# Tim Sullivan

### Welcome!

I am Assistant Professor in Predictive Modelling in the Mathematics Institute and School of Engineering at the University of Warwick and Research Group Leader for Uncertainty Quantification at the Zuse Institute Berlin. I have wide interests in uncertainty quantification the broad sense, understood as the meeting point of numerical analysis, applied probability and statistics, and scientific computation. On this site you will find information about how to contact me, my research, publications, and teaching activities.

### Joining the University of Warwick

It is a pleasure to announce that I have accepted an Assistant Professorship in Predictive Modelling at the University of Warwick, to be held jointly between the Mathematics Institute and the School of Engineering, in particular the Warwick Centre for Predictive Modelling.

Published on Monday 6 April 2020 at 14:00 UTC #warwick #wcpm

### Birzhan Ayanbayev joins the UQ Group

It is a pleasure to announce that Birzhan Ayanbayev will join the UQ research group as a postdoctoral researcher with effect from 28 February 2020. He will be working on the DFG-funded project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems”.

Published on Friday 28 February 2020 at 12:00 UTC #group #job #dfg #map-estimators #ayanbayev

### Geodesic analysis in Kendall's shape space now in J. Math. Imaging Vis.

The article “Geodesic analysis in Kendall's shape space with epidemiological applications” by Esfandiar Nava-Yazdani, Christoph von Tycowicz, Hans-Christian Hege, and myself has just appeared online in the Journal of Mathematical Imaging and Vision.

E. Nava-Yazdani, H.-C. Hege, T. J. Sullivan, and C. von Tycowicz. “Geodesic analysis in Kendall's shape space with epidemiological applications.” Journal of Mathematical Imaging and Vision 1–11, 2020. doi:10.1007/s10851-020-00945-w

Abstract. We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application, we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.

Published on Tuesday 18 February 2020 at 14:00 UTC #publication #ch15 #shape-trajectories #nava-yazdani #von-tycowicz #hege

### Preprint: A rigorous theory of conditional mean embeddings

Ilja Klebanov, Ingmar Schuster, and I have just uploaded a preprint of our recent work “A rigorous theory of conditional mean embeddings” to the arXiv. In this work we take a close mathematical look at the method of conditional mean embedding. In this approach to non-parametric inference, a random variable $$Y \sim \mathbb{P}_{Y}$$ in a set $$\mathcal{Y}$$ is represented by its kernel mean embedding, the reproducing kernel Hilbert space element

$$\displaystyle \mu_{Y} = \int_{\mathcal{Y}} \psi(y) \, \mathrm{d} \mathbb{P}_{Y} (y) \in \mathcal{G},$$

and conditioning with respect to an observation $$x$$ of a related random variable $$X \sim \mathbb{P}_{X}$$ in a set $$\mathcal{X}$$ with RKHS $$\mathcal{H}$$ is performed using the Woodbury formula

$$\displaystyle \mu_{Y|X = x} = \mu_Y + (C_{XX}^{\dagger} C_{XY})^\ast \, (\varphi(x) - \mu_X) .$$

Here $$\psi \colon \mathcal{Y} \to \mathcal{G}$$ and $$\varphi \colon \mathcal{X} \to \mathcal{H}$$ are the canonical feature maps and the $$C$$'s denote the appropriate centred (cross-)covariance operators of the embedded random variables $$\psi(Y)$$ in $$\mathcal{G}$$ and $$\varphi(X)$$ in $$\mathcal{H}$$.

Our article aims to provide rigorous mathematical foundations for this attractive but apparently naïve approach to conditional probability, and hence to Bayesian inference.

Abstract. Conditional mean embeddings (CME) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the kernel mean embeddings of the respective probability distributions. Both centered and uncentered covariance operators have been used to define CMEs in the existing literature. In this paper, we develop a mathematically rigorous theory for both variants, discuss the merits and problems of either, and significantly weaken the conditions for applicability of CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian conditioning in Hilbert spaces.

Published on Tuesday 3 December 2019 at 07:00 UTC #publication #preprint #mathplus #tru2 #rkhs #mean-embedding #klebanov #schuster

### Postdoc position: Analysis of MAP estimators

There is still an opening for a full-time two-year postdoctoral research position in the UQ group at the Freie Universität Berlin. This position will be associated to the project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems” funded by the DFG.

This project aims to advance the state of the art in rigorous mathematical understanding of MAP estimators in infinite-dimensional statistical inverse problems. In particular, the research in this project will connect the “small balls” approach of Dashti et al. (2013) to the calculus of variations and hence properly link the variational and fully Bayesian points of view on inverse problems. This is an exciting opportunity for someone well-versed in the calculus of variations and tools such as Γ-convergence to make an impact on fundamental questions of non-parametric statistics and inverse problems or, vice versa, for someone with a statistical inverse problems background to advance the rigorous state of the art for such methods.

Prospective candidates are encouraged to contact me with informal enquiries. Formal applications are to be sent by post or email, by 23 December 2019, under the heading MAP-Analysis, and should include a cover letter, a scientific CV including list of publications and research statement, and the contact details of two professional references.

Published on Monday 25 November 2019 at 08:00 UTC #group #job #fu-berlin #inverse-problems #dfg #map-estimators