# Recent seminars

## 20/01/2022, Thursday, 17:00–18:00 Europe/Lisbon — Online

Anders Hansen, Faculty of Mathematics and Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Why things don’t work — On the extended Smale's 9th and 18th problems (the limits of AI) and methodological barriers

The alchemists wanted to create gold, Hilbert wanted an algorithm to solve Diophantine equations, researchers want to make deep learning robust in AI, MATLAB wants (but fails) to detect when it provides wrong solutions to linear programs etc. Why does one not succeed in so many of these fundamental cases? The reason is typically methodological barriers. The history of science is full of methodological barriers — reasons for why we never succeed in reaching certain goals. In many cases, this is due to the foundations of mathematics. We will present a new program on methodological barriers and foundations of mathematics, where — in this talk — we will focus on two basic problems: (1) The instability problem in deep learning: Why do researchers fail to produce stable neural networks in basic classification and computer vision problems that can easily be handled by humans — when one can prove that there exist stable and accurate neural networks? Moreover, AI algorithms can typically not detect when they are wrong, which becomes a serious issue when striving to create trustworthy AI. The problem is more general, as for example MATLAB's linprog routine is incapable of certifying correct solutions of basic linear programs. Thus, we’ll address the following question: (2) Why are algorithms (in AI and computations in general) incapable of determining when they are wrong? These questions are deeply connected to the extended Smale’s 9th and 18th problems on the list of mathematical problems for the 21st century.

### Video

Hansen slides.pdf

## 13/01/2022, Thursday, 17:00–18:00 Europe/Lisbon — Online

Dan Roberts, MIT, Center for Theoretical Physics
The Principles of Deep Learning Theory

Deep learning is an exciting approach to modern artificial intelligence based on artificial neural networks. The goal of this talk is to provide a blueprint — using tools from physics — for theoretically analyzing deep neural networks of practical relevance. This task will encompass both understanding the statistics of initialized deep networks and determining the training dynamics of such an ensemble when learning from data.

This talk is based on a book, The Principles of Deep Learning Theory, co-authored with Sho Yaida and based on research also in collaboration with Boris Hanin. It will be published next year by Cambridge University Press.

### Video

Roberts slides.pdf

## 09/12/2021, Thursday, 17:00–18:00 Europe/Lisbon — Online

Pier Luigi Dragotti, Department of Electrical and Electronic Engineering, Imperial College, London
Computational Imaging for Art investigation and for Neuroscience

The revolution in sensing, with the emergence of many new imaging techniques, offers the possibility of gaining unprecedented access tothe physical world, but this revolution can only bear fruit through the skilful interplay between the physical and computational worlds. This is the domain of computational imaging which advocates that, to develop effective imaging systems, it will be necessary to go beyond the traditional decoupled imaging pipeline where device physics, image processing and the end-user application are considered separately. Instead, we need to rethink imaging as an integrated sensing and inference model. In this talk we cover two research areas where computational imaging is likely to have an impact.

We first focus on the heritage sector which is experiencing a digital revolution driven in part by the increasing use of non-invasive, non-destructive imaging techniques. These new imaging methods provide a way to capture information about an entire painting and can give us information about features at or below the surface of the painting. We focus on Macro X-Ray Fluorescence (XRF) scanning which is a technique for the mapping of chemical elements in paintings. After describing in broad terms the working of this device, a method that can process XRF scanning data from paintings is introduced. The method is based on connecting the problem of extracting elemental maps in XRF data to Prony's method, a technique broadly used in engineering to estimate frequencies of a sum of sinusoids. The results presented show the ability of our method to detect and separate weak signals related to hidden chemical elements in the paintings. We then discuss results on the Leonardo’s The Virgin of the Rocks and show that our algorithm is able to reveal, more clearly than ever before, the hidden drawings of a previous composition that Leonardo then abandoned for the painting that we can now see.

In the second part of the talk, we focus on two-photon microscopy and neuroscience. To understand how networks of neurons process information, it is essential to monitor their activity in living tissue. Multi-photon microscopy is unparalleled in its ability to image cellular activity and neural circuits, deep in living tissue, at single-cell resolution. However, in order to achieve step changes in our understanding of brain function, large-scale imaging studies of neural populations are needed and this can be achieved only by developing computational tools that can enhance the quality of the data acquired and can scan 3-D volumes quickly. In this talk we introduce light-field microscopy and present a method to localize neurons in 3-D. The method is based on the use of proper sparsity priors, novel optimization strategies and machine learning.

This is joint work with A. Foust, P. Song, C. Howe, H. Verinaz, J. Huang and Y.Su from Imperial College London, and C. Higgitt and N. Daly from The National Gallery in London

### Video

Dragotti slides.pdf

## 02/12/2021, Thursday, 17:00–18:00 Europe/Lisbon — Online

Soledad Villar, Mathematical Institute for Data Science at Johns Hopkins University
Equivariant machine learning structure like classical physics

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality $d$. The key observation is that nonlinear $O(d)$-equivariant (and related-group-equivariant) functions can be expressed in terms of a lightweight collection of scalars–scalar products and scalar contractions of the scalar, vector, and tensor inputs. These results demonstrate theoretically that gauge-invariant deep learning models for classical physics with good scaling for large problems are feasible right now.