Wednesday, March 29, 2023
Time  Items 

All day 

1pm 
03/29/2023  1:00pm Gaussian process regression is widely used in geostatistics, timeseries analysis, and machine learning. It infers an unknown continuous function in a principled fashion from noisy measurements at $N$ scattered data points. The prior on the function is Gaussian, with covariance given by some userchosen translationally invariant kernel. Yet $N$ has been limited to about $10^6$, even with modern lowrank methods. Focusing on low spatial dimension (13), we present a GP regression method using kernel approximation by an equispaced quadrature grid in the Fourier domain. This enables the iterative solution of a smaller Toeplitz linear system, exploiting both the FFT and the nonuniform FFT to give ${\mathcal O}(N)$ cost. The result is often one to two orders of magnitude faster than state of the art methods, and enables cheap massivescale regressions. For example, for a 2D Matérn3/2 kernel and $N = 10^9$ points, the posterior mean function is found to 3digit accuracy in two minutes on a desktop. Joint work with Philip Greengard (Columbia) and Manas Rachh (Flatiron Institute) Location:
AKW 200

4pm 
03/29/2023  4:15pm Borel and Dwork gave conditions on when a nice power series with rational number coefficients comes from a rational function in terms of meromorphic convergence radii at all places. Such a criterion was used in Dwork’s proof of the rationality of zeta functions of varieties over finite fields. Later, the work of Andre, Bost and many others generalized the rationality criterion of Borel–Dwork and deduced many applications in the arithmetic of differential equations and elliptic curves. In this talk, we will discuss some further refinements and generalizations of the criteria of Andre and Bost and their applications to the unbounded denominators conjecture for modular forms, and irrationality of 2adic zeta value at 5 and some other linear independence problems. This is joint work with Frank Calegari and Vesselin Dimitrov. Location:
LOM 214
