Low-rank completion and recovery of correlation matrices - Ramlall, Chetan K (Division of Actuarial Science, 2020)
In the pursuit of efficient methods of dimension reduction for multi-factor correlation systems and for sparsely populated and partially observed matrices, the problem of matrix completion within a low-rank framework is of particular significance. This dissertation presents the methods of spectral completion and convex relaxation, which have been successfully applied to the particular problem of lowrank completion and recovery of valid correlation matrices. Numerical testing was performed on the classical exponential and noisy Toeplitz parametrisations and, in addition, to real datasets comprising of FX rates and stock price data. In almost all instances, the method of convex relaxation performed better than spectral methods and achieved the closest and best-fitted low-rank approximations to the true, optimal low-rank matrices (for some rank-n). Furthermore, a dependence was found to exist on which correlation pairs were used as inputs, with the accuracy of the approximations being, in general, directly proportional to the number of input correlations provided to the algorithms.