Related papers: Mathematical conjecture generation using machine i…

Modeling uncertainty in deep neural networks, despite recent important advances, is still an open problem. Bayesian neural networks are a powerful solution, where the prior over network weights is a design choice, often a normal…

In this paper, we introduce some new iterative optimisation algorithms on Riemannian manifolds and Hilbert spaces which have good global convergence guarantees to local minima. More precisely, these algorithms have the following properties:…

Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate…

Bayesian inference was once a gold standard for learning with neural networks, providing accurate full predictive distributions and well calibrated uncertainty. However, scaling Bayesian inference techniques to deep neural networks is…

When optimization theorists consider optimization problems in infinite dimensional spaces, they need to deal with closed convex subsets(usually cones) which mostly have empty interior. These subsets often prevent optimization theorists from…

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer…

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of…

Machine learning provides algorithms that can learn from data and make inferences or predictions on data. Bayesian networks are a class of graphical models that allow to represent a collection of random variables and their condititional…

The present document is the draft of a book which presents an introduction to infinite-dimensional differential geometry beyond Banach manifolds. As is well known the usual calculus breaks down in this setting. Hence, we replace it by the…

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…

Any act of problem-solving combines prior knowledge, local search, and a third element that is less often discussed: the extraction of information from search to update understanding. I propose a model of mathematical problem-solving as a…

We introduce a novel approach to inference on parameters that take values in a Riemannian manifold embedded in a Euclidean space. Parameter spaces of this form are ubiquitous across many fields, including chemistry, physics, computer…

Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science and statistical physics. While there has been success evidencing these gaps from the failure of restricted…

The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…

A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to…

A stochastic algorithm is proposed, finding the set of generalized means associated to a probability measure on a compact Riemannian manifold M and a continuous cost function on the product of M by itself. Generalized means include p-means…

Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call "mathematical data science". In this paradigm, one studies mathematical objects collectively rather than…

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a…

Across machine learning (ML) sub-disciplines, researchers make explicit mathematical assumptions in order to facilitate proof-writing. We note that, specifically in the area of fairness-accuracy trade-off optimization scholarship, similar…

We propose probabilistic task modelling -- a generative probabilistic model for collections of tasks used in meta-learning. The proposed model combines variational auto-encoding and latent Dirichlet allocation to model each task as a…