More Like this

1. Consistency of Sparse-Group Lasso Graphical Model Selection for Time Series

   https://doi.org/10.1109/IEEECONF51394.2020.9443298  

   Tugnait, Jitendra K. ( November 2020 , 2020 54th Asilomar Conference on Signals, Systems, and Computers)

   null (Ed.)

   We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A p-variate Gaussian time series graphical model associated with an undirected graph with p vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph. A sparse-group lasso-based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the inverse power spectral density (PSD) of the data via optimization of a sparse-group lasso based penalized log-likelihood cost function that is formulated in the frequency-domain. The CIG is then inferred from the estimated inverse PSD. In this paper we establish sufficient conditions for consistency of the inverse PSD estimator resulting from the sparse-group graphical lasso-based approach. 

   more » « less
2. Sparse-Group Log-Sum Penalized Graphical Model Learning For Time Series

   https://doi.org/10.1109/ICASSP43922.2022.9747446  

   Tugnait, Jitendra K. ( May 2022 , ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) |)

   We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the sparse inverse power spectral density (PSD) of the data. The CIG is then inferred from the estimated inverse PSD. In this paper we investigate use of a sparse-group log-sum penalty (LSP) instead of sparse-group lasso penalty. An alternating direction method of multipliers (ADMM) approach for iterative optimization of the non-convex problem is presented. We provide sufficient conditions for local convergence in the Frobenius norm of the inverse PSD estimators to the true value. This results also yields a rate of convergence. We illustrate our approach using numerical examples utilizing both synthetic and real data. 

   more » « less
3. A study of optical scattering modelling for mixed-phase polar stratospheric clouds

   https://doi.org/10.5194/amt-16-419-2023  

   Cairo, Francesco ; Deshler, Terry ; Di Liberto, Luca ; Scoccione, Andrea ; Snels, Marcel ( January 2023 , Atmospheric Measurement Techniques)

   Abstract. Scattering codes are used to study the optical properties of polar stratospheric clouds (PSCs). Particle backscattering and depolarization coefficients can be computed with available scattering codes once the particle size distribution (PSD) is known and a suitable refractive index is assumed. However, PSCs often appear as external mixtures of supercooled ternary solution (STS) droplets, solid nitric acid trihydrate (NAT) and possibly ice particles, making the assumption of a single refractive index and a single morphology to model the scatterers questionable.Here we consider a set of 15 coincident measurements of PSCs above McMurdo Station, Antarctica, using ground-based lidar, a balloon-borne optical particle counter (OPC) and in situ observations taken by a laser backscattersonde and OPC during four balloon stratospheric flights from Kiruna, Sweden. This unique dataset of microphysical and optical observations allows us to test the performances of optical scattering models when both spherical and aspherical scatterers of different composition and, possibly, shapes are present. We consider particles as STS if their radius is below a certain threshold value Rth and NAT or possibly ice if it is above it. The refractive indices are assumed known from the literature. Mie scattering is used for the STS, assumed spherical. Scattering from NAT particles, considered spheroids of different aspect ratio (AR), is treated with T-matrix results where applicable. The geometric-optics–integral-equation approach is used whenever the particle size parameter is too large to allow for a convergence of the T-matrix method.The parameters Rth and AR of our model have been varied between 0.1 and 2 µm and between 0.3 and 3, respectively, and the calculated backscattering coefficient and depolarization were compared with the observed ones. The best agreement was found for Rth between 0.5 and 0.8 µm and for AR less than 0.55 and greater than 1.5.To further constrain the variability of AR within the identified intervals, we have sought an agreement with the experimental data by varying AR on a case-by-case basis and further optimizing the agreement by a proper choice of AR smaller than 0.55 and greater than 1.5 and Rth within the interval 0.5 and 0.8 µm. The ARs identified in this way cluster around the values 0.5 and 2.5.The comparison of the calculations with the measurements is presented and discussed. The results of this work help to set limits to the variability of the dimensions and asphericity of PSC solid particles, within the limits of applicability of our model based on the T-matrix theory of scattering and on assumptions on a common particle shape in a PSD and a common threshold radius for all the PSDs. 

   more » « less
4. New Results on Graphical Modeling of High-Dimensional Dependent Time Series

   https://doi.org/10.1109/IEEECONF53345.2021.9723239  

   Tugnait, Jitendra K. ( October 2021 , 2021 55th Asilomar Conference on Signals, Systems, and Computers)

   We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the sparse inverse power spectral density (PSD) of the data via optimization of a sparse-group lasso based penalized log-likelihood cost function that is formulated in the frequency-domain. The CIG is then inferred from the estimated inverse PSD. Optimization in the previous approach was performed using an alternating minimization (AM) approach whose performance depends upon choice of a penalty parameter. In this paper we investigate an alternating direction method of multipliers (ADMM) approach for optimization to mitigate dependence on the penalty parameter. We also investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using synthetic and real data. Comparisons with the "usual" i.i.d. modeling of time series for graph estimation are also provided. 

   more » « less
5. Implicit Balancing and Regularization: Generalization and Convergence Guarantees for Overparameterized Asymmetric Matrix Sensing

   Soltanolkotabi, Mahdi ; Stöger, Dominik ; Xie, Changzhi ( July 2023 , Proceedings of Thirty Sixth Conference on Learning Theory)

   Recently, there has been significant progress in understanding the convergence and generalization properties of gradient-based methods for training overparameterized learning models. However, many aspects including the role of small random initialization and how the various parameters of the model are coupled during gradient-based updates to facilitate good generalization, remain largely mysterious. A series of recent papers have begun to study this role for non-convex formulations of symmetric Positive Semi-Definite (PSD) matrix sensing problems which involve reconstructing a low-rank PSD matrix from a few linear measurements. The underlying symmetry/PSDness is crucial to existing convergence and generalization guarantees for this problem. In this paper, we study a general overparameterized low-rank matrix sensing problem where one wishes to reconstruct an asymmetric rectangular low-rank matrix from a few linear measurements. We prove that an overparameterized model trained via factorized gradient descent converges to the low-rank matrix generating the measurements. We show that in this setting, factorized gradient descent enjoys two implicit properties: (1) coupling of the trajectory of gradient descent where the factors are coupled in various ways throughout the gradient update trajectory and (2) an algorithmic regularization property where the iterates show a propensity towards low-rank models despite the overparameterized nature of the factorized model. These two implicit properties in turn allow us to show that the gradient descent trajectory from small random initialization moves towards solutions that are both globally optimal and generalize well. 

   more » « less