Online learning with matrix parameters
Manfred K. Warmuth, Dima Kuzmin · 2009
This thesis develops and analyzes several online learning algorithms that maintain a density matrix as their parameter. A density matrix is a symmetric positive definite matrix of trace one. Its eigenvalues form a probability vector and it can be seen as a generalization of finite probability distribution. In Chapter 3, density matrices are used to maintain uncertainty information over directions and an algorithm is developed that tracks the direction of smallest variance. In Chapter 4 this algorithm is extended to maintain uncertainty information over k-dimensional subspaces, which results in an online PCA algorithm. Finally, Chapter 5 considers the analogy between probability vectors and density matrices in more detail, and drawing on ideas from quantum mechanics develops a generalized probability calculus for density matrices. Central to this calculus is a certain generalization of Bayes rule to the matrix domain.