### Working paper

## Tensor Rank bounds for Point Singularities in ℝ^3

For degenerate Hybrid Stochastic Systems with full Dependence of all components exponential convergence to the stationary regime has been established.

We introduce T-Basis, a novel concept for a compact representation of a set of tensors, each of an arbitrary shape, which is often seen in Neural Networks. Each of the tensors in the set is modelled using Tensor Rings, though the concept is applicable to other Tensor Networks as well. Owing its name to the T-shape of nodes in diagram notation of Tensor Rings, T-Basis is simply a list of equally shaped three-dimensional tensors, used to represent Tensor Ring nodes. Such representation allows us to parameterize the tensor set with a small number of parameters (coefficients of the T-Basis tensors), scaling logarithmically with the size of each tensor in the set, and linearly with the dimensionality of T-Basis. We evaluate the proposed approach on the task of neural network compression, and demonstrate that it reaches high compression rates at acceptable performance drops. Finally, we analyze memory and operation requirements of the compressed networks, and conclude that T-Basis networks are equally well suited for training and inference in resource-constrained environments, as well as usage on the edge devices.

The aim of this paper is to propose a robust numerical solver, which is capable of efficiently solving a three-dimensional elliptic problem in a data-sparse quantized tensor format. In particular, we use the combined Tucker and quantized tensor train format (TQTT), which allows us to use astronomically large grid sizes. However, due to the ill-conditioning of discretized differential operators, so fine grids lead to numerical instabilities. To obtain a robust solver, we utilize the well-known alternating direction implicit method and modify it to avoid multiplication by differential operators. So as to make the method efficient, we derive an explicit TQTT representation of the iteration matrix and quantized tensor train (QTT) representations of the inverses of symmetric tridiagonal Toeplitz matrices as an auxiliary result. As an application, we consider accurate solution of elliptic problems with singular potentials arising in electronic Schroedinger’s equation.

For degenerate mechanical systems with switching exponential convergence to the stationary regime has been established.