Accuracy of approximate projection to the semidefinite cone

P. J. Goulart, Y. Nakatsukasa and N. Rontsis

@article{GNR:2020,
  author = {P. J. Goulart and Y. Nakatsukasa and N. Rontsis},
  title = {Accuracy of approximate projection to the semidefinite cone},
  journal = {Linear Algebra and Its Applications},
  year = {2020},
  volume = {594},
  pages = {177-192},
  url = {https://doi.org/10.1016/j.laa.2020.02.014},
  doi = {10.1016/j.laa.2020.02.014}
}

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.