Tensor reconstruction beyond constant rank

Q: What does this research mean for the field?

Efficient randomized algorithms can reconstruct polynomials computed by multilinear and set-multilinear depth-3 arithmetic circuits, enabling the first efficient tensor rank finding and optimal decomposition for tensors of super-constant rank. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Shir Peleg; Ben Lee Volk

doi:10.1007/s00037-026-00295-w

August 20, 2026Computational ComplexityOpen Access

Tensor reconstruction beyond constant rank

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Key Points

Develop efficient algorithms to compute tensor rank, find optimal tensor decompositions, and reconstruct depth-3 arithmetic circuits beyond constant rank settings.
Analyzed circuit complexity across finite fields and general fields with coefficient bit-complexity constraints.
Established the first efficient algorithm for computing tensor rank and optimal rank-one tensor decompositions given black-box access to tensors of super-constant rank.
Achieved a time complexity of poly(n, d, c) k^(k^(k^(k^O(k)))) for reconstructing both multilinear and set-multilinear Sigma^[k] Prod^[d] Sigma circuits.

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Implication

Theoretical analysis uncovers randomized algorithms for tensor decomposition and depth-3 circuit reconstruction, highlighting efficient polynomial reconstruction beyond constant rank.

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Peleg et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c7e0 https://doi.org/10.1007/s00037-026-00295-w

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Tensor reconstruction beyond constant rank

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Discussion

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Tensor reconstruction beyond constant rank

Key Points

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Implication

Discussion

Cite This Study

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