New Hardness Results for Low-Rank Matrix Completion

The low-rank matrix completion problem asks whether a given real matrix with missing values can be completed so that the resulting matrix has low rank or is close to a low-rank matrix. The completed matrix is often required to satisfy additional structural constraints, such as positive semi-definiteness or a bounded infinity norm. The problem arises in various research fields, including machine learning, statistics, and theoretical computer science, and has broad real-world applications. This paper presents new $\mathsf{NP}$-hardness results for low-rank matrix completion problems. We show that for every sufficiently large integer $d$ and any real number $\varepsilon \in [ 2^{-O(d)},\frac{1}{7}]$, given a partial matrix $A$ with exposed values of magnitude at most $1$ that admits a positive semi-definite completion of rank $d$, it is $\mathsf{NP}$-hard to find a positive semi-definite matrix that agrees with each given value of $A$ up to an additive error of at most $\varepsilon$, even when the rank is allowed to exceed $d$ by a multiplicative factor of $O (\frac{1}{\varepsilon ^2 \cdot \log(1/\varepsilon)} )$. This strengthens a result of Hardt, Meka, Raghavendra, and Weitz (COLT, 2014), which applies to multiplicative factors smaller than $2$ and to $\varepsilon$ that decays polynomially in $d$. We establish similar $\mathsf{NP}$-hardness results for the case where the completed matrix is constrained to have a bounded infinity norm (rather than be positive semi-definite), for which all previous hardness results rely on complexity assumptions related to the Unique Games Conjecture. Our proofs involve a novel notion of nearly orthonormal representations of graphs, the concept of line digraphs, and bounds on the rank of perturbed identity matrices.