Harald Oberhauser

  Professor of Mathematics and Tutorial Fellow
  Mathematical Institute
  University of Oxford
  Email oberhauser at maths.ox.ac.uk
  Phone +44 1865 615176
  Address Mathematical Institute, Andrew Wiles Building, Woodstock Road, OX2 6GG

Research

I am interested in probability theory, stochastic processes, and their applications. In particular, I enjoy developing ideas from pure mathematics into tools for applications, and vice-versa, in developing mathematics inspired by questions that arise in applications. Most of my recent research does this in situations where stochastic analysis intersects with statistics, machine learning, algebra, geometry, or topology.

Recent Stuff (in reversed chronological order of the arXiv posting)

  • Sampling-based Nyström Approximation and Kernel Quadrature with Satoshi Hayakawa, Terry Lyons. ICML 2023
  • Hypercontractivity meets Random Convex Hulls with Satoshi Hayakawa, Terry Lyons. Proc. Royal Society A 2023+
  • Fast Bayesian Inference via Kernel Recombination with Masaki Adachi, Satoshi Hayakawa, Martin Jørgensen, Michael A. Osborne. NeuRIPS 2022
  • Capturing Graphs with Hypo-Elliptic Diffusions with Csaba Toth, Darrick Lee, Celia Hacker. NeuRIPS 2022
  • A Topological Approach to Mapping Space Signatures with Chad Giusti, Darrick Lee, Vidit Nanda. Submitted
  • Proper Scoring Rules, Gradients, Divergences, and Entropies for Paths and Time Series with Patric Bonnier. Submitted
  • Markov Chain Approximations to SDEs by Recombination on Lattice Trees with Francesco Cosentino, Alessandro Abate. Submitted
  • Tangent Space and Dimension Estimation with the Wasserstein Distance with Uzu Lim, Vidit Nanda. Submitted
  • Positively Weighted Kernel Quadrature via Subsampling with Satoshi Hayakawa, Terry Lyons. NeuRIPS 2022
  • Grid-Free Computation of Probabilistic Safety with Malliavin Calculus with Francesco Cosentino, Alessandro Abate. IEEE Trans. on Aut. Control 2023+
  • Neural SDEs as Infinite-Dimensional GANs with Patrick Kidger, James Foster, Xuechen Li, Terry Lyons. ICML 2021
  • Nonlinear Independent Component Analysis for Continuous Time Signals with Alexander Schell. Annals of Statistics 2023+
  • Estimating the Probability that a Vector is in the Convex Hull with Satoshi Hayakawa, Terry Lyons. Prob. Theory and Rel. Fields 2023+
  • The shifted ODE method for underdamped Langevin MCMC with James Foster, Terry Lyons. Submitted
  • Seq2Tens: An Efficient Representation of Sequences by Low-Rank Tensor Projections with Csaba Toth, Patric Bonnier. ICLR 2021
  • Carathéodory Sampling for Stochastic Gradient Descent with Francesco Cosentino, Alessandro Abate. Submitted
  • A Randomized Algorithm to Reduce the Support of Discrete Measures with Francesco Cosentino, Alessandro Abate. NeuRIPs 2020 (spotlight paper)
  • Adapted Topologies and Higher Rank Signatures with Chong Liu, Patric Bonnier. Annals of Applied Probability 2022+
  • Signature Cumulants, Ordered Partitions, and Independence of Stochastic Processes with Patric Bonnier. Bernoulli 2019
  • Bayesian Learning from Sequential Data using Gaussian Processes with Signature Covariances with Csaba Toth. ICML 2020
  • A Free Boundary Characterisation of the Root Barrier for Markov Processes with Paul Gassiat, Christina Zou. Prob. Theory and Rel. Fields 2021
  • An optimal polynomial approximation of Brownian motion with James Foster. SIAM Journal on Numerical Analysis 2020
  • Signature Moments to Characterize Laws of Stochastic Processes with Ilya Chevyrev. Journal of Machine Learning Research 2022
  • Persistence Paths and Signature Features in Topological Data Analysis with Ilya Chevyrev, Vidit Nanda. Trans. Pattern Anal. and Machine Int. 2020
  • Kernels for Sequentially Ordered Data with Franz Kiraly. Journal of Machine Learning Research 2019
  • ...for earlier stuff see arXiv

    • Software used in some of the above papers

  • KSig a package for GPU-accelerated computation of the signature kernel and MMD for stochastic processes. Written by Csaba Toth
  • Python code for constructing Positively weighted Kernel Quadrature Formulas. Written by Satoshi Hayakawa
  • Python code for measure reduction via recombination. Written by Francesco Cosentino

    • Supervision

    DPhil (=PhD) students James Foster (DPhil 20), Francesco Cosentino (DPhil'21), Alexander Schell (DPhil'22), Patric Bonnier, Csaba Toth, Uzu Lim, Satoshi Hayakawa
    Postdoctoral Researchers James Foster (until 09/22), Darrick Lee

    I am always looking for prospective DPhil (PhD) students and postdoctoral researcher with shared interests. There are two standard routes for DPhil applications: the CDT in Random Dynamical Systems and the usual DPhil program in mathematics. Both result in a DPhil degree but the CDT has several courses in the first year. You are welcome to get in touch before you apply but please be aware that application deadlines are relatively early. For postdoctoral positions, there are several funding possibilities and it's essential to get in touch early since this has to go through internal Oxford admin before we can submit to a funding body.

    Teaching

    HT22 C8.2 Stochastic Analysis and Partial Differential Equations

    Affiliations

    I'm a Tutorial Fellow at St. Hugh's College. My research is supported by DataSig, the Turing Institute, and CIMDA.