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Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an l-th order tensor in (Rd)⊗l of rank r (where r≪d), can variants of gradient descent find a rank m decomposition where m>r? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least m=Ω(dl−1), while a variant of gradient descent can find an approximate tensor when m=O∗(r2.5llogd). Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
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This content will become publicly available on December 27, 2025
Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm
Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate tensor generated during the contractions is approximated as a low-rank binary tree tensor network. The proposed algorithm has the flexibility to incorporate a large portion of the environment when performing low-rank approximations, which can lead to high accuracy for a given rank. Here, the environment refers to the remaining set of tensors in the network, and low-rank approximations with larger environments can generally provide higher accuracy. For contracting tensor networks defined on lattices, the proposed algorithm can be viewed as a generalization of the standard boundary-based algorithms. In addition, the algorithm includes a cost-efficient density matrix algorithm for approximating a tensor network with a general graph structure into a tree structure, whose computational cost is asymptotically upper-bounded by that of the standard algorithm that uses canonicalization. Experimental results indicate that the proposed technique outperforms previously proposed approximate tensor network contraction algorithms for multiple problems in terms of both accuracy and efficiency.
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- Award ID(s):
- 1942995
- PAR ID:
- 10629138
- Publisher / Repository:
- Quantum Journal
- Date Published:
- Journal Name:
- Quantum
- Volume:
- 8
- ISSN:
- 2521-327X
- Page Range / eLocation ID:
- 1580
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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