An inductive bias from quantum mechanics: learning order effects with non-commuting measurements

There are two major approaches to building good machine learning algorithms: feeding lots of data into large models, or picking a model class with an ''inductive bias'' that suits the structure of the data. When taking the second approach as a starting point to design quantum algorithms for machine learning, it is important to understand how mathematical structures in quantum mechanics can lead to useful inductive biases in quantum models. In this work, we bring a collection of theoretical evidence from the Quantum Cognition literature to the field of Quantum Machine Learning to investigate how non-commutativity of quantum observables can help to learn data with ''order effects'', such as the changes in human answering patterns when swapping the order of questions in a survey. We design a multi-task learning setting in which a generative quantum model consisting of sequential learnable measurements can be adapted to a given task -- or question order -- by changing the order of observables, and we provide artificial datasets inspired by human psychology to carry out our investigation. Our first experimental simulations show that in some cases the quantum model learns more non-commutativity as the amount of order effect present in the data is increased, and that the quantum model can learn to generate better samples for unseen question orders when trained on others - both signs that the model architecture suits the task

Modelling arithmetic strategies

This thesis examines children's arithmetic strategies and their relation to the concepts of commutativity and associativity. Two complementary methods were used in this research: empirical studies and computational models.Empirical studies were carried out to identify the strategies children used for solving problems like 3 + 4, and 3 + 4 + 7, and the conceptual knowledge associated with them. Their understanding of subtraction problems where the minuend is less than the subtrahend (e.g. 6 - 8) was also considered. A study with 105 subjects revealed a variety of strategies and information about children's knowledge of commutativity and associativity. Four levels of performance of commutativity were also identified. A longitudinal study was carried out with 12 children in order to obtain details of children's changes in strategy, and to double check the results obtained in the main study. The strategies observed to be used by children over a 20 month period parallel those found in previous studies, which show a general transition to more efficient methods. However, the longitudinal study revealed that development of such arithmetic strategies is a slow process. Furthermore, the studies indicated that knowledge of commutativity is a prerequisite for associativity.Models of the observed strategies have been implemented in the form of production rules in a computer program called PALM. The process of implementation highlighted features of children's problem solving that had not been 'detected during the studies.In addition to models that describe the space of strategies, a model of learning has been implemented for the transition from procedural knowledge of commutativity to that of associativity. The model is capable of generalizing its inbuilt knowledge, for instance, its ability to solve 2-term arithmetic expressions, to allow it to solve more complex problems, such as 3-term arithmetic expressions. A further model has been constructed for learning arithmetic strategies that are more efficient than those already represented in the program. It learns specific rules by adding conditions for efficient problem solving to its previous general rules

Mean Field Limits for Interacting Diffusions in a Two-Scale Potential

In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in~\cite{DuncanPavliotis2016}. We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques