Approximate Projections onto the Positive Semidefinite Cone Using Randomization

Abstract: This paper presents two novel algorithms for approximately projecting symmetric matrices onto the Positive Semidefinite (PSD) cone using Randomized Numerical Linear Algebra (RNLA). Classical PSD projection methods rely on full-rank deterministic eigen-decomposition, which can be computationally prohibitive for large-scale problems. Our approach leverages RNLA to construct low-rank matrix approximations before projection, significantly reducing the required numerical resources. The first algorithm utilizes random sampling to generate a low-rank approximation, followed by a standard eigen-decomposition on this smaller matrix. The second algorithm enhances this process by introducing a scaling approach that aligns the leading-order singular values with the positive eigenvalues, ensuring that the low-rank approximation captures the essential information about the positive eigenvalues for PSD projection. Both methods offer a trade-off between accuracy and computational speed, supported by probabilistic error bounds. To further demonstrate the practical benefits of our approach, we integrate the randomized projection methods into a first-order Semi-Definite Programming (SDP) solver. Numerical experiments, including those on SDPs derived from Sum-of-Squares (SOS) programming problems, validate the effectiveness of our method, especially for problems that are infeasible with traditional deterministic methods.

• The paper introduces two randomized algorithms that efficiently approximate PSD cone projections, reducing the need for full-rank eigen-decompositions.
• It employs random sampling and a scaled approach to balance computational efficiency with approximation accuracy using RNLA techniques.
• Numerical tests on SDP solvers demonstrate significant computational savings for large-scale symmetric matrices, indicating effective applications in scientific computing.

Approximate Projections onto the Positive Semidefinite Cone Using Randomization

This paper introduces two algorithms for approximating projections of symmetric matrices onto the Positive Semidefinite (PSD) cone using techniques from Randomized Numerical Linear Algebra (RNLA). Traditional projections rely on full-rank eigen-decomposition, which becomes computationally expensive for large-scale matrices. The proposed methods aim to mitigate this cost by leveraging low-rank approximations.

Methodology

The authors present two distinct approaches:

1. Random Sampling Approach:
   • A random sampling technique generates a low-rank approximation of the matrix.
   • An eigen-decomposition is performed on this reduced-size matrix, thus decreasing computational complexity.
   • This method provides a balance between computational efficiency and approximation accuracy.
2. Scaled Random Sampling Approach:
   • Enhances the basic randomization by aligning leading-order singular values with positive eigenvalues.
   • A scaling factor is introduced, concentrating on capturing critical positive eigenvalues, thus improving the preservation of PSD properties.

Both algorithms offer probabilistic error bounds under spectral and Frobenius norms, facilitating a trade-off between speed and accuracy.

Numerical Results

The proposed methods were integrated into a first-order Semi-Definite Programming (SDP) solver. Tests, particularly on SDPs from Sum-of-Squares (SOS) programming, demonstrated significant effectiveness, especially compared to traditional methods which were infeasible for large instances.

Implications and Future Work

The introduction of efficient randomized projections reduces computational requirements for PSD projections, a common task in numerical linear algebra and convex optimization. The algorithms are particularly well-suited for scenarios where traditional methods become computationally prohibitive due to matrix size.

These advancements have the potential to impact various applications in scientific computing and data science, where large PSD matrix projections are required. Future developments could explore adaptive rank-revealing strategies, further enhancing scalability and robustness.

Conclusion

The paper provides a significant contribution to approximating PSD projections, showing that RNLA techniques can be effectively applied to large-scale problems. By sacrificing small amounts of precision, these methods offer substantial computational savings, paving the way for more extensive and complex problem-solving capabilities in numerical optimization and programming.