Daniel P. Palomar

Ziping Zhao, Daniel P. Palomar

In this paper, the multiple-input multiple-output (MIMO) transmit beampattern matching problem is considered. The problem is formulated to approximate a desired transmit beampattern (i.e., an energy distribution in space and frequency) and to minimize the cross-correlation of signals reflected back to the array by considering different practical waveform constraints at the same time. Due to the nonconvexity of the objective function and the waveform constraints, the optimization problem is highly nonconvex. An efficient one-step method is proposed to solve this problem based on the majorization-minimization (MM) method. The performance of the proposed algorithms compared to the state-of-art algorithms is shown through numerical simulations.

Jiaxi Ying, José Vinícius de M. Cardoso, Daniel P. Palomar

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

Xiwen Wang, Jiaxi Ying, Daniel P. Palomar

This paper studies the problem of learning the large-scale Gaussian graphical models that are multivariate totally positive of order two ($\text{MTP}_2$). By introducing the concept of bridge, which commonly exists in large-scale sparse graphs, we show that the entire problem can be equivalently optimized through (1) several smaller-scaled sub-problems induced by a \emph{bridge-block decomposition} on the thresholded sample covariance graph and (2) a set of explicit solutions on entries corresponding to bridges. From practical aspect, this simple and provable discipline can be applied to break down a large problem into small tractable ones, leading to enormous reduction on the computational complexity and substantial improvements for all existing algorithms. The synthetic and real-world experiments demonstrate that our proposed method presents a significant speed-up compared to the state-of-the-art benchmarks.

Arnau Vilella, Jasin Machkour, Michael Muma et al.

Controlling the false discovery rate (FDR) in high-dimensional variable selection requires balancing rigorous error control with statistical power. Existing methods with provable guarantees are often overly conservative, creating a persistent gap between the realized false discovery proportion (FDP) and the target FDR level. We introduce a learning-augmented enhancement of the T-Rex Selector framework that narrows this gap. Our approach replaces the analytical FDP estimator with a neural network trained solely on diverse synthetic datasets, enabling a substantially tighter and more accurate approximation of the FDP. This refinement allows the procedure to operate much closer to the desired FDR level, thereby increasing discovery power while maintaining effective approximate control. Through extensive simulations and a challenging synthetic genome-wide association study (GWAS), we demonstrate that our method achieves superior detection of true variables compared to existing approaches.

Jasin Machkour, Daniel P. Palomar, Michael Muma

In high-dimensional data analysis, such as financial index tracking or biomedical applications, it is crucial to select the few relevant variables while maintaining control over the false discovery rate (FDR). In these applications, strong dependencies often exist among the variables (e.g., stock returns), which can undermine the FDR control property of existing methods like the model-X knockoff method or the T-Rex selector. To address this issue, we have expanded the T-Rex framework to accommodate overlapping groups of highly correlated variables. This is achieved by integrating a nearest neighbors penalization mechanism into the framework, which provably controls the FDR at the user-defined target level. A real-world example of sparse index tracking demonstrates the proposed method's ability to accurately track the S&P 500 index over the past 20 years based on a small number of stocks. An open-source implementation is provided within the R package TRexSelector on CRAN.

Jasin Machkour, Michael Muma, Daniel P. Palomar

We propose the Terminating-Random Experiments (T-Rex) selector, a fast variable selection method for high-dimensional data. The T-Rex selector controls a user-defined target false discovery rate (FDR) while maximizing the number of selected variables. This is achieved by fusing the solutions of multiple early terminated random experiments. The experiments are conducted on a combination of the original predictors and multiple sets of randomly generated dummy predictors. A finite sample proof based on martingale theory for the FDR control property is provided. Numerical simulations confirm that the FDR is controlled at the target level while allowing for high power. We prove that the dummies can be sampled from any univariate probability distribution with finite expectation and variance. The computational complexity of the proposed method is linear in the number of variables. The T-Rex selector outperforms state-of-the-art methods for FDR control in numerical experiments and on a simulated genome-wide association study (GWAS), while its sequential computation time is more than two orders of magnitude lower than that of the strongest benchmark methods. The open source R package TRexSelector containing the implementation of the T-Rex selector is available on CRAN.

Andrei Buciulea, Jiaxi Ying, Antonio G. Marques et al.

This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a graph-learning formulation that combines the strengths of graphical lasso with a more encompassing model. Specifically, we assume that the precision matrix can take any polynomial form of the sought graph, allowing for increased flexibility in modeling nodal relationships. Given the resulting complexity and nonconvexity of the resulting optimization problem, we (i) propose a low-complexity algorithm that alternates between estimating the graph and precision matrices, and (ii) characterize its convergence. We evaluate the performance of PGL through comprehensive numerical simulations using both synthetic and real data, demonstrating its superiority over several alternatives. Overall, this approach presents a significant advancement in graph learning and holds promise for various applications in graph-aware signal analysis and beyond.

Amirhossein Javaheri, Arash Amini, Farokh Marvasti et al.

Learning a graph from data is the key to taking advantage of graph signal processing tools. Most of the conventional algorithms for graph learning require complete data statistics, which might not be available in some scenarios. In this work, we aim to learn a graph from incomplete time-series observations. From another viewpoint, we consider the problem of semi-blind recovery of time-varying graph signals where the underlying graph model is unknown. We propose an algorithm based on the method of block successive upperbound minimization (BSUM), for simultaneous inference of the signal and the graph from incomplete data. Simulation results on synthetic and real time-series demonstrate the performance of the proposed method for graph learning and signal recovery.

Amirhossein Javaheri, Daniel P. Palomar

There are various approaches to graph learning for data clustering, incorporating different spectral and structural constraints through diverse graph structures. Some methods rely on bipartite graph models, where nodes are divided into two classes: centers and members. These models typically require access to data for the center nodes in addition to observations from the member nodes. However, such additional data may not always be available in many practical scenarios. Moreover, popular Gaussian models for graph learning have demonstrated limited effectiveness in modeling data with heavy-tailed distributions, which are common in financial markets. In this paper, we propose a clustering method based on a bipartite graph model that addresses these challenges. First, it can infer clusters from incomplete data without requiring information about the center nodes. Second, it is designed to effectively handle heavy-tailed data. Numerical experiments using real financial data validate the efficiency of the proposed method for data clustering.

Yifan Yu, Shengjie Xiu, Daniel P. Palomar

State-space models are pivotal for dynamic system analysis but often struggle with outlier data that deviates from Gaussian distributions, frequently exhibiting skewness and heavy tails. This paper introduces a robust extension utilizing the asymmetric Laplace distribution, specifically tailored to capture these complex characteristics. We propose an efficient variational Bayes algorithm and a novel single-loop parameter estimation strategy, significantly enhancing the efficiency of the filtering, smoothing, and parameter estimation processes. Our comprehensive experiments demonstrate that our methods provide consistently robust performance across various noise settings without the need for manual hyperparameter adjustments. In stark contrast, existing models generally rely on specific noise conditions and necessitate extensive manual tuning. Moreover, our approach uses far fewer computational resources, thereby validating the model's effectiveness and underscoring its potential for practical applications in fields such as robust control and financial modeling.

Amirhossein Javaheri, Jiaxi Ying, Daniel P. Palomar et al.

Graph models provide efficient tools to capture the underlying structure of data defined over networks. Many real-world network topologies are subject to change over time. Learning to model the dynamic interactions between entities in such networks is known as time-varying graph learning. Current methodology for learning such models often lacks robustness to outliers in the data and fails to handle heavy-tailed distributions, a common feature in many real-world datasets (e.g., financial data). This paper addresses the problem of learning time-varying graph models capable of efficiently representing heavy-tailed data. Unlike traditional approaches, we incorporate graph structures with specific spectral properties to enhance data clustering in our model. Our proposed method, which can also deal with noise and missing values in the data, is based on a stochastic approach, where a non-negative vector auto-regressive (VAR) model captures the variations in the graph and a Student-t distribution models the signal originating from this underlying time-varying graph. We propose an iterative method to learn time-varying graph topologies within a semi-online framework where only a mini-batch of data is used to update the graph. Simulations with both synthetic and real datasets demonstrate the efficacy of our model in analyzing heavy-tailed data, particularly those found in financial markets.

Jasin Machkour, Arnaud Breloy, Michael Muma et al.

Sparse principal component analysis (PCA) aims at mapping large dimensional data to a linear subspace of lower dimension. By imposing loading vectors to be sparse, it performs the double duty of dimension reduction and variable selection. Sparse PCA algorithms are usually expressed as a trade-off between explained variance and sparsity of the loading vectors (i.e., number of selected variables). As a high explained variance is not necessarily synonymous with relevant information, these methods are prone to select irrelevant variables. To overcome this issue, we propose an alternative formulation of sparse PCA driven by the false discovery rate (FDR). We then leverage the Terminating-Random Experiments (T-Rex) selector to automatically determine an FDR-controlled support of the loading vectors. A major advantage of the resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical experiments and a stock market data example demonstrate a significant performance improvement.

Jian-Feng Cai, José Vinícius de M. Cardoso, Daniel P. Palomar et al.

We study the problem of estimating precision matrices in Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_2$). The precision matrix in such a distribution is an M-matrix. This problem can be formulated as a sign-constrained log-determinant program. Current algorithms are designed using the block coordinate descent method or the proximal point algorithm, which becomes computationally challenging in high-dimensional cases due to the requirement to solve numerous nonnegative quadratic programs or large-scale linear systems. To address this issue, we propose a novel algorithm based on the two-metric projection method, incorporating a carefully designed search direction and variable partitioning scheme. Our algorithm substantially reduces computational complexity, and its theoretical convergence is established. Experimental results on synthetic and real-world datasets demonstrate that our proposed algorithm provides a significant improvement in computational efficiency compared to the state-of-the-art methods.

Jiaxi Ying, José Vinícius de M. Cardoso, Daniel P. Palomar

We consider the problem of learning a sparse graph under the Laplacian constrained Gaussian graphical models. This problem can be formulated as a penalized maximum likelihood estimation of the Laplacian constrained precision matrix. Like in the classical graphical lasso problem, recent works made use of the $\ell_1$-norm regularization with the goal of promoting sparsity in Laplacian constrained precision matrix estimation. However, we find that the widely used $\ell_1$-norm is not effective in imposing a sparse solution in this problem. Through empirical evidence, we observe that the number of nonzero graph weights grows with the increase of the regularization parameter. From a theoretical perspective, we prove that a large regularization parameter will surprisingly lead to a complete graph, i.e., every pair of vertices is connected by an edge. To address this issue, we introduce the nonconvex sparsity penalty, and propose a new estimator by solving a sequence of weighted $\ell_1$-norm penalized sub-problems. We establish the non-asymptotic optimization performance guarantees on both optimization error and statistical error, and prove that the proposed estimator can recover the edges correctly with a high probability. To solve each sub-problem, we develop a projected gradient descent algorithm which enjoys a linear convergence rate. Finally, an extension to learn disconnected graphs is proposed by imposing additional rank constraint. We propose a numerical algorithm based on based on the alternating direction method of multipliers, and establish its theoretical sequence convergence. Numerical experiments involving synthetic and real-world data sets demonstrate the effectiveness of the proposed method.

José Vinícius de Miranda Cardoso, Daniel P. Palomar

We investigate the problem of learning undirected graphical models under Laplacian structural constraints from the point of view of financial market data. We show that Laplacian constraints have meaningful physical interpretations related to the market index factor and to the conditional correlations between stocks. Those interpretations lead to a set of guidelines that users should be aware of when estimating graphs in financial markets. In addition, we propose algorithms to learn undirected graphs that account for stylized facts and tasks intrinsic to financial data such as non-stationarity and stock clustering.

Esa Ollila, Daniel P. Palomar, Frederic Pascal

A popular regularized (shrinkage) covariance estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward its grand mean. In this paper, a more general approach is considered in which the SCM is replaced by an M-estimator of scatter matrix and a fully automatic data adaptive method to compute the optimal shrinkage parameter with minimum mean squared error is proposed. Our approach permits the use of any weight function such as Gaussian, Huber's, or $t$ weight functions, all of which are commonly used in M-estimation framework. Our simulation examples illustrate that shrinkage M-estimators based on the proposed optimal tuning combined with robust weight function do not loose in performance to shrinkage SCM estimator when the data is Gaussian, but provide significantly improved performance when the data is sampled from a heavy-tailed distribution.

Sandeep Kumar, Jiaxi Ying, Jos'e Vin'icius de M. Cardoso et al.

Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. It is well known that structured graph learning from observed samples is an NP-hard combinatorial problem. In this paper, we first show that for a set of important graph families it is possible to convert the structural constraints of structure into eigenvalue constraints of the graph Laplacian matrix. Then we introduce a unified graph learning framework, lying at the integration of the spectral properties of the Laplacian matrix with Gaussian graphical modeling that is capable of learning structures of a large class of graph families. The proposed algorithms are provably convergent and practically amenable for large-scale semi-supervised and unsupervised graph-based learning tasks. Extensive numerical experiments with both synthetic and real data sets demonstrate the effectiveness of the proposed methods. An R package containing code for all the experimental results is available at https://cran.r-project.org/package=spectralGraphTopology.

Sandeep Kumar, Ketan Rajawat, Daniel P. Palomar

In this two-part work, we propose an algorithmic framework for solving non-convex problems whose objective function is the sum of a number of smooth component functions plus a convex (possibly non-smooth) or/and smooth (possibly non-convex) regularization function. The proposed algorithm incorporates ideas from several existing approaches such as alternate direction method of multipliers (ADMM), successive convex approximation (SCA), distributed and asynchronous algorithms, and inexact gradient methods. Different from a number of existing approaches, however, the proposed framework is flexible enough to incorporate a class of non-convex objective functions, allow distributed operation with and without a fusion center, and include variance reduced methods as special cases. Remarkably, the proposed algorithms are robust to uncertainties arising from random, deterministic, and adversarial sources. The part I of the paper develops two variants of the algorithm under very mild assumptions and establishes first-order convergence rate guarantees. The proof developed here allows for generic errors and delays, paving the way for different variance-reduced, asynchronous, and stochastic implementations, outlined and evaluated in part II.

Ziping Zhao, Daniel P. Palomar

In this paper, the estimation problem for sparse reduced rank regression (SRRR) model is considered. The SRRR model is widely used for dimension reduction and variable selection with applications in signal processing, econometrics, etc. The problem is formulated to minimize the least squares loss with a sparsity-inducing penalty considering an orthogonality constraint. Convex sparsity-inducing functions have been used for SRRR in literature. In this work, a nonconvex function is proposed for better sparsity inducing. An efficient algorithm is developed based on the alternating minimization (or projection) method to solve the nonconvex optimization problem. Numerical simulations show that the proposed algorithm is much more efficient compared to the benchmark methods and the nonconvex function can result in a better estimation accuracy.

Ziping Zhao, Daniel P. Palomar

In econometrics and finance, the vector error correction model (VECM) is an important time series model for cointegration analysis, which is used to estimate the long-run equilibrium variable relationships. The traditional analysis and estimation methodologies assume the underlying Gaussian distribution but, in practice, heavy-tailed data and outliers can lead to the inapplicability of these methods. In this paper, we propose a robust model estimation method based on the Cauchy distribution to tackle this issue. In addition, sparse cointegration relations are considered to realize feature selection and dimension reduction. An efficient algorithm based on the majorization-minimization (MM) method is applied to solve the proposed nonconvex problem. The performance of this algorithm is shown through numerical simulations.

Konstantinos Benidis, Ying Sun, Prabhu Babu et al.

The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty term into the objective function that encourages a sparse solution. However, the resulting methods achieve sparsity at the expense of sacrificing the orthogonality property. In this paper, we develop a new method to estimate dominant sparse eigenvectors without trading off their orthogonality. The problem is highly non-convex and hard to handle. We apply the MM framework where we iteratively maximize a tight lower bound (surrogate function) of the objective function over the Stiefel manifold. The inner maximization problem turns out to be a rectangular Procrustes problem, which has a closed form solution. In addition, we propose a method to improve the covariance estimation problem when its underlying eigenvectors are known to be sparse. We use the eigenvalue decomposition of the covariance matrix to formulate an optimization problem where we impose sparsity on the corresponding eigenvectors. Numerical experiments show that the proposed eigenvector extraction algorithm matches or outperforms existing algorithms in terms of support recovery and explained variance, while the covariance estimation algorithms improve significantly the sample covariance estimator.

Ying Sun, Prabhu Babu, Daniel P. Palomar

This paper considers the problem of robustly estimating a structured covariance matrix with an elliptical underlying distribution with known mean. In applications where the covariance matrix naturally possesses a certain structure, taking the prior structure information into account in the estimation procedure is beneficial to improve the estimation accuracy. We propose incorporating the prior structure information into Tyler's M-estimator and formulate the problem as minimizing the cost function of Tyler's estimator under the prior structural constraint. First, the estimation under a general convex structural constraint is introduced with an efficient algorithm for finding the estimator derived based on the majorization minimization (MM) algorithm framework. Then, the algorithm is tailored to several special structures that enjoy a wide range of applications in signal processing related fields, namely, sum of rank-one matrices, Toeplitz, and banded Toeplitz structure. In addition, two types of non-convex structures, i.e., the Kronecker structure and the spiked covariance structure, are also discussed, where it is shown that simple algorithms can be derived under the guidelines of MM. Numerical results show that the proposed estimator achieves a smaller estimation error than the benchmark estimators at a lower computational cost.

Junxiao Song, Prabhu Babu, Daniel P. Palomar

In this paper, we consider an $\ell_{0}$-norm penalized formulation of the generalized eigenvalue problem (GEP), aimed at extracting the leading sparse generalized eigenvector of a matrix pair. The formulation involves maximization of a discontinuous nonconcave objective function over a nonconvex constraint set, and is therefore computationally intractable. To tackle the problem, we first approximate the $\ell_{0}$-norm by a continuous surrogate function. Then an algorithm is developed via iteratively majorizing the surrogate function by a quadratic separable function, which at each iteration reduces to a regular generalized eigenvalue problem. A preconditioned steepest ascent algorithm for finding the leading generalized eigenvector is provided. A systematic way based on smoothing is proposed to deal with the "singularity issue" that arises when a quadratic function is used to majorize the nondifferentiable surrogate function. For sparse GEPs with special structure, algorithms that admit a closed-form solution at every iteration are derived. Numerical experiments show that the proposed algorithms match or outperform existing algorithms in terms of computational complexity and support recovery.