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

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.