Statistics Department Hosted Seminar Series by Professor Yves Atchade: Jie Chen, Linear-Complexity Kernel Matrix Computation and Its Use in Gaussian Processes
Professor Yves Atchade is hosting a bi-weekly seminar series called "Statistical Computing" which will discuss how statistical methods are implemented, and to explore computational techniques with potential applications in statistics.
Gaussian processes are the cornerstone of statistical analysis in many application areas. However, most of the applications are limited by the expensive dense matrix computations with the covariance matrix (e.g., the Cholesky factorization). With several decades of constant improvement in the implementation of numerical linear algebra libraries, dense matrices still face the fundamental scalability barrier---storage is O(n^2) and most of the required computation is O(n^2) to O(n^3)---which limits their use for a large n. We exploit a recursively low-rank property ubiquitously seen in covariance matrices and develop a data structure that requires only O(n) storage. Moreover, the structure enables the development of O(n) algorithms for many matrix operations of practical interests, including matrix-vector multiplication, matrix inversion, determinant calculation, and square-root factorization. We demonstrate numerical behavior of the algorithms and their use in Gaussian process analysis.