Sample Complexity of Low-rank Tensor Recovery from Uniformly Random Entries
Hiroki Hamaguchi, Shin-ichi Tanigawa
TL;DR
This work establishes an information-theoretic bound for exact recovery of generic high-order tensors from a small number of uniformly random entries. By embedding the tensor completion problem into a graph-rigidity framework via Segre varieties and $d$-secants, it leverages sharp thresholds for Erdős–Rényi hypergraphs and identifiability criteria to show that a dimension $n\log n + d n\log\log n + o(n\log\log n)$ suffices (and is tight up to the second term) for constant rank $d$ and order $k$. The approach reveals a deep connection between algebraic geometry, combinatorial rigidity, and tensor analysis, providing a rigorous information-theoretic limit that clarifies gaps with computationally efficient methods. The results hinge on geometric projections of Segre varieties and introduce a polymatroid framework to certify $d$-identifiability, with potential implications for understanding limits of polynomial-time tensor completion.
Abstract
We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and $k$ are constant and $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The bound is tight up to the coefficient of the second leading term and improves on the existing $O(n^{\frac{k}{2}}{\rm polylog}(n))$ upper bound for order $k$ tensors. The bound is obtained by showing that the projection of the Segre variety to a random axis-parallel linear subspace preserves $d$-identifiability with high probability if the dimension of the subspace is $n\log n+dn\log \log n +o(n\log \log n) $ and $n$ is sufficiently large.