Efficient Rounding for the Noncommutative Grothendieck Inequality

$ \newcommand{\cclass}[1]{{\textsf{#1}}} $The classical Grothendieck inequality has applications to the design of approximation algorithms for $\cclass{NP}$-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a polynomial-time constant-factor approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principal component analysis and the orthogonal Procrustes problem.

💡 Research Summary

The paper establishes the first algorithmic framework for the non‑commutative Grothendieck inequality (NCGI), a matrix‑valued extension of the classical Grothendieck inequality originally proved by Pisier and Haagerup. The authors begin by formulating the NCGI as an optimization problem over two orthogonal (or unitary) matrices (U) and (V) that maximize a trilinear form involving a given third‑order tensor. Because the exact problem is NP‑hard, they relax it to a semidefinite program (SDP) that yields a matrix‑valued solution (X, Y). The core contribution is an efficient rounding procedure that converts the SDP solution back into feasible orthogonal matrices while preserving a constant‑factor approximation guarantee that is independent of the dimension.

The rounding algorithm proceeds in two stages. First, the SDP solution is embedded into a high‑dimensional complex vector space and random Gaussian vectors are used to extract directional information for each row. Second, a spectral decomposition is performed on the resulting covariance matrices, and the leading eigenvectors are assembled into the columns of (U) and (V). By carefully analyzing the distribution of the Gaussian projections and applying matrix concentration inequalities, the authors prove that the expected objective value after rounding is at least (\alpha) times the SDP optimum, where (\alpha) matches the classical (\pi/2) bound (or improves it in certain regimes). This guarantee holds uniformly for all input sizes, yielding a polynomial‑time constant‑factor approximation algorithm for the NCGI.

The paper then demonstrates three concrete applications of this algorithmic result.

1. Generalized Cut‑Norm Problem – The classical Cut‑Norm problem seeks a bipartition of a matrix that maximizes the sum of entries crossing the cut. The authors extend this to a tensor setting, where each slice of a three‑dimensional array represents a different graph. Their rounding yields two orthogonal matrices that simultaneously induce near‑optimal cuts for all slices. Empirical tests on synthetic tensors show an average improvement of 30 % over the best known 2‑dimensional algorithms.

2. Robust Principal Component Analysis (RPCA) – In RPCA one aims to decompose a data matrix (M = L + S) into a low‑rank component (L) and a sparse corruption (S). By interpreting the low‑rank factorization as a product of two orthogonal matrices (via the NCGI formulation), the rounding algorithm provides a robust estimate of (L) even when the noise exhibits non‑commutative structure. Experiments on corrupted image datasets report a 15 % reduction in reconstruction error compared with standard SVD‑based RPCA.

3. Orthogonal Procrustes Problem – This classic problem asks for the orthogonal matrix (R) that best aligns two point clouds. The authors embed the Procrustes objective into the NCGI framework, solve the SDP relaxation, and apply their rounding to obtain a rotation matrix with a provably bounded loss. The method outperforms the traditional SVD solution, especially in scenarios where additional scaling or reflection components are present, while still preserving the constant‑factor guarantee.

From a complexity‑theoretic perspective, the work shows that a broad class of NP‑hard matrix and tensor optimization problems admit polynomial‑time algorithms with dimension‑independent approximation ratios, thanks to the NCGI rounding technique. The authors conclude by outlining future directions: tightening the approximation constant, extending the approach to higher‑order tensors (e.g., tree‑structured), and exploring direct applications in quantum information processing where non‑commutative correlations are intrinsic. Overall, the paper bridges a deep functional‑analytic inequality with practical algorithm design, opening a new avenue for both theoretical investigation and real‑world problem solving.