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1. A super-resolution framework for tensor decomposition

   https://doi.org/https://doi.org/10.1093/imaiai/iaac002  

   Li, Qiuwei; Ashley, Ashley; Shen, Lixin; Tang, Gongguo (January 2022, Information and inference)

   This work considers a super-resolution framework for overcomplete tensor decomposition. Specifically, we view tensor decomposition as a super-resolution problem of recovering a sum of Dirac measures on the sphere and solve it by minimizing a continuous analog of the ℓ1 norm on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Similar to the separation condition in the super-resolution problem, by explicitly constructing a dual certificate, we develop incoherence conditions of the tensor factors so that they form the unique optimal solution of the continuous analog of ℓ1 norm minimization. Remarkably, the derived incoherence conditions are satisfied with high probability by random tensor factors uniformly distributed on the sphere, implying global identifiability of random tensor factors. 

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2. Hyperspectral Super-Resolution Via Coupled Tensor Factorization: Identifiability and Algorithms

   https://doi.org/10.1109/ICASSP.2018.8462525  

   Kanatsoulis, Charilaos I.; Fu, Xiao; Sidiropoulos, Nicholas D.; Ma, Wing-Kin (April 2018, 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   This work focuses on the problem of fusing a hyperspectral image (HSI) and a multispectral image (MSI) to produce a super-resolution image that admits high spatial and spectral resolutions. Existing algorithms are mostly based on joint low-rank factorization of the ma-tricized HSI and MSI. This framework is effective to some extent, but several challenges remain. First, it is unclear whether or not the super-resolution image is identifiable in theory under this framework, while identifiability usually plays an essential role in such estimation problems. Second, most algorithms assume that the degradation operators from the super-resolution image to the HSI and MSI are known or can be easily estimated - which is hardly true in practice. In this work, we propose a novel coupled tensor decomposition method that can effectively circumvent these issues. The proposed approach guarantees the identifiability of the super-resolution image under realistic conditions. The method can work even without knowing the spatial degradation operator, which could be hard to accurately estimate in practice. Simulations using AVIRIS Cuprite data are employed to demonstrate the effectiveness of the proposed approach. 

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3. 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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4. Some aspects of Ricci flow on the 4-sphere

   https://doi.org/10.53733/152  

   Chang, Sun-Yung Alice; Chen, Eric (September 2021, New Zealand Journal of Mathematics)

   In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere. 

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5. Approximately low-rank recovery from noisy and local measurements by convex program

   https://doi.org/10.1093/imaiai/iaad013  

   Lee, Kiryung; Sharma, Rakshith Srinivasa; Junge, Marius; Romberg, Justin (May 2023, Information and Inference: A Journal of the IMA)

   Abstract Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $$\ell _1$$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey’s empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications. 

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