function. We also conduct various numerical experiments on linear and nonlinear measurements to demonstrate the performance of StoNT.

The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data., Comment: 29 pages, 2 figures

Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. As opposed to binary matrix factorization which uses standard arithmetic, BMF uses the Boolean OR and Boolean AND operations to perform matrix products, which leads to lower reconstruction errors. BMF is an NP-hard problem. In this paper, we first propose an alternating optimization (AO) strategy that solves the subproblem in one factor matrix in BMF using an integer program (IP). We also provide two ways to initialize the factors within AO. Then, we show how several solutions of BMF can be combined optimally using another IP. This allows us to come up with a new algorithm: it generates several solutions using AO and then combines them in an optimal way. Experiments show that our algorithms (available on gitlab) outperform the state of the art on medium-scale problems., Comment: 6 pages, submitted to the MLSP workshop

In this paper, we study the following nonlinear matrix decomposition (NMD) problem: given a sparse nonnegative matrix $X$, find a low-rank matrix $\Theta$ such that $X \approx f(\Theta)$, where $f$ is an element-wise nonlinear function. We focus on the case where $f(\cdot) = \max(0, \cdot)$, the rectified unit (ReLU) non-linear activation. We refer to the corresponding problem as ReLU-NMD. We first provide a brief overview of the existing approaches that were developed to tackle ReLU-NMD. Then we introduce two new algorithms: (1) aggressive accelerated NMD (A-NMD) which uses an adaptive Nesterov extrapolation to accelerate an existing algorithm, and (2) three-block NMD (3B-NMD) which parametrizes $\Theta = WH$ and leads to a significant reduction in the computational cost. We also propose an effective initialization strategy based on the nuclear norm as a proxy for the rank function. We illustrate the effectiveness of the proposed algorithms (available on gitlab) on synthetic and real-world data sets., Comment: 6 pages, submitted to the MLSP workshop

Graph learning from signals is a core task in Graph Signal Processing (GSP). One of the most commonly used models to learn graphs from stationary signals is SpecT. However, its practical formulation rSpecT is known to be sensitive to hyperparameter selection and, even worse, to suffer from infeasibility. In this paper, we give the first condition that guarantees the infeasibility of rSpecT and design a novel model (LogSpecT) and its practical formulation (rLogSpecT) to overcome this issue. Contrary to rSpecT, the novel practical model rLogSpecT is always feasible. Furthermore, we provide recovery guarantees of rLogSpecT, which are derived from modern optimization tools related to epi-convergence. These tools could be of independent interest and significant for various learning problems. To demonstrate the advantages of rLogSpecT in practice, a highly efficient algorithm based on the linearized alternating direction method of multipliers (L-ADMM) is proposed. The subproblems of L-ADMM admit closed-form solutions and the convergence is guaranteed. Extensive numerical results on both synthetic and real networks corroborate the stability and superiority of our proposed methods, underscoring their potential for various graph learning applications.

This paper considers the robust phase retrieval problem, which can be cast as a nonsmooth and nonconvex optimization problem. We propose a new inexact proximal linear algorithm with the subproblem being solved inexactly. Our contributions are two adaptive stopping criteria for the subproblem. The convergence behavior of the proposed methods is analyzed. Through experiments on both synthetic and real datasets, we demonstrate that our methods are much more efficient than existing methods, such as the original proximal linear algorithm and the subgradient method., Comment: 23 pages

Principal Component Analysis (PCA) is a fundamental data preprocessing tool in the world of machine learning. While PCA is often thought of as a dimensionality reduction method, the purpose of PCA is actually two-fold: dimension reduction and uncorrelated feature learning. Furthermore, the enormity of the dimensions and sample size in the modern day datasets have rendered the centralized PCA solutions unusable. In that vein, this paper reconsiders the problem of PCA when data samples are distributed across nodes in an arbitrarily connected network. While a few solutions for distributed PCA exist, those either overlook the uncorrelated feature learning aspect of the PCA, tend to have high communication overhead that makes them inefficient and/or lack `exact' or `global' convergence guarantees. To overcome these aforementioned issues, this paper proposes a distributed PCA algorithm termed FAST-PCA (Fast and exAct diSTributed PCA). The proposed algorithm is efficient in terms of communication and is proven to converge linearly and exactly to the principal components, leading to dimension reduction as well as uncorrelated features. The claims are further supported by experimental results., Comment: 16 pages (two-column version); substantially revised version, including expanded comparisons with other works

In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behavior of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.

This paper examines the separation of wireless communication and radar signals, thereby guaranteeing cohabitation and acting as a panacea to spectrum sensing. First, considering that the channel impulse response was known by the receivers (communication and radar), we showed that the optimizing beamforming weights mitigate the interference caused by signals and improve the physical layer security (PLS) of the system. Furthermore, when the channel responses were unknown, we designed an interference filter as a low-complex noise and interference cancellation autoencoder. By mitigating the interference on the legitimate users, the PLS was guaranteed. Results showed that even for a low signal-to-noise ratio, the autoencoder produces low root-mean-square error (RMSE) values., Comment: 8 pages, 8 figures

Due to the significant computational challenge of training large-scale graph neural networks (GNNs), various sparse learning techniques have been exploited to reduce memory and storage costs. Examples include \textit{graph sparsification} that samples a subgraph to reduce the amount of data aggregation and \textit{model sparsification} that prunes the neural network to reduce the number of trainable weights. Despite the empirical successes in reducing the training cost while maintaining the test accuracy, the theoretical generalization analysis of sparse learning for GNNs remains elusive. To the best of our knowledge, this paper provides the first theoretical characterization of joint edge-model sparse learning from the perspective of sample complexity and convergence rate in achieving zero generalization error. It proves analytically that both sampling important nodes and pruning neurons with the lowest-magnitude can reduce the sample complexity and improve convergence without compromising the test accuracy. Although the analysis is centered on two-layer GNNs with structural constraints on data, the insights are applicable to more general setups and justified by both synthetic and practical citation datasets.

Principal Subspace Analysis (PSA) -- and its sibling, Principal Component Analysis (PCA) -- is one of the most popular approaches for dimensionality reduction in signal processing and machine learning. But centralized PSA/PCA solutions are fast becoming irrelevant in the modern era of big data, in which the number of samples and/or the dimensionality of samples often exceed the storage and/or computational capabilities of individual machines. This has led to the study of distributed PSA/PCA solutions, in which the data are partitioned across multiple machines and an estimate of the principal subspace is obtained through collaboration among the machines. It is in this vein that this paper revisits the problem of distributed PSA/PCA under the general framework of an arbitrarily connected network of machines that lacks a central server. The main contributions of the paper in this regard are threefold. First, two algorithms are proposed in the paper that can be used for distributed PSA/PCA, with one in the case of data partitioned across samples and the other in the case of data partitioned across (raw) features. Second, in the case of sample-wise partitioned data, the proposed algorithm and a variant of it are analyzed, and their convergence to the true subspace at linear rates is established. Third, extensive experiments on both synthetic and real-world data are carried out to validate the usefulness of the proposed algorithms. In particular, in the case of sample-wise partitioned data, an MPI-based distributed implementation is carried out to study the interplay between network topology and communications cost as well as to study the effects of straggler machines on the proposed algorithms., Comment: 16 pages; Final accepted version; To appear in IEEE Transactions on Signal and Information Processing Over Networks

Dictionary learning is a widely used unsupervised learning method in signal processing and machine learning. Most existing works of dictionary learning are in an offline manner. There are mainly two offline ways for dictionary learning. One is to do an alternative optimization of both the dictionary and the sparse code; the other way is to optimize the dictionary by restricting it over the orthogonal group. The latter one is called orthogonal dictionary learning which has a lower complexity implementation, hence, it is more favorable for lowcost devices. However, existing schemes on orthogonal dictionary learning only work with batch data and can not be implemented online, which is not applicable for real-time applications. This paper proposes a novel online orthogonal dictionary scheme to dynamically learn the dictionary from streaming data without storing the historical data. The proposed scheme includes a novel problem formulation and an efficient online algorithm design with convergence analysis. In the problem formulation, we relax the orthogonal constraint to enable an efficient online algorithm. In the algorithm design, we propose a new Frank-Wolfe-based online algorithm with a convergence rate of O(ln t/t^(1/4)). The convergence rate in terms of key system parameters is also derived. Experiments with synthetic data and real-world sensor readings demonstrate the effectiveness and efficiency of the proposed online orthogonal dictionary learning scheme., Comment: This paper has been accepted by IEEE Transactions on Neural Networks and Learning Systems as a regular paper

In this paper, we propose an algorithmic framework, dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous ADMM that use specific surrogate functions in the MM step, but also lead to new efficient ADMM schemes. To the best of our knowledge, in the nonconvex nonsmooth setting, ADMM used in combination with the MM principle to update each block of variables, and ADMM combined with \emph{inertial terms for the primal variables} have not been studied in the literature. Under standard assumptions, we prove the subsequential convergence and global convergence for the generated sequence of iterates. We illustrate the effectiveness of iADMM on a class of nonconvex low-rank representation problems., Comment: 35 pages, several parts of the paper clarified, additional experiments on a regularized NMF problem

Quantum computing, leveraging principles of quantum mechanics, represents a transformative approach in computational methodologies, offering significant enhancements over traditional classical systems. This study tackles the complex and computationally demanding task of simulating power system transients through solving differential algebraic equations (DAEs). We introduce two novel Quantum Neural Networks (QNNs): the Sinusoidal-Friendly QNN and the Polynomial-Friendly QNN, proposing them as effective alternatives to conventional simulation techniques. Our application of these QNNs successfully simulates two small power systems, demonstrating their potential to achieve good accuracy. We further explore various configurations, including time intervals, training points, and the selection of classical optimizers, to optimize the solving of DAEs using QNNs. This research not only marks a pioneering effort in applying quantum computing to power system simulations but also expands the potential of quantum technologies in addressing intricate engineering challenges., Comment: 10 pages, 11 figures

In the problem of quickest change detection (QCD), a change occurs at some unknown time in the distribution of a sequence of independent observations. This work studies a QCD problem where the change is either a bad change, which we aim to detect, or a confusing change, which is not of our interest. Our objective is to detect a bad change as quickly as possible while avoiding raising a false alarm for pre-change or a confusing change. We identify a specific set of pre-change, bad change, and confusing change distributions that pose challenges beyond the capabilities of standard Cumulative Sum (CuSum) procedures. Proposing novel CuSum-based detection procedures, S-CuSum and J-CuSum, leveraging two CuSum statistics, we offer solutions applicable across all kinds of pre-change, bad change, and confusing change distributions. For both S-CuSum and J-CuSum, we provide analytical performance guarantees and validate them by numerical results. Furthermore, both procedures are computationally efficient as they only require simple recursive updates.

Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points $[\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n] = \mathbf{X} \in \mathbb{R}^{d \times n}$ and a vector $\mathbf{y} \in \mathbb{R}^d$, the goal is to find coefficients $\mathbf{w} \in \mathbb{R}^n$ so that $\mathbf{X} \mathbf{w} \approx \mathbf{y}$, subject to some desired structure on $\mathbf{w}$. In this work we seek $\mathbf{w}$ that forms a local reconstruction of $\mathbf{y}$ by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of $\mathbf{X}$ that are close to $\mathbf{y}$ when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of $\mathbf{X}$ have a unique Delaunay triangulation, the optimal coefficients' number of non-zero entries is upper bounded by $d+1$, thereby providing local sparse solutions when $d \ll n$. Under the same condition we also show that for any $\mathbf{y}$ contained in the convex hull of $\mathbf{X}$ there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing $\mathbf{y}$. This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of $\mathbf{X}$. We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex., Comment: 26 pages, 8 figures

In this work, we approach the problem of multi-task submodular optimization with the perspective of local distributional robustness, within the neighborhood of a reference distribution which assigns an importance score to each task. We initially propose to introduce a regularization term which makes use of the relative entropy to the standard multi-task objective. We then demonstrate through duality that this novel formulation itself is equivalent to the maximization of a submodular function, which may be efficiently carried out through standard greedy selection methods. This approach bridges the existing gap in the optimization of performance-robustness trade-offs in multi-task subset selection. To numerically validate our theoretical results, we test the proposed method in two different setting, one involving the selection of satellites in low Earth orbit constellations in the context of a sensor selection problem, and the other involving an image summarization task using neural networks. Our method is compared with two other algorithms focused on optimizing the performance of the worst-case task, and on directly optimizing the performance on the reference distribution itself. We conclude that our novel formulation produces a solution that is locally distributional robust, and computationally inexpensive., Comment: 35 pages, 7 figures. A preliminary version of this article was presented at the 2023 Allerton Conference on Communication, Control, and Computing. This version was submitted to IEEE Transactions on Signal Processing

Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as the product of two smaller nonnegative matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, and with penalty terms in the objective function. By disjoint, we mean that each variable appears in at most one equality constraint. Our MU satisfy the set of constraints after each update of the variables during the optimization process, while guaranteeing that the objective function decreases monotonically. We showcase this framework on three NMF models, and show that it competes favorably the state of the art: (1)~$\beta$-NMF with sum-to-one constraints on the columns of $H$, (2) minimum-volume $\beta$-NMF with sum-to-one constraints on the columns of $W$, and (3) sparse $\beta$-NMF with $\ell_2$-norm constraints on the columns of $W$., Comment: We have corrected an error in Table 1 and in Section 2.3: the updates are not correct for $\beta \in (1,2)$; see the paper for more details. We have highlighted the changes in red color

In this paper, we introduce TITAN, a novel inerTIal block majorizaTion minimizAtioN framework for non-smooth non-convex optimization problems. To the best of our knowledge, TITAN is the first framework of block-coordinate update method that relies on the majorization-minimization framework while embedding inertial force to each step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavy-ball and Nesterov-type accelerations for block proximal gradient methods as special cases. By choosing various surrogate functions, such as proximal, Lipschitz gradient, Bregman, quadratic, and composite surrogate functions, and by varying the extrapolation operator, TITAN produces a rich set of inertial block-coordinate update methods. We study sub-sequential convergence as well as global convergence for the generated sequence of TITAN. We illustrate the effectiveness of TITAN on two important machine learning problems, namely sparse non-negative matrix factorization and matrix completion., Comment: 42 pages, we have clarified several aspects of the paper

Federated learning (FL) has emerged as an instance of distributed machine learning paradigm that avoids the transmission of data generated on the users' side. Although data are not transmitted, edge devices have to deal with limited communication bandwidths, data heterogeneity, and straggler effects due to the limited computational resources of users' devices. A prominent approach to overcome such difficulties is FedADMM, which is based on the classical two-operator consensus alternating direction method of multipliers (ADMM). The common assumption of FL algorithms, including FedADMM, is that they learn a global model using data only on the users' side and not on the edge server. However, in edge learning, the server is expected to be near the base station and have direct access to rich datasets. In this paper, we argue that leveraging the rich data on the edge server is much more beneficial than utilizing only user datasets. Specifically, we show that the mere application of FL with an additional virtual user node representing the data on the edge server is inefficient. We propose FedTOP-ADMM, which generalizes FedADMM and is based on a three-operator ADMM-type technique that exploits a smooth cost function on the edge server to learn a global model parallel to the edge devices. Our numerical experiments indicate that FedTOP-ADMM has substantial gain up to 33\% in communication efficiency to reach a desired test accuracy with respect to FedADMM, including a virtual user on the edge server., Comment: accepted to IEEE Journal of Selected Topics in Signal Processing, 2022

This paper focuses on complete dictionary learning problem, where the goal is to reparametrize a set of given signals as linear combinations of atoms from a learned dictionary. There are two main challenges faced by theoretical and practical studies of dictionary learning: the lack of theoretical guarantees for practically-used heuristic algorithms, and their poor scalability when dealing with huge-scale datasets. Towards addressing these issues, we show that when the dictionary to be learned is orthogonal, that an alternating minimization method directly applied to the nonconvex and discrete formulation of the problem exactly recovers the ground truth. For the huge-scale, potentially online setting, we propose a minibatch version of our algorithm, which can provably learn a complete dictionary from a huge-scale dataset with minimal sample complexity, linear sparsity level, and linear convergence rate, thereby negating the need for any convex relaxation for the problem. Our numerical experiments showcase the superiority of our method compared with the existing techniques when applied to tasks on real data.

Convergence (virtual) bidding is an important part of two-settlement electric power markets as it can effectively reduce discrepancies between the day-ahead and real-time markets. Consequently, there is extensive research into the bidding strategies of virtual participants aiming to obtain optimal bids to submit to the day-ahead market. In this paper, we introduce a price-based general stochastic optimization framework to obtain optimal convergence bid curves. Within this framework, we develop a computationally tractable linear programming-based optimization model, which produces bid prices and volumes simultaneously. We also show that different approximations and simplifications in the general model lead naturally to state-of-the-art convergence bidding approaches, such as self-scheduling and opportunistic approaches. Our general framework also provides a straightforward way to compare the performance of these models, which is demonstrated by numerical experiments on the California (CAISO) market.

Multigraph matching is a recent variant of the graph matching problem. In this framework, the optimization procedure considers several graphs and enforces the consistency of the matches along the graphs. This constraint can be formalized as a cycle consistency across the pairwise permutation matrices, which implies the definition of a universe of vertex~\citep{pachauri2013solving}. The label of each vertex is encoded by a sparse vector and the dimension of this space corresponds to the rank of the bulk permutation matrix, the matrix built from the aggregation of all the pairwise permutation matrices. The matching problem can then be formulated as a non-convex quadratic optimization problem (QAP) under constraints imposed on the rank and the permutations. In this paper, we introduce a novel kernelized multigraph matching technique that handles vectors of attributes on both the vertices and edges of the graphs, while maintaining a low memory usage. We solve the QAP problem using a projected power optimization approach and propose several projectors leading to improved stability of the results. We provide several experiments showing that our method is competitive against other unsupervised methods.

Imposing explicit constraints is relatively new but increasingly pressing in deep learning, stimulated by, e.g., trustworthy AI that performs robust optimization over complicated perturbation sets and scientific applications that need to respect physical laws and constraints. However, it can be hard to reliably solve constrained deep learning problems without optimization expertise. The existing deep learning frameworks do not admit constraints. General-purpose optimization packages can handle constraints but do not perform auto-differentiation and have trouble dealing with nonsmoothness. In this paper, we introduce a new software package called NCVX, whose initial release contains the solver PyGRANSO, a PyTorch-enabled general-purpose optimization package for constrained machine/deep learning problems, the first of its kind. NCVX inherits auto-differentiation, GPU acceleration, and tensor variables from PyTorch, and is built on freely available and widely used open-source frameworks. NCVX is available at https://ncvx.org, with detailed documentation and numerous examples from machine/deep learning and other fields., Comment: Accepted by the NeurIPS Workshop on Optimization for Machine Learning (OPT 2022). arXiv admin note: text overlap with arXiv:2111.13984

Empirical evaluation of deep learning models against adversarial attacks entails solving nontrivial constrained optimization problems. Popular algorithms for solving these constrained problems rely on projected gradient descent (PGD) and require careful tuning of multiple hyperparameters. Moreover, PGD can only handle $\ell_1$, $\ell_2$, and $\ell_\infty$ attack models due to the use of analytical projectors. In this paper, we introduce a novel algorithmic framework that blends a general-purpose constrained-optimization solver PyGRANSO, With Constraint-Folding (PWCF), to add reliability and generality to robustness evaluation. PWCF 1) finds good-quality solutions without the need of delicate hyperparameter tuning, and 2) can handle general attack models, e.g., general $\ell_p$ ($p \geq 0$) and perceptual attacks, which are inaccessible to PGD-based algorithms., Comment: 5 pages, 1 figure, 3 tables, accepted by the 14th International OPT Workshop on Optimization for Machine Learning, and submitted to the 2023 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2023)

The AC Optimal Power Flow (AC-OPF) is a key building block in many power system applications. It determines generator setpoints at minimal cost that meet the power demands while satisfying the underlying physical and operational constraints. It is non-convex and NP-hard, and computationally challenging for large-scale power systems. Motivated by the increased stochasticity in generation schedules and increasing penetration of renewable sources, this paper explores a deep learning approach to deliver highly efficient and accurate approximations to the AC-OPF. In particular, the paper proposes an integration of deep neural networks and Lagrangian duality to capture the physical and operational constraints. The resulting model, called OPF-DNN, is evaluated on real case studies from the French transmission system, with up to 3,400 buses and 4,500 lines. Computational results show that OPF-DNN produces highly accurate AC-OPF approximations whose costs are within 0.01% of optimality. OPF-DNN generates, in milliseconds, solutions that capture the problem constraints with high fidelity.

Emerging applications of machine learning in numerous areas involve continuous gathering of and learning from streams of data. Real-time incorporation of streaming data into the learned models is essential for improved inference in these applications. Further, these applications often involve data that are either inherently gathered at geographically distributed entities or that are intentionally distributed across multiple machines for memory, computational, and/or privacy reasons. Training of models in this distributed, streaming setting requires solving stochastic optimization problems in a collaborative manner over communication links between the physical entities. When the streaming data rate is high compared to the processing capabilities of compute nodes and/or the rate of the communications links, this poses a challenging question: how can one best leverage the incoming data for distributed training under constraints on computing capabilities and/or communications rate? A large body of research has emerged in recent decades to tackle this and related problems. This paper reviews recently developed methods that focus on large-scale distributed stochastic optimization in the compute- and bandwidth-limited regime, with an emphasis on convergence analysis that explicitly accounts for the mismatch between computation, communication and streaming rates. In particular, it focuses on methods that solve: (i) distributed stochastic convex problems, and (ii) distributed principal component analysis, which is a nonconvex problem with geometric structure that permits global convergence. For such methods, the paper discusses recent advances in terms of distributed algorithmic designs when faced with high-rate streaming data. Further, it reviews guarantees underlying these methods, which show there exist regimes in which systems can learn from distributed, streaming data at order-optimal rates., Comment: 45 pages, 9 figures; preprint of a journal paper published in Proceedings of the IEEE (Special Issue on Optimization for Data-driven Learning and Control)

This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-\`a-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of constant rank, we demonstrate that a principled convex program achieves near-optimal statistical accuracy, in terms of both the Euclidean loss and the $\ell_{\infty}$ loss. All of this happens even when nearly a constant fraction of observations are corrupted by outliers with arbitrary magnitudes. The key analysis idea lies in bridging the convex program in use and an auxiliary nonconvex optimization algorithm, and hence the title of this paper., Comment: accepted to the Annals of Statistics

This paper considers the problem of estimating the principal eigenvector of a covariance matrix from independent and identically distributed data samples in streaming settings. The streaming rate of data in many contemporary applications can be high enough that a single processor cannot finish an iteration of existing methods for eigenvector estimation before a new sample arrives. This paper formulates and analyzes a distributed variant of the classical Krasulina's method (D-Krasulina) that can keep up with the high streaming rate of data by distributing the computational load across multiple processing nodes. The analysis shows that---under appropriate conditions---D-Krasulina converges to the principal eigenvector in an order-wise optimal manner; i.e., after receiving $M$ samples across all nodes, its estimation error can be $O(1/M)$. In order to reduce the network communication overhead, the paper also develops and analyzes a mini-batch extension of D-Krasulina, which is termed DM-Krasulina. The analysis of DM-Krasulina shows that it can also achieve order-optimal estimation error rates under appropriate conditions, even when some samples have to be discarded within the network due to communication latency. Finally, experiments are performed over synthetic and real-world data to validate the convergence behaviors of D-Krasulina and DM-Krasulina in high-rate streaming settings., Comment: 37 pages, 11 figures; preprint of a journal submission

This paper presents for the first time, to our knowledge, a framework for verifying neural network behavior in power system applications. Up to this moment, neural networks have been applied in power systems as a black-box; this has presented a major barrier for their adoption in practice. Developing a rigorous framework based on mixed integer linear programming, our methods can determine the range of inputs that neural networks classify as safe or unsafe, and are able to systematically identify adversarial examples. Such methods have the potential to build the missing trust of power system operators on neural networks, and unlock a series of new applications in power systems. This paper presents the framework, methods to assess and improve neural network robustness in power systems, and addresses concerns related to scalability and accuracy. We demonstrate our methods on the IEEE 9-bus, 14-bus, and 162-bus systems, treating both N-1 security and small-signal stability., Comment: published in IEEE Transactions on Smart Grid (https://ieeexplore.ieee.org/abstract/document/9141308)

In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes $\mu$, the strategy is stable both in terms of mean-square error and average bit rate: by reducing $\mu$, it is possible to keep the estimation errors small (on the order of $\mu$) without increasing indefinitely the bit rate as $\mu\rightarrow 0$. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.

In a great number of tasks in science and engineering, the goal is to infer an unknown image from a small number of measurements collected from a known forward model describing certain sensing or imaging modality. Due to resource constraints, this task is often extremely ill-posed, which necessitates the adoption of expressive prior information to regularize the solution space. Score-based diffusion models, due to its impressive empirical success, have emerged as an appealing candidate of an expressive prior in image reconstruction. In order to accommodate diverse tasks at once, it is of great interest to develop efficient, consistent and robust algorithms that incorporate unconditional score functions of an image prior distribution in conjunction with flexible choices of forward models. This work develops an algorithmic framework for employing score-based diffusion models as an expressive data prior in general nonlinear inverse problems. Motivated by the plug-and-play framework in the imaging community, we introduce a diffusion plug-and-play method (DPnP) that alternatively calls two samplers, a proximal consistency sampler based solely on the likelihood function of the forward model, and a denoising diffusion sampler based solely on the score functions of the image prior. The key insight is that denoising under white Gaussian noise can be solved rigorously via both stochastic (i.e., DDPM-type) and deterministic (i.e., DDIM-type) samplers using the unconditional score functions. We establish both asymptotic and non-asymptotic performance guarantees of DPnP, and provide numerical experiments to illustrate its promise in solving both linear and nonlinear image reconstruction tasks. To the best of our knowledge, DPnP is the first provably-robust posterior sampling method for nonlinear inverse problems using unconditional diffusion priors.

With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.

Iteratively reweighted L1 (IRL1) algorithm is a common algorithm for solving sparse optimization problems with nonconvex and nonsmooth regularization. The development of its acceleration algorithm, often employing Nesterov acceleration, has sparked significant interest. Nevertheless, the convergence and complexity analysis of these acceleration algorithms consistently poses substantial challenges. Recently, Anderson acceleration has gained prominence owing to its exceptional performance for speeding up fixed-point iteration, with numerous recent studies applying it to gradient-based algorithms. Motivated by the powerful impact of Anderson acceleration, we propose an Anderson-accelerated IRL1 algorithm and establish its local linear convergence rate. We extend this convergence result, typically observed in smooth settings, to a nonsmooth scenario. Importantly, our theoretical results do not depend on the Kurdyka-Lojasiewicz condition, a necessary condition in existing Nesterov acceleration-based algorithms. Furthermore, to ensure global convergence, we introduce a globally convergent Anderson accelerated IRL1 algorithm by incorporating a classical nonmonotone line search condition. Experimental results indicate that our algorithm outperforms existing Nesterov acceleration-based algorithms.

Matrix sensing problems exhibit pervasive non-convexity, plaguing optimization with a proliferation of suboptimal spurious solutions. Avoiding convergence to these critical points poses a major challenge. This work provides new theoretical insights that help demystify the intricacies of the non-convex landscape. In this work, we prove that under certain conditions, critical points sufficiently distant from the ground truth matrix exhibit favorable geometry by being strict saddle points rather than troublesome local minima. Moreover, we introduce the notion of higher-order losses for the matrix sensing problem and show that the incorporation of such losses into the objective function amplifies the negative curvature around those distant critical points. This implies that increasing the complexity of the objective function via high-order losses accelerates the escape from such critical points and acts as a desirable alternative to increasing the complexity of the optimization problem via over-parametrization. By elucidating key characteristics of the non-convex optimization landscape, this work makes progress towards a comprehensive framework for tackling broader machine learning objectives plagued by non-convexity., Comment: Accepted by AISTATS 2024