The paper generalizes Cartan Geometry using Polacek and Siegel's approach.
problem Formulating sigma model dynamics in a covariant way.
method Using Polacek and Siegel's generalised curvature and torsion approach within the generalised metric formalism.
result Almost all higher generalised tensors correspond to covariant derivatives of the generalised Riemann tensor.
Develops a new Bayesian inference method for discrete data.
problem Computational challenges in discrete state spaces, especially intractable likelihoods.
method Uses a discrete Fisher divergence to update beliefs about model parameters, circumventing the intractable normalising constant.
result Establishes statistical properties of the generalised posterior and proposes a calibration approach.
Study generalised Einstein metrics on Lie groups, classifying various types.
problem Classify left-invariant generalised Einstein metrics on Lie groups.
method New algebraic reformulation, classification based on Lie bracket and scalar product.
result Full classification of solvable and almost Abelian generalised Einstein Lie groups.
We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkahler manifolds. We consider how various generalisations of the invariants might prove useful for the study of non-compact hyperkahler manifolds, of quaternionic-Kahler manifolds, and of relation…
Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB - algebroids. In this p…
We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving N supersymmetries in dimensions D≥4 correspond precisely to integrable generalised GN structures, where GN is the generalised structure group defined by the Killing spinors. In other word…
We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened 3-manifolds. In particul…
Defines a new process for financial modeling.
problem Developing a new stochastic process for financial applications.
method Introduces a fractional Cox-Ingersoll-Ross process and proves its properties.
result The process has unique solutions and is strictly positive for certain Hurst parameters.
The paper improves generalization bounds using interpolation between various divergences.
problem Improving generalization bounds in machine learning.
method Derives new PAC-Bayes generalization bounds based on (f,Γ)-divergence and interpolates between various divergences. result Connects derived bounds to earlier statistical learning results and provides practical training objectives.
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…
A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a presc…
The paper connects DNN generalization to node SNR using information theory.
problem Exploring the reasons behind DNN generalization performance.
method Using information theory, the paper derives SNR expressions for DNN nodes and uses them to quantify weight optimization.
result Good SNR performance in DNN nodes correlates with good generalization.
Decouples homotopy quotients of generalised configuration spaces on surfaces.
problem Homological stability of generalised configuration spaces on surfaces.
method Analyzes actions of diffeomorphism groups and uses homotopy quotients.
result Decouples theorem for homology of homotopy quotients on surfaces.
The study shows symmetry improves machine learning generalization.
problem Improving machine learning generalization through symmetry.
method Using an averaging operator to prove equivariance reduces test risk.
result Equivariant predictors reduce test risk compared to non-equivariant ones.
Study on approximability and generalization in machine learning.
problem Understanding how approximation affects learning and generalization in machine learning.
method Introducing a notion of sensitivity to analyze the impact of approximation operators on predictors and proving upper bounds on generalization.
result Proven that approximable target concepts are learnable with fewer labelled samples and sufficient unlabelled data.
This paper explains double descent in linear neural networks, identifying new factors.
problem Understanding double descent in linear neural networks.
method Gradient flow derivation and necessary conditions for double descent.
result Singular values of input-output covariance matrix are important for double descent in two-layer models.
Diffusion models adapt to data geometry through log-domain smoothing.
problem Understanding why diffusion models generalize well across diverse domains.
method Investigating the role of score matching and log-domain smoothing in diffusion models.
result Log-domain smoothing adapts the diffusion model to the data manifold.
Introduces a new phase space for 2D supersymmetric sigma models.
problem Developing a new Hamiltonian formulation for 2D supersymmetric sigma models.
method Introduces a phase space with spinorial momenta and derives a covariant Hamiltonian formulation.
result Shows the existence of additional supersymmetries in the new formulation.
This review compares GAMs and neural networks on real-world tabular data.
problem Comparing the performance and characteristics of GAMs and neural networks in tabular data applications.
method Systematic review following PRISMA guidelines, extracting and analysing key attributes from 143 papers and 430 datasets.
result No consistent evidence of superiority for either GAMs or neural networks, with performance trade-offs depending on dataset characteristics.
We define (p,q) hermitian geometry as the target space geometry of the two dimensional (p,q) supersymmetric sigma model. This includes generalised Kähler geometry for (2,2), generalised hyperkähler geometry for (4,2), strong Kähler with torsion geometry for (2,1) and strong hyperkähler with torsion geometry f…
Novel GLMMNet model tackles high-cardinality categorical features in actuarial applications.
problem Inadequate encoding methods for high-cardinality categorical features in actuarial data.
method Generalised Linear Mixed Model Neural Network (GLMMNet) integrating a generalised linear mixed model in a deep learning framework.
result GLMMNet often outperforms or performs comparably with entity embedded neural networks, providing transparency.
This work uncovers algorithm-dependent regularisation in diffusion models.
problem Understanding and improving generalisation in high-dimensional diffusion models.
method Algorithmic stability and score stability analysis.
result Identifies multiple sources of implicit regularisation unique to diffusion models.
In this work we consider optimal stopping problems with conditional convex risk measures called optimised certainty equivalents. Without assuming any kind of time-consistency for the underlying family of risk measures, we derive a novel representation for the solution of the optimal stopping problem. In particular, we …
Paper proves non-zero generalization boost for equivariant models.
problem Improving generalization in machine learning models.
method Analyzes simplest case of linear models, focusing on invariant/equivariant properties.
result First provably non-zero improvement in generalization for invariant/equivariant models.
A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and P^2-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with part…
This paper presents a cross-country comparison of significant predictors of small business failure between Italy and the UK. Financial measures of profitability, leverage, coverage, liquidity, scale and non-financial information are explored, some commonalities and differences are highlighted. Several models are consid…
Motivated by the definition of the smooth manifold structure on a suitable mapping space, we consider the general problem of how to transfer local properties from a smooth space to an associated mapping space. This leads to the notion of smoothly local properties. In realising the definition of a local property at a pa…
The paper expands cluster algebra formulae to non-orientable surfaces and proves positivity.
problem Proving positivity for quasi-cluster algebras from non-orientable surfaces.
method Generalizing Musiker, Schiffler, and Williams' expansion formulae to principal laminations and quasi-triangulations.
result Positivity for quasi-cluster algebras is proven with respect to any choice of coefficients.
Strong Kähler with Torsion is the target space geometry of (2,1) and (2,0) supersymmetric nonlinear sigma models. We discuss how it can be represented in terms of Generalised Complex Geometry in analogy to the Gualtieri map from the geometry of (2,2) supersymmetric nonlinear sigma models to Generalised Kähler Geo…
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space L(4k,2k−1) and the generalised quaternionic space S3/Q4k have complexity $k,…
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of J-reflection groups. result Link groups of torus necklaces are precisely braid groups of J-reflection groups, with meridians as braid reflections. In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathem…
Study on generalisation in random feature learning and hidden manifold models.
problem Generalisation in high-dimensional learning problems.
method Replica method from statistical physics for asymptotic generalisation performance.
result Closed-form expression for generalisation performance in various high-dimensional settings.
Geometry-aware noise improves model generalization on complex manifolds.
problem Improving model generalization on highly curved data manifolds.
method Add geometry-aware noise to input space, projecting Gaussian noise onto tangent space of manifold and mapping it via geodesic curve.
result Geometry-aware noise leads to improved generalization and robustness on highly curved manifolds.
Generalizes beam models to include curvature and torsion.
problem Modeling curvature and torsion in unidimensional structures.
method Generalized Euler-Bernoulli and Timoshenko beam models using non-holonomic kinematics.
result Generalized beams exhibit curvature and torsion, with torsion in Timoshenko model.
mcanalysis quantifies menstrual cycle effects in health data.
problem Lack of standardised statistical methods for menstrual cycle research.
method Fourier-basis generalised additive model (GAM) pipeline.
result Nine out of 15 health outcomes showed significant association with menstrual cycle.
Cross validation residuals are well known for the ordinary least squares model. Here leave-M-out cross validation is extended to generalised least squares. The relationship between cross validation residuals and Cook's distance is demonstrated, in terms of an approximation to the difference in the generalised residual …
Learning with auxiliary tasks can improve the ability of a primary task to generalise. However, this comes at the cost of manually labelling auxiliary data. We propose a new method which automatically learns appropriate labels for an auxiliary task, such that any supervised learning task can be improved without requiri…
In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2,3,5)-distributions determined by a single function of the form F(q), the vanishing condition for the curvature invar…
Constructs a unique Levi-Civita connection for generalised metrics.
problem Non-uniqueness of generalised Levi-Civita connections.
method Geometrically constructs a canonical generalised Levi-Civita connection.
result Decomposes the generalised Riemann curvature tensor in terms of classical geometric data.
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
New method improves policies by combining Markov and non-Markov strategies.
problem Improving policies in reinforcement learning.
method Geometric Policy Composition (GPI) using a geometric horizon model (GHM).
result GPI can improve non-Markov policies by composing GHMs of base Markov policies.
The paper applies generalised geometry to semi-Riemannian immersions and hypersurfaces.
problem Analyzing semi-Riemannian immersions and hypersurfaces using generalised geometry.
method Develops the pullback of generalised metrics and divergence operators, introduces generalised exterior curvature, and derives Gauß-Codazzi equations.
result Establishes the constraint equations for the initial value formulation of the generalised Einstein equations.
New bounds link flat minima to good generalisation in overparameterized models.
problem Understanding the relationship between flat minima and generalisation in overparameterized machine learning models.
method Combining PAC-Bayes, Poincaré, and Log-Sobolev inequalities to derive generalisation bounds involving gradient terms.
result Flat minima positively influence generalisation performance, highlighting the benefits of the optimisation phase.
Paper uses SLT to improve model selection for SHM.
problem Model selection for SHM using data-based systems.
method Utilizes Statistical Learning Theory to rigorously estimate generalisation.
result Incorporating domain knowledge improves model generalisation.
New approach to T-duality using Courant algebroids.
problem Developing a new framework for T-duality.
method Relational description of Courant algebroids and weakened isometries.
result Existence and uniqueness of T-dual backgrounds.
Introduces Conditional Action Trees to simplify RL action spaces.
problem Challenges in RL with large, complex action spaces.
method Structures action spaces and reduces complexity through Conditional Action Trees.
result Demonstrates effectiveness in reducing action space and improving decision making.
We define and examine the notion of a Killing section of a Riemannian Lie algebroid as a natural generalisation of a Killing vector field. We show that the various expression for a vector field to be Killing naturally generalise to the setting of Lie algebroids. As an application we examine the internal symmetries of a…