Tree Tensor Networks Demonstrate Superior Efficiency for Compressing Multivariate Functions
Researchers have shown that tree tensor networks (TTNs) can more efficiently compress and represent multidimensional functions compared to the commonly used tensor train format. The study provides direct constructions for elementary functions and interpolative methods for complex ones, with applications to solving nonlinear Fredholm equations. This work advances computational methods for handling high-dimensional data in physics and numerical analysis.
A new arXiv preprint demonstrates that tree tensor networks offer a more efficient compressed representation for multivariate functions than standard tensor train approaches. The authors provide explicit constructions of elementary functions as TTNs and develop generalized tensor cross interpolation algorithms for more complicated functions. Through numerical examples across a range of multidimensional functions, they show that structured tree tensor networks require significantly fewer parameters than tensor trains while maintaining accuracy. The research extends these methods to develop a TTN-based solver for multidimensional nonlinear Fredholm equations. This work builds on the established use of one-dimensional tensor networks (tensor trains and matrix product states) in quantum physics and numerical analysis, expanding their applicability to more complex function approximation problems.
What's missing
The preprint does not discuss computational complexity comparisons (time and memory requirements) between TTN and tensor train methods, nor does it provide guidance on when practitioners should choose TTN over tensor train approaches. Additionally, the practical scalability of the proposed methods to very high-dimensional problems (beyond those demonstrated) remains unclear.
What different sources said
- arXiv physicsCenter
Compressing multivariate functions with tree tensor networks
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