**Project Heads**

Martin Eigel, Claudia Schillings, Gabriele Steidl

**Project Members**

Robert Gruhlke

**Project Duration**

01.01.2023 − 31.12.2024

**Located at**

FU Berlin

**External Website**

**Related Publications
**

Eigel, M., Gruhlke, R., & Sommer, D. (2023). Less interaction with forward models in Langevin dynamics. *arXiv preprint arXiv:2212.11528*.

Eigel, M., Gruhlke, R., Kirstein M., Schillings C. & Sommer, D. (2023). Diffusion Generative Modelling by directly solving the Hamilton-Jacobi-Bellman equation of Stochastic optimal control using Tensor Networks. (in preparation)

Gruhlke, R., Moser, D. (2023). Automatic differentiation within hierachical tensor formats. (in preparation)

Gruhlke, R., Miranda, C., Nouy, A. & Trunschke, P. (2023). Optimal Sampling for alternating steepest descent on tensor networks. (in preparation)

**Related Media
**

Density trajectory from unimodal standard normal Gaussian to an non-symmetric multimodal density of non Gaussian-mixture type. The trajectory is defined through an Ornstein-Uhlenbeck process and its time-reverse counterpart process. The drift term in the reverse process is defined upon the score, which is obtained through solution of the Hamilton-Jacobi-Bellman equation. The latter is obtained through Hopf-Cole transformation of the Fokker-Planck equation associated to the forward Ornstein-Uhlenbeck process.

Many objective functions of minimization problems can be expressed in terms of the expectation of a loss. A common solution strategy is to minimize a corresponding empirical mean estimate. Unfortunately, the deviation of the exact and empirical minimizer then depends on the sample size. As an alternative, we empirically project the gradient of the exact objective function onto the tangent space. Descent is ensured by optimal weighted least squares approximation within an alternating minimization scheme.

The main goal of this project is the minimization of objective functions defined on high-dimensional Euclidean tensor spaces. For the case, that the objective function allows for cheap evaluations on the Riemannian manifold of tensors given in hierachical tree-based low-rank format, we construct a cheap approach to obtain Riemannian Gradients based on Automatic differentiation. Examples of such type include (empirical) regression or completion problems.

This approach in turn overcomes the curse of dimensionality arising when computing Riemannian gradients as projection of (non traceable) Euclidean gradients to the tangential space.

Low-rank tensor formats define a non-linear approximation class of tensors, in particular they are multilinear and are a subclass of tensor networks, multigraphs with edge identities with additional dangling edges representing the indices of the full tensor.

This type of topology allows for efficient (sub)-contractions required to define local projections that define the degrees of freedom in the Riemannian gradient.