Monday, November 23, 2009
Applied and Computational Mathematics Seminar: Matrix Perturbation and Manifold-based Dimension Reduction.
Mon, 11/23/2009 - 1:00pm, Skiles 255
Xiaoming Huo , Georgia Tech (School of ISyE), email Organizer: Sung Ha Kang
Many algorithms were proposed in the past ten years on utilizing manifold structure for dimension reduction. Interestingly, many algorithms ended up with computing for eigen-subspaces. Applying theorems from matrix perturbation, we study the consistency and rate of convergence of some manifold-based learning algorithm. In particular, we studied local tangent space alignment (Zhang & Zha 2004) and give a worst-case upper bound on its performance. Some conjectures on the rate of convergence are made. It's a joint work with a former student, Andrew Smith.
Geometry Topology Seminar: Geometry, computational complexity and algebraic number fields
Mon, 11/23/2009 - 2:00pm, Skiles 269
Hong-Van Le, Mathematical Institute of Academy of Sciences of the Czech Republic Organizer: Thang Le
In 1979 Valiant gave algebraic analogs to algorithmic complexity problem such as $P \not = NP$. His central conjecture concerns the determinantal complexity of the permanents. In my lecture I shall propose geometric and algebraic methods to attack this problem and other lower bound problems based on the elusive functions approach by Raz. In particular I shall give new algorithms to get lower bounds for determinantal complexity of polynomials over $Q$, $R$ and $C$.
Algebra Seminar: Certified numerical polynomial homotopy continuation
Mon, 11/23/2009 - 3:30pm, Skiles 255
Anton Leykin, Georgia Tech Organizer: Matt Baker
This talk will start with an introduction to the area of numerical algebraic geometry. The homotopy continuation algorithms that it currently utilizes are based on heuristics: in general their results are not certified. Jointly with Carlos Beltran, using recent developments in theoretical complexity analysis of numerical computation, we have implemented a practical homotopy tracking algorithm that provides the status of a mathematical proof to its approximate numerical output.