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1. Tensor Algebras in Finite Tensor Categories

   https://doi.org/10.1093/imrn/rnz332  

   Etingof, Pavel ; Kinser, Ryan ; Walton, Chelsea ( December 2019 , International Mathematics Research Notices)

   This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions that preserve the ascending filtration on tensor algebras and the second for actions that preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories $\textsf{Vec}_{G}^{\omega }$ and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra.

    

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2. Bayesian tensor-on-tensor regression with efficient computation

   https://doi.org/10.4310/23-SII786  

   Wang, Kunbo ; Xu, Yanxun ( January 2024 , Statistics and Its Interface)
3. Optimal High-Order Tensor SVD via Tensor-Train Orthogonal Iteration

   https://doi.org/10.1109/TIT.2022.3152733  

   Zhou, Yuchen ; Zhang, Anru R. ; Zheng, Lili ; Wang, Yazhen ( June 2022 , IEEE Transactions on Information Theory)
4. Cost-efficient Gaussian Tensor Network Embeddings for Tensor-structured Inputs

   Ma, Linjian ; Solomonik, Edgar ( January 2023 , Advances in neural information processing systems)
5. Comb Tensor Networks

   Chepiga, Natalia ; White, Steven R ( June 2019 , Physical review)

   In this paper we propose a special type of a tree tensor network that has the geometry of a comb—a one-dimensional (1D) backbone with finite 1D teeth projecting out from it. This tensor network is designed to provide an effective description of higher-dimensional objects with special limited interactions or, alternatively, one-dimensional systems composed of complicated zero-dimensional objects. We provide details on the best numerical procedures for the proposed network, including an algorithm for variational optimization of the wave function as a comb tensor network and the transformation of the comb into a matrix product state. We compare the complexity of using a comb versus alternative matrix product state representations using density matrix renormalization group algorithms. As an application, we study a spin-1 Heisenberg model system which has a comb geometry. In the case where the ends of the teeth are terminated by spin-1/2 spins, we find that Haldane edge states of the teeth along the backbone form a critical spin-1/2 chain, whose properties can be tuned by the coupling constant along the backbone. By adding next-nearest-neighbor interactions along the backbone, the comb can be brought into a gapped phase with a long-range dimerization along the backbone. The critical and dimerized phases are separated by a Kosterlitz-Thouless phase transition, the presence of which we confirm numerically. Finally, we show that when the teeth contain an odd number of spins and are not terminated by spin-1/2's, a special type of comb edge states emerge. 

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