Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals

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.

FDR-Controlled Portfolio Optimization for Sparse Financial Index Tracking

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.

High-Dimensional False Discovery Rate Control for Dependent Variables

Algorithms that ensure reproducible findings from large-scale, high-dimensional data are pivotal in numerous signal processing applications. In recent years, multivariate false discovery rate (FDR) controlling methods have emerged, providing guarantees even in high-dimensional settings where the number of variables surpasses the number of samples. However, these methods often fail to reliably control the FDR in the presence of highly dependent variable groups, a common characteristic in fields such as genomics and finance. To tackle this critical issue, we introduce a novel framework that accounts for general dependency structures. Our proposed dependency-aware T-Rex selector integrates hierarchical graphical models within the T-Rex framework to effectively harness the dependency structure among variables. Leveraging martingale theory, we prove that our variable penalization mechanism ensures FDR control. We further generalize the FDR-controlling framework by stating and proving a clear condition necessary for designing both graphical and non-graphical models that capture dependencies. Additionally, we formulate a fully integrated optimal calibration algorithm that concurrently determines the parameters of the graphical model and the T-Rex framework, such that the FDR is controlled while maximizing the number of selected variables. Numerical experiments and a breast cancer survival analysis use-case demonstrate that the proposed method is the only one among the state-of-the-art benchmark methods that controls the FDR and reliably detects genes that have been previously identified to be related to breast cancer. An open-source implementation is available within the R package TRexSelector on CRAN.

Sparse PCA with False Discovery Rate Controlled Variable Selection

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.

Joint Signal Recovery and Graph Learning from Incomplete Time-Series

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.

Discerning and Enhancing the Weighted Sum-Rate Maximization Algorithms in Communications

Weighted sum-rate (WSR) maximization plays a critical role in communication system design. This paper examines three optimization methods for WSR maximization, which ensure convergence to stationary points: two block coordinate ascent (BCA) algorithms, namely, weighted sum-minimum mean-square error (WMMSE) and WSR maximization via fractional programming (WSR-FP), along with a minorization-maximization (MM) algorithm, WSR maximization via MM (WSR-MM). Our contributions are threefold. Firstly, we delineate the exact relationships among WMMSE, WSR-FP, and WSR-MM, which, despite their extensive use in the literature, lack a comprehensive comparative study. By probing the theoretical underpinnings linking the BCA and MM algorithmic frameworks, we reveal the direct correlations between the equivalent transformation techniques, essential to the development of WMMSE and WSR-FP, and the surrogate functions pivotal to WSR-MM. Secondly, we propose a novel algorithm, WSR-MM+, harnessing the flexibility of selecting surrogate functions in MM framework. By circumventing the repeated matrix inversions in the search for optimal Lagrange multipliers in existing algorithms, WSR-MM+ significantly reduces the computational load per iteration and accelerates convergence. Thirdly, we reconceptualize WSR-MM+ within the BCA framework, introducing a new equivalent transform, which gives rise to an enhanced version of WSR-FP, named as WSR-FP+. We further demonstrate that WSR-MM+ can be construed as the basic gradient projection method. This perspective yields a deeper understanding into its computational intricacies. Numerical simulations corroborate the connections between WMMSE, WSR-FP, and WSR-MM and confirm the efficacy of the proposed WSR-MM+ and WSR-FP+ algorithms.

Learning Large-Scale MTP$_2$ Gaussian Graphical Models via Bridge-Block Decomposition

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.

A Fast Successive QP Algorithm for General Mean-Variance Portfolio Optimization

The mean and variance of portfolio returns are the standard quantities to measure the expected return and risk of a portfolio. Efficient portfolios that provide optimal trade-offs between mean and variance warrant consideration. To express a preference among these efficient portfolios, investors have put forward many mean-variance portfolio (MVP) formulations which date back to the classical Markowitz portfolio. However, most existing algorithms are highly specialized to particular formulations and cannot be generalized for broader applications. Therefore, a fast and unified algorithm would be extremely beneficial. In this paper, we first introduce a general MVP problem formulation that can fit most existing cases by exploring their commonalities. Then, we propose a widely applicable and provably convergent successive quadratic programming algorithm (SCQP) for the general formulation. The proposed algorithm can be implemented based on only the QP solvers and thus is computationally efficient. In addition, a fast implementation is considered to accelerate the algorithm. The numerical results show that our proposed algorithm significantly outperforms the state-of-the-art ones in terms of convergence speed and scalability.

Adaptive Estimation of $\text{MTP}_2$ Graphical Models

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. Such models have received increasing attention in recent years, and have shown interesting properties, e.g., the maximum likelihood estimator exists with as little as two observations regardless of the underlying dimension. In this paper, we propose an adaptive estimation method, which consists of multiple stages: In the first stage, we solve an $\ell_1$-regularized maximum likelihood estimation problem, which leads to an initial estimate; in the subsequent stages, we iteratively refine the initial estimate by solving a sequence of weighted $\ell_1$-regularized problems. We further establish the theoretical guarantees on the estimation error, which consists of optimization error and statistical error. The optimization error decays to zero at a linear rate, indicating that the estimate is refined iteratively in subsequent stages, and the statistical error characterizes the statistical rate. The proposed method outperforms state-of-the-art methods in estimating precision matrices and identifying graph edges, as evidenced by synthetic and financial time-series data sets.

Efficient and Scalable High-Order Portfolios Design via Parametric Skew-t Distribution

Since Markowitz's mean-variance framework, optimizing a portfolio that maximizes the profit and minimizes the risk has been ubiquitous in the financial industry. Initially, profit and risk were measured by the first two moments of the portfolio's return, a.k.a. the mean and variance, which are sufficient to characterize a Gaussian distribution. However, it is broadly believed that the first two moments are not enough to capture the characteristics of the returns' behavior, which have been recognized to be asymmetric and heavy-tailed. Although there is ample evidence that portfolio designs involving the third and fourth moments, i.e., skewness and kurtosis, will outperform the conventional mean-variance framework, they are non-trivial. Specifically, in the classical framework, the memory and computational cost of computing the skewness and kurtosis grow sharply with the number of assets. To alleviate the difficulty in high-dimensional problems, we consider an alternative expression for high-order moments based on parametric representations via a generalized hyperbolic skew-t distribution. Then, we reformulate the high-order portfolio optimization problem as a fixed-point problem and propose a robust fixed-point acceleration algorithm that solves the problem in an efficient and scalable manner. Empirical experiments also demonstrate that our proposed high-order portfolio optimization framework is of low complexity and significantly outperforms the state-of-the-art methods by 2 to 4 orders of magnitude.