Tue, 21 May 2019

14:30 - 15:00
L5

A Model-Based Derivative-Free Approach to Black-Box Adversarial Examples in Deep Learning

Giuseppe Ughi
(Oxford)
Abstract

Neural Network algorithms have achieved unprecedented performance in image recognition over the past decade. However, their application in real world use-cases, such as self driving cars, raises the question of whether it is safe to rely on them.

We generally associate the robustness of these algorithms with how easy it is to generate an adversarial example: a tiny perturbation of an image which leads it to be misclassified by the Neural Net (which classifies the original image correctly). Neural Nets are strongly susceptible to such adversarial examples, but when the architecture of the target neural net is unknown to the attacker it becomes more difficult to generate these examples efficiently.

In this Black-Box setting, we frame the generation of an adversarial example as an optimisation problem solvable via derivative free optimisation methods. Thus, we introduce an algorithm based on the BOBYQA model-based method and compare this to the current state of the art algorithm.

Mon, 20 May 2019
15:45
L6

Rational cobordisms and integral homology

Paolo Aceto
(Oxford)
Abstract

We prove that every rational homology cobordism class in the subgroup generated
by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in
any other element in the same class. As a consequence we show that several natural maps to
the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility
condition between the determinants of certain 2-bridge knots and other knots in the same
concordance class. This is joint work with Daniele Celoria and JungHwan Park.

Mon, 13 May 2019
15:45
L6

On operads with homological stability

Tom Zeman
(Oxford)
Abstract

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined
topological operads with homological stability (OHS) and proved that the
group completion of an algebra over an OHS is weakly equivalent to an
infinite loop space.

In this talk, I shall outline a construction which to an algebra A over
an OHS associates a new infinite loop space. Under mild conditions on
the operad, this space is equivalent as an infinite loop space to the
group completion of A. This generalises a result of Wahl on the
equivalence of the two infinite loop space structures constructed by
Tillmann on the classifying space of the stable mapping class group. I
shall also talk about an application of this construction to stable
moduli spaces of high-dimensional manifolds in thesense of Galatius and
Randal-Williams.

Tue, 07 May 2019

14:00 - 14:30
L5

Sharp error bounds for Ritz vectors and approximate singular vectors

Yuji Nakatsukasa
(Oxford)
Abstract

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan sin-theta theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the sin-theta theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.

Tue, 30 Apr 2019

14:30 - 15:00
L3

Exponential integrators for stiff PDEs

Lloyd Nick Trefethen
(Oxford)
Abstract

Many time-dependent PDEs -- KdV, Burgers, Gray-Scott, Allen-Cahn, Navier-Stokes and many others -- combine a higher-order linear term with a lower-order nonlinear term.  This talk will review the method of exponential integrators for solving such problems with better than 2nd-order accuracy in time.

Tue, 04 Jun 2019

14:30 - 15:00
L5

The dual approach to non-negative super-resolution: impact on primal reconstruction accuracy

Bogdan Toader
(Oxford)
Abstract

We study the problem of super-resolution using TV norm minimisation, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. A practical approach is to solve the dual problem. In this talk, we study the stability of solutions with respect to the solutions to the dual problem. In particular, we establish a relationship between perturbations in the dual variable and the primal variables around the optimiser. This is achieved by applying a quantitative version of the implicit function theorem in a non-trivial way.

Tue, 04 Jun 2019

14:00 - 14:30
L5

Decentralised Sparse Multi-Task Regression

Dominic Richards
(Oxford)
Abstract

We consider a sparse multi-task regression framework for fitting a collection of related sparse models. Representing models as nodes in a graph with edges between related models, a framework that fuses lasso regressions with the total variation penalty is investigated. Under a form of generalised restricted eigenvalue assumption, bounds on prediction and squared error are given that depend upon the sparsity of each model and the differences between related models. This assumption relates to the smallest eigenvalue restricted to the intersection of two cone sets of the covariance matrix constructed from each of the agents' covariances. In the case of a grid topology high-probability bounds are given that match, up to log factors, the no-communication setting of fitting a lasso on each model, divided by the number of agents.  A decentralised dual method that exploits a convex-concave formulation of the penalised problem is proposed to fit the models and its effectiveness demonstrated on simulations. (Joint work with Sahand Negahban and Patrick Rebeschini)

Tue, 25 Jun 2019

15:30 - 16:30
L4

Global analytic geometry and Hodge theory

Kobi Kremnizer
(Oxford)
Abstract

In this talk I will describe how to make sense of the function $(1+t)^x$ over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz. 

Tue, 21 May 2019

15:30 - 16:30
L4

Equivariant Hilbert scheme of points on K3 surfaces and modular forms

Adam Gyenge
(Oxford)
Abstract

Let $X$ be a K3 surface and let $Z_X(q)$ be the generating series of the topological Euler characteristics of the Hilbert scheme of points on $X$. It is known that $q/Z_X(q)$ equals the discriminant form $\Delta(\tau)$ after the change of variables $q=e^{2 \pi i \tau}$. In this talk we consider the equivariant generalization of this result, when a finite group $G$ acts on $X$ symplectically. Mukai and Xiao has shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of $q/Z_X(q)$ in each of the 81 cases turns out to be a cusp form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.

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