Functional tensor trains plus BSDE regression solve the HJB score PDE, yielding a fast low-rank sampler that outperforms neural diffusion methods on multimodal targets.
Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions
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abstract
This paper is concerned with convergence estimates for fully discrete tree tensor network approximations of high-dimensional functions from several model classes. For functions having standard or mixed Sobolev regularity, new estimates generalizing and refining known results are obtained, based on notions of linear widths of multivariate functions. In the main results of this paper, such techniques are applied to classes of functions with compositional structure, which are known to be particularly suitable for approximation by deep neural networks. As shown here, such functions can also be approximated by tree tensor networks without a curse of dimensionality -- however, subject to certain conditions, in particular on the depth of the underlying tree. In addition, a constructive encoding of compositional functions in tree tensor networks is given.
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Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling
Functional tensor trains plus BSDE regression solve the HJB score PDE, yielding a fast low-rank sampler that outperforms neural diffusion methods on multimodal targets.