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Tensor trains are a versatile tool to compress and work with high-dimensional data and functions. In this work we introduce the Streaming Tensor Train Approximation (STTA), a new class of algorithms for approximating a given tensor $\mathcal T$ in the tensor train format. STTA accesses $\mathcal T$ exclusively via two-sided random sketches of the original data, making it streamable and easy to implement in parallel -- unlike existing deterministic and randomized tensor train approximations. This property also allows STTA to conveniently leverage structure in $\mathcal T$, such as sparsity and various low-rank tensor formats, as well as linear combinations thereof. When Gaussian random matrices are used for sketching, STTA is admissible to an analysis that builds and extends upon existing results on the generalized Nystr\"om approximation for matrices. Our results show that STTA can be expected to attain a nearly optimal approximation error if the sizes of the sketches are suitably chosen. A range of numerical experiments illustrates the performance of STTA compared to existing deterministic and randomized approaches., Comment: 21 pages, code available at https://github.com/RikVoorhaar/tt-sketch

This work proposes a novel general-purpose estimator for supervised machine learning (ML) based on tensor trains (TT). The estimator uses TTs to parametrize discretized functions, which are then optimized using Riemannian gradient descent under the form of a tensor completion problem. Since this optimization is sensitive to initialization, it turns out that the use of other ML estimators for initialization is crucial. This results in a competitive, fast ML estimator with lower memory usage than many other ML estimators, like the ones used for the initialization., Comment: 23 pages

We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for $\mathfrak{g} = \mathfrak{sl}_3$, we obtain the H*$(\mathfrak{u}, \mathbb{C}) = \mathbb{C}[\mathcal{N}]$-module structure of HH*$(\mathfrak{u}_{\lambda})$., Comment: 25 pages, 34 figures

We present a computer algorithm to explicitly compute the BGG resolution and its cohomology. We give several applications, in particular computation of various sheaf cohomology groups on flag varieties. An implementation of the algorithm is available at https://github.com/RikVoorhaar/bgg-cohomology.

We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2-connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The definition of the 2-category of 2-group torsors is new, and we develop the tools necessary for computations in this 2-category. We prove a version of the non-Abelian Stokes Theorem and the Ambrose-Singer Theorem for 2-transport. This definition motivated by the fact that principal $G$-bundles with connection are equivalent to functors from the thin fundamental groupoid to the category of $G$-torsors. In the same lines we deduce a notion of principal 2-bundle with 2-connection, and show it is equivalent to our notion 2-transport functors. This gives a stricter notion than appears in the literature, which is more concrete. It allows for computations of 2-holonomy which will be exploited in a companion paper to define Wilson surface observables. Furthermore this notion can be generalized to a concrete but strict notion of $n$-transport for arbitrary $n$.