Orbifold equivalent potentials

To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numerical invariants, the left and right quantum dimensions. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories

Binary matrix factorisation and completion via integer programming

Binary matrix factorisation is an essential tool for identifying discrete patterns in binary data. In this paper we consider the rank-k binary matrix factorisation problem (k-BMF) under Boolean arithmetic: we are given an n × m binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension n × k and k × m respectively, which minimise the distance between X and the Boolean product of A and B in the squared Frobenius distance. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times the zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations

Constrained Low-Rank Matrix/Tensor Factorisation

University of Technology Sydney. Faculty of Engineering and Information Technology.Constrained low-rank matrix and tensor factorisation (MF/TF) have been widely used in machine learning and data analytics. Studies on the way of modelling constraints and the solution of optimisation task in general can provide theoretical supports for applications like image clustering, recommender systems and data compression. This thesis studies three algorithms of constrained low-rank MF/TF. Imposing constraints on each feature vector of factor matrices is a common practice in many constrained low-rank MF algorithms. However, in many real scenarios, the relationships among features can influence the factorisation results as well. In order to better characterise the relationships among features, a novel MF algorithm, Relative Pairwise Relationship Constrained Non-negative Matrix Factorisation, is proposed. It places soft constraints over relative pairwise distances amongst features as regularisations to retain expected relationships after factorisation. It conforms to the so-called \multiplicative update rules" and detailed convergence proofs are provided. Experiments on both synthetic and real datasets have verified that imposing such constraints can keep most expected relationships unchanged after factorisation. Directly adopted on tensor data, low-rank TF can effectively avoid the information loss caused by matricisation. The relationships among features of factor matrices in TF have practical meanings in many real scenarios. To describe such relative relationships in low-rank TF, this thesis proposes Relative Pairwise Relationship Constrained Non-negative Tensor Factorisation. It deals with both Candecomp/Parafac and Tucker decomposition schemes and both squared Euclidean distance and divergence measures. The utilisation of tensor factorisation matricisation equation simplifies the update rules and greatly improves the computation efficiency. Experiments have demonstrated that the proposed algorithm can achieve higher accuracy when adopted on tensor applications. There exists a problem of acquiring out-of-bounds and fluctuating values over predictions when applying low-rank MF on recommender systems. The commonly used solutions, truncation and imposing penalties, can cause the decrease in the number of effective predictions and affect the recommendation accuracy. This thesis creatively proposes Magnitude Bounded Matrix Factorisation to handle the above problem by imposing magnitude constraints for the first time. It first converts the original quadratically constrained quadratic programming task to an unconstrained one which is then solved by the well-known stochastic gradient descent. An acceleration approach for improving computation efficiency, an extracting method for magnitude constraints and a variant of MBMF for non-negative data are also introduced. Experiments have demonstrated that the algorithm is superior to existing bounding algorithms on both computing efficiency and recommendation performance

Pushing forward matrix factorisations

We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the convolution of kernels defining integral functors between categories of matrix factorisations. We give an elementary proof of a formula for the Chern character of the convolution generalising the Hirzebruch-Riemann-Roch formula of Polishchuk and Vaintrob.Comment: 43 pages, comments welcom

B-type defects in Landau-Ginzburg models

We consider Landau-Ginzburg models with possibly different superpotentials glued together along one-dimensional defect lines. Defects preserving B-type supersymmetry can be represented by matrix factorisations of the difference of the superpotentials. The composition of these defects and their action on B-type boundary conditions is described in this framework. The cases of Landau-Ginzburg models with superpotential W=X^d and W=X^d+Z^2 are analysed in detail, and the results are compared to the CFT treatment of defects in N=2 superconformal minimal models to which these Landau-Ginzburg models flow in the IR.Comment: 50 pages, 2 figure

Orbifold equivalence: structure and new examples

Orbifold equivalence is a notion of symmetry that does not rely on group actions. Among other applications, it leads to surprising connections between hitherto unrelated singularities. While the concept can be defined in a very general category-theoretic language, we focus on the most explicit setting in terms of matrix factorisations, where orbifold equivalences arise from defects with special properties. Examples are relatively difficult to construct, but we uncover some structural features that distinguish orbifold equivalences -- most notably a finite perturbation expansion. We use those properties to devise a search algorithm, then present some new examples including Arnold singularities.Comment: 34 pages, web-link to Singular code provide