Core Concepts

Efficient algorithms with theoretical guarantees can recover the underlying graph structure, i.e., the permutation mapping tensor indices to the vertices, for one-dimensional tensor networks in the tensor ring and tensor train formats.

Abstract

The content discusses the problem of recovering the underlying graph structure, i.e., the permutation mapping tensor indices to vertices, for one-dimensional tensor networks in the tensor ring (TR) and tensor train (TT) formats.
The key highlights are:
The TR and TT formats represent high-order tensors using a group of 3-tensors ordered as a loop or a path, respectively. The underlying graph structure, specified by a permutation, is often unknown and needs to be recovered.
The authors propose polynomial-time algorithms to efficiently recover the underlying permutation for TR and TT formats. The algorithms compare the matricization ranks after downsampling the tensor, with complexity O(d log d) for d-th order tensors.
The authors prove that their algorithms can almost surely recover the correct permutation when tensor entries can be observed without noise. They further establish the robustness of the algorithms against observational noise.
The theoretical results are validated through numerical experiments. The content provides a rigorous analysis of the proposed algorithms, including complexity bounds, almost sure correctness in the noiseless case, and robustness guarantees.

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Key Insights Distilled From

by Ziang Chen,J... at **arxiv.org** 04-04-2024

Deeper Inquiries

To extend the proposed algorithms for approximate recovery, one could introduce a tolerance parameter that allows for deviations from the exact permutation. By adjusting the criteria for correctness in the algorithms, one can accommodate permutations that are close to the true underlying graph but not an exact match. This flexibility would enable the algorithms to handle cases where the observed tensor entries do not perfectly align with any permutation in the equivalence class. Additionally, by incorporating error bounds or allowing for increased bond dimensions, the algorithms can be adapted to find the best possible approximation to the true permutation that represents the target tensor.

The tensor network recovery problem has significant implications in various fields such as machine learning, scientific computing, and theoretical chemistry. In machine learning, the ability to recover the underlying graph structure for tensors is crucial for tasks like dimensionality reduction, feature extraction, and data representation. By accurately identifying the correct graph or permutation, tensor network decomposition tasks can be optimized, leading to more efficient and effective machine learning models.
In scientific computing, tensor networks are used to encode complex multidimensional data and facilitate computations in quantum physics, materials science, and other scientific domains. Recovering the underlying graph in tensor networks is essential for accurately representing physical systems, simulating quantum states, and analyzing large datasets. The performance of tensor network algorithms heavily relies on the choice of the graph, making the recovery of the correct structure a critical step in scientific computations.
In theoretical chemistry, tensor networks play a vital role in modeling molecular systems, quantum chemical calculations, and studying chemical reactions. The ability to recover the underlying graph for tensors enables researchers to analyze molecular properties, predict chemical behaviors, and simulate complex chemical processes. By accurately recovering the graph structure, tensor network decomposition techniques can enhance the understanding of chemical systems and improve computational simulations in theoretical chemistry.

The ideas behind the proposed algorithms for recovering the underlying graph structure in one-dimensional tensor networks can be extended to other tensor network formats, such as tree tensor networks and projected entangled pair states (PEPS). While the specific implementation details may vary, the fundamental concept of comparing matricization ranks after downsampling can be applied to these more complex tensor network structures.
For tree tensor networks, which involve hierarchical tensor formats, one could adapt the divide-and-conquer approach to determine the correct relative positions of indices based on the observed tensor entries. By considering the connectivity of nodes in the tree structure, algorithms similar to those proposed for one-dimensional tensor networks can be developed to recover the underlying graph for tree tensor networks.
Similarly, for projected entangled pair states (PEPS), which are used in quantum many-body physics and quantum information theory, the concept of comparing ranks of matricizations can be utilized to recover the correct graph structure. By analyzing the relationships between tensors in the PEPS format, algorithms can be designed to identify the optimal permutation that maps tensor indices to the vertices of the given graph in PEPS.
Overall, the principles underlying the proposed algorithms for one-dimensional tensor network recovery can be generalized and adapted to recover the underlying graph structure for various tensor network formats, extending their applicability to a broader range of multidimensional array representations.

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