Tensor completion is an extension of matrix completion aimed at recovering a multiway data tensor by leveraging a given subset of its entries (observations) and the pattern of observation. The low-rank assumption is key in establishing a relationship between the observed and unobserved entries of the tensor. The low-rank tensor completion problem is typically solved using numerical optimization techniques, where the rank information is used either implicitly (in the rank minimization approach) or explicitly (in the error minimization approach). Current theories concerning these techniques often study probabilistic recovery guarantees under conditions such as random uniform observations and incoherence requirements. However, if an observation pattern exhibits some low-rank structure that can be exploited, more efficient algorithms with deterministic recovery guarantees can be designed by leveraging this structure. This work shows how to use only standard linear algebra operations to compute the tensor train decomposition of a specific type of ``fiber-wise" observed tensor, where some of the fibers of a tensor (along a single specific mode) are either fully observed or entirely missing, unlike the usual entry-wise observations. From an application viewpoint, this setting is relevant when it is easier to sample or collect a multiway data tensor along a specific mode (e.g., temporal). The proposed completion method is fast and is guaranteed to work under reasonable deterministic conditions on the observation pattern. Through numerical experiments, we showcase interesting applications and use cases that illustrate the effectiveness of the proposed approach.

Sampling from probability densities is a common challenge in fields such as Uncertainty Quantification (UQ) and Generative Modelling (GM). In GM in particular, the use of reverse-time diffusion processes depending on the log-densities of Ornstein-Uhlenbeck forward processes are a popular sampling tool. In Berner et al. [2022] the authors point out that these log-densities can be obtained by solution of a \textit{Hamilton-Jacobi-Bellman} (HJB) equation known from stochastic optimal control. While this HJB equation is usually treated with indirect methods such as policy iteration and unsupervised training of black-box architectures like Neural Networks, we propose instead to solve the HJB equation by direct time integration, using compressed polynomials represented in the Tensor Train (TT) format for spatial discretization. Crucially, this method is sample-free, agnostic to normalization constants and can avoid the curse of dimensionality due to the TT compression. We provide a complete derivation of the HJB equation's action on Tensor Train polynomials and demonstrate the performance of the proposed time-step-, rank- and degree-adaptive integration method on a nonlinear sampling task in 20 dimensions.