# Dr Matthew Thorpe

I can be emailed at m dot thorpe at maths dot cam dot ac dot uk.

## Career

**2017-current:**Research Fellow, Cantab Capital Institute for the Mathematics of Information, Department of Applied Mathematics and Theoretical Physics, University of Cambridge**2015-2017:**Postdoctoral Associate, Department of Mathematical Sciences, Carnegie Mellon University**2012-2015:**PhD Student, Mathematics Institute, University of Warwick

## Research

Matthew is a research fellow in the Cantab Capital Institute for the Mathematics of Information. His research interests are in discrete-to-continuum limits in graphical problems, particularly variational problems that arise from applications in machine learning, and optimal transport distances and their applications to signal and image analysis.

Before joining Cambridge Matthew was a postdoctoral associate at Carnegie Mellon University working with Dejan Slepčev on optimal transport problems (NSF grant number 1421502). Prior to that Matthew was a PhD student in the maths and statistics doctoral training centre (MASDOC) at Warwick under the supervision of Florian Theil and Adam Johansen. The focus of the PhD was on applying variational methods to statistical inference problems.

More information on his research is given below under research projects.

## Teaching

In the Lent term (2018-2019) I am teaching a part III course on the introduction to optimal transport. We meet on Mondays, Wednesdays and Fridays in MR14 at 12pm. Material can be found here:

- syllabus,
- lecture notes (I have now made all my planned changes - I will only update these further if mistakes & typos are brought to my attention) [small edits (marked in dark blue) 10/05/2019: pages 31, 34, 48 & 89],
- exam guidance,
- example sheet 1 questions and solutions,
- example sheet 2 questions and solutions,
- example sheet 3 questions and solutions,
- example sheet 4 questions and solutions; a cleaner solution to ex 4.1 (as suggested in class) can be found here.

The exercise classes are scheduled as follows

- exercise class 1, Thursday, 7th February, 4:00pm to 5:30pm, MR11,
- exercise class 2, Thursday, 21st February, 4:00pm to 5:30pm, MR11,
- exercise class 3, Tuesday, 5th March, 4:00pm to 5:30pm, MR13,
- exercise class 4/revision class, Friday, 17th May, 2:00pm to 3:30pm, MR5.

## Publications

### Preprints

- M. Thorpe and Y. van Gennip,
*Deep Limits of Residual Neural Networks*, 2018. Arxiv. - M. Thorpe and D. Slepčev,
*Transportation L*, 2017. Arxiv.^{p}Distances: Properties and Extensions

### Accepted

- M. M. Dunlop, D. Slepčev, A. M. Stuart and M. Thorpe,
*Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms*, Applied and Computational Harmonic Analysis, 2018. Arxiv. - D. Slepčev and M. Thorpe,
*Analysis of p-Laplacian Regularization in Semi-Supervised Learning*, SIAM Journal on Mathematical Analysis, 2019. Arxiv. - R. Cristoferi and M. Thorpe,
*Large Data Limit for a Phase Transition Model with the p-Laplacian on Point Clouds*, European Journal of Applied Mathematics, 2018. Arxiv.

### Published

- M. Thorpe and F. Theil,
*Asymptotic Analysis of the Ginzburg-Landau Functional on Point Clouds*, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 149(2), 387-427, 2019. Article. Arxiv. - M. Thorpe and A. M. Johansen,
*Pointwise Convergence in Probability of**General Smoothing Splines*, the Annals of the Institute of Statistical Mathematics, 70(4): 717-744, 2018. Article. Arxiv. - S. Park and M. Thorpe,
*Representing and Learning High Dimensional Data with the Optimal Transport Map from a Probabilistic Viewpoint*, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2018. Article. - M. Thorpe, S. Park, S. Kolouri, G. K. Rohde and D. Slepčev,
*A Transportation Lp Distance for Signal Analysis*, Journal of Mathematical Imaging and Vision, 59(2):187-210, 2017. Article. Arxiv. - S. Kolouri, S. Park, M. Thorpe, D. Slepčev and G. K. Rohde,
*Optimal Mass Transport: Signal Processing and Machine-Learning Applications*, IEEE Signal Processing Magazine, 34(4):43-59, 2017. Article. Arxiv. - M. Thorpe and A. M. Johansen,
*Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using k-Means Type Optimization*, Electronic Journal of Statistics, 10(2):3693-3722, 2016. Article. Arxiv. - M. Thorpe,
*Variational Methods for Geometric Statistical Inference*, PhD Thesis, 2015. - M. Thorpe, F. Theil, A. M. Johansen and N. Cade,
*Convergence of the k-Means Minimization Problem Using Gamma-Convergence*, SIAM Journal on Applied Mathematics, 75(6): 2444-2474, 2015. Article. Arxiv. - A. Gkiokas, A. Cristea and M. Thorpe,
*Self-Reinforced Meta Learning for Belief Generation*, Research and Development in Intelligent Systems XXXI, Springer International Publishing, 185-190, 2014. Article.

## Links to Colloborators

- Luca Calatroni
- Jeff Calder
- Riccardo Cristoferi
- Olly Crook
- Mihai Cucuringu
- Matthew Dunlop
- Nicolas Garcia Trillos
- Yves van Gennip
- Tim Hurst
- Adam Johansen
- Lisa Kreusser
- Sam Power
- Filip Rindler
- Gustavo Rohde
- Carola-Bibiane Schönlieb
- Dejan Slepčev
- Elizabeth Soileux
- Andrew Stuart
- He Sun
- Daniel Tenbrinck
- Florian Theil
- Kostas Zygalakis

## Other Links

- Maths and Statistics Doctoral Training Centre at the University of Warwick, MASDOC.
- Centre for Nonlinear Anaysis at Carnegie Mellon University, CNA.
- Cantab Capital Institute for the Mathematics of Information at the University of Cambridge, CCIMI.
- Cambridge Image Analysis, CIA.
- Isaac Newton Institute, INI.
- Turing Gateway to Mathematics, TGM.
- The Alan Turing Institute.
- The Smith Institute.
- NoMADS.

## Research Projects

### Large Data Limits in Graphical Modelling

Graphical models are a powerful data driven tool for extracting information from large data sets with a-priori unknown relationships. Given a graph *G=(V,E,W)* with nodes *V={x _{i}}_{i=1,...,n}*, edges

*E={e*

_{ij}}_{ij∈Γ}, and edge weights

*W={w*, a typical problem falls into one of three categories: regression, classification, or clustering. The latter two can be regarded as labelling problems (with and without training data respectively). A labelling problem concerns finding a function

_{ij}}_{∈Γ}*u:V→ K*where

*K*is the set of classes (usually discrete), and a regression problem concerns finding the function

*u:V→ ℝ*; that is, in some sense, the best fit to a training data set

*{(x*

_{i},y_{i})}_{i∈I}where

*I⊂{1,2,...,n}*. I am interested in problems where

*u*is defined as the solution to a variational problem, i.e.

*u*minimises

*E*for some functional

*E*. In particular I am interested in large data limits (when the number of data points

*n→ ∞*). Under appropriate normalisation and localisation we can often find a PDE limit. See references [9,10,11] for more details.

*Optimal Transport Distances for Image and Signal Analysis** *

*Optimal Transport Distances for Image and Signal Analysis*

Optimal transport can be used to define a distance (or psuedo distance) between images and signals. Let *μ* and *ν* be two images/signals which we interpret as a measure with domains *U* and *V* respectively (note that this formulation allows one to treat both discrete and continuous images/signals simultaneously). Optimal transport distances (in the Monge formulation) are based on the cost of transport maps *T:U→V* where *T* pushes the measure *μ* to *ν*. When there is more than one map *T* that pushes *μ* to *ν* (as will usually be the case) we select the optimal map. How one defines optimal gives rise to different optimal transport based distances. The most common is to use L^{p} costs and consider the map *T* which minimises *||*Id*-T|| _{Lp}* amongst all such transport maps. Examples of such distances include the Wasserstein or earth movers distance. The requirement that

*T*pushes the measure

*μ*to

*ν*necessitates that the images/signals are treated as measures. This is actually quite restrictive since one has to assume positivity, single channels (e.g. greyscale images), and unit mass. Ther are ad-hoc methods to deal with all these limitations, for example by renormalising, but at the loss of key properties, for example renormalising may dampen features. A framework I have been working on considers images/signals as a coupling of a function with a measure. The function contains the amplitude and the measure contains the support and can be used to emphasise certain features (often we just consider the uniform measure over the support). We define the distance between two images

*(f, μ)*and

*(g,ν)*as the optimal transport distance between the measure

*μ*raised to the graph of

*f*and

*ν*raised to the graph of

*g*. This formulation keeps key advantages of optimal transport whilst being applicable to functions hence being more widely accessible. The distance is called the TL

^{p}distance and my work in this area can be found in references [6,7,13], an overview of optimal transport and how it relates to signal processing can be found in [5].

* *

*Large Data Limits in Data Association-Smoothing Problems*

*Large Data Limits in Data Association-Smoothing Problems** *

Variational problems are well studied in statistics, for example maximum likelihood or maximum a-posteriori estimation. My interest is in infinite dimensional, i.e. functional, settings. One well known estimation technique is the k-means method. This can be adapted to solve the data association-smoothing problem. The data association-smoothing problem concerns recovering *k* curves *{y _{j}}_{j=1,...,k}*, where

*y*

_{j}:[0,1]*→ ℝ*, from a data set

*{(t*

_{i},x_{i})}_{i=1,...,n}. The data association refers to the fact that the data set is unlabelled, i.e. the association of each data point

*(t*to curve

_{i},x_{i})*y*

_{j}is unknown. With suitable modifications the k-means method can adapted to address this problem. More information can be found in references [2,4]. When

*k=1*there is no data association and the problem reduces to smoothing which is well studied in the context of splines. My results in this area can be found in [8].