Dense kernel matrices arise frequently in many scientific computing and machine learning applications. For large-scale applications, it is common to exploit low-rank property associated with kernel matrices to accelerate the computation. Various fast algorithms have been developed under certain assumptions on the dataset and the kernel function in the past. In this talk, we consider the low rank approximation problem in the most general setting. The proposed scheme selects a small subset of data points that accounts for the geometry of the given data and the numerical stability of the resulting factorization. The entire procedure is performed over the given data and does not require access to the kernel matrix or points outside the give data. The algorithm scales as O(r2(m + n)) for computing a rank-r approximation to an m × n kernel matrix. The proposed method outperforms existing state-of-the-art methods on various kernel functions and datasets ranging from three dimensions to over a hundred dimensions.

Join URL: https://uci.zoom.us/j/93282432501

In this talk, I will introduce a recent work in showing positivity of the Lyapunov exponent for Schr\"odinger operators with potentials generated by hyperbolic dynamics. Specifically, we showed that if the base dynamics is a subshift of finite type with an ergodic measure admitting a local product structure and if it has a fixed point, then for all nonconstant H\"older continuous potentials, the set of energies with zero Lyapunov exponent is a discrete set. If the potentials are locally constant or globally fiber bunched, then the set of zero Lyapunov exponent is finite. We also showed that for generic such potentials, we have full positivity in the general case and uniform postivity in the special cases. Such hyperbolic dynamics include expanding maps such as the doubling map on the unit circle, or Anosov diffeomorphism such as the Arnold's Cat map on 2-dimensional torus. It also can be applied to Markov chains whose special cases include the i.i.d. random variable. This is a joint with A. Avila and D. Damanik.