Factorization-free Orthogonal Projection onto the Positive Semidefinite Cone with Composite Polynomial Filtering

TL;DR

This paper introduces a factorization-free, polynomial filtering-based method for projecting onto the PSD cone, enabling fast GPU implementation with low-precision arithmetic and outperforming traditional eigenvalue decomposition methods.

Contribution

The authors develop a novel composite polynomial filtering approach for PSD projection that avoids matrix factorization, significantly improving computational efficiency on GPUs.

Findings

Achieves a 10x speed-up over eigenvalue decomposition routines.

Maintains acceptable accuracy with low-precision arithmetic.

Efficiently computes projections for large-scale matrices.

Abstract

We propose a factorization-free method for orthogonal projection onto the positive semidefinite (PSD) cone, leveraging composite polynomial filtering. Inspired by recent advances in homomorphic encryption, our approach approximates the PSD cone projection operator using a carefully optimized composite polynomial evaluated exclusively via matrix-matrix multiplications. This approach enables efficient GPU implementations with low-precision arithmetic, significantly outperforming the classical PSD cone projection using state-of-the-art GPU-based eigenvalue decomposition solvers. Specifically, our method achieves a consistent relative error of $10^{-3}$ in half-precision arithmetic with only 22 matrix-matrix multiplications, providing roughly a $10\times$ speed-up over NVIDIA's cuSOLVER routines on various large-scale matrices. In single-precision arithmetic with emulation on B200 GPUs, our approach maintains competitive accuracy while achieving up to a $2\times$ speed-up. Consequently, for a $10,000 \times 10,000$ dense symmetric matrix, our method requires approximately $55$ ms in half-precision and $400$ ms in single-precision arithmetic on B200 GPUs. Integration into a first-order semidefinite programming solver confirms that our low-precision projections reliably yield solutions of moderate accuracy.