🤖 AI Summary
This study systematically characterizes the approximation capacity of tree tensor networks (TTNs) for multivariate functions, addressing two central problems: the approximation rates of TTNs for classical smooth function classes, and the intrinsic structure of their attainable approximation classes. Methodologically, it integrates tensor network theory, approximation theory, and function space embedding analysis. The contributions are threefold: (i) it establishes that TTNs achieve near-optimal *h*-uniform and *h*-adaptive approximation rates; (ii) it identifies the TTN approximation class as a quasi-Banach space strictly containing—yet not contained in—the classical isotropic, anisotropic, and mixed smoothness spaces; and (iii) it constructs a rigorous theoretical framework for universal approximation by TTNs, demonstrating expressive power comparable to deep ReLU networks and establishing continuous embeddings from multiple smoothness spaces into the TTN approximation class.
📝 Abstract
We study the approximation of multivariate functions with tensor networks (TNs). The main conclusion of this work is an answer to the following two questions: ``What are the approximation capabilities of TNs?"and"What is an appropriate model class of functions that can be approximated with TNs?"To answer the former, we show that TNs can (near to) optimally replicate $h$-uniform and $h$-adaptive approximation, for any smoothness order of the target function. Tensor networks thus exhibit universal expressivity w.r.t. isotropic, anisotropic and mixed smoothness spaces that is comparable with more general neural networks families such as deep rectified linear unit (ReLU) networks. Put differently, TNs have the capacity to (near to) optimally approximate many function classes -- without being adapted to the particular class in question. To answer the latter, as a candidate model class we consider approximation classes of TNs and show that these are (quasi-)Banach spaces, that many types of classical smoothness spaces are continuously embedded into said approximation classes and that TN approximation classes are themselves not embedded in any classical smoothness space.