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1. Quenched large deviation principles for random projections of lpn balls

   https://doi.org/10.1016/j.jfa.2025.110937  

   Lopatto, Patrick; Ramanan, Kavita; Xie, Xiaoyu (September 2025, Journal of Functional Analysis)

   Let (kn)n∈N be a sequence of positive integers growing to infinity at a sublinear rate, kn → ∞ and kn/n → 0 as n → ∞. Given a sequence of n-dimensional random vectors {Y (n)}n∈N belonging to a certain class, which includes uniform distributions on suitably scaled ℓnp -balls or ℓnp -spheres, p ≥ 2, and product distributions with sub-Gaussian marginals, we study the large deviations behavior of the corresponding sequence of kn-dimensional orthogonal projections. For almost every sequence of projection matrices, we establish a large deviation principle (LDP) for the corresponding sequence of projections, with a fairly explicit rate function that does not depend on the sequence of projection matrices. As corollaries, we also obtain quenched LDPs for sequences of ℓ2-norms and ℓ∞-norms of the coordinates of the projections. Past work on LDPs for projections with growing dimension has mainly focused on the annealed setting, where one also averages over the random projection matrix, chosen from the Haar measure, in which case the coordinates of the projection are exchangeable. The quenched setting lacks such symmetry properties, and gives rise to significant new challenges in the setting of growing projection dimension. Along the way, we establish new Gaussian approximation results on the Stiefel manifold that may be of independent interest. Such LDPs are of relevance in asymptotic convex geometry, statistical physics and high-dimensional statistics. 

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2. On Sparse High-Dimensional Graph Estimation from Multi-Attribute Data

   https://doi.org/10.1109/IEEECONF60004.2024.10942781  

   Tugnait, Jitendra K (October 2024, IEEE)

   We consider the problem of inferring the conditional independence graph (CIG) of high-dimensional Gaussian vectors from multi-attribute data. Most existing methods for graph estimation are based on single-attribute models where one associates a scalar random variable with each node. In multi-attribute graphical models, each node represents a random vector. In this paper we provide a unified theoretical analysis of multi-attribute graph learning using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the precision matrix to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. 

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3. Conditional Independence Graph Estimation from Multi-Attribute Dependent Time Series

   https://doi.org/10.1109/MLSP58920.2024.10734734  

   Tugnait, Jitendra K (September 2024, IEEE)

   Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. All existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node. In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We illustrate our approach using numerical examples utilizing both synthetic and real data. 

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4. Persistent cup product structures and related invariants

   https://doi.org/10.1007/s41468-023-00138-5  

   Mémoli, Facundo; Stefanou, Anastasios; Zhou, Ling (October 2023, Journal of Applied and Computational Topology)

   Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LS-category invariant. 

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5. On Conditional Independence Graph Learning From Multi-Attribute Gaussian Dependent Time Series

   https://doi.org/10.1109/OJSP.2025.3578807  

   Tugnait, Jitendra K (January 2025, IEEE Open Journal of Signal Processing)

   Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. Existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node.In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function formulated in the frequency domain using the discrete Fourier transform of the time-domain data. We consider both convex (sparse-group lasso) and non-convex(log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. We also empirically investigate selection of the tuning parameters based on the Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data. 

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