The κ-generalised distribution fits daily stock returns well.
problem Stock returns are often heavy-tailed, not normally distributed.
method Used the κ-generalised distribution with a Monte-Carlo goodness of fit test. result The κ-generalised distribution fits historic daily stock returns well for a significant proportion of analyzed stocks. Extends nonlinear theory of distributional geometry.
problem Developing a theory for nonsmooth differential geometry.
method Extending Colombeau theory to tensor fields, introducing Lie derivative and covariant derivative, defining generalised metric.
result Preserves Einstein equations and curvature of cones in nonsmooth geometry.
Local equivalence shown between specific distributions and flat Cartan distribution.
problem Establishing local equivalence between specific distributions and flat Cartan distribution.
method Change of coordinates mapping specific distributions to flat Cartan distribution.
result Local equivalence between maximally symmetric (2,3,5)-distributions and flat Cartan distribution. Develops a new global theory of generalised functions on manifolds.
problem Lack of a global theory for generalised functions on manifolds.
method Constructs a global theory based on smoothing operators and generalised Lie derivative.
result Generalised Lie derivative commutes with embedding of distributions.
Improved neural networks for relational reasoning by projecting high-dimensional data to low-dimensional manifolds.
problem Out-of-distribution generalization in complex relational reasoning tasks.
method Neuroscience-inspired inductive-biased module projecting high-dimensional object representations to low-dimensional manifolds.
result Significantly better out-of-distribution generalization performance on relational reasoning tasks.
PAC-Bayes bound requires prior to place mass on high-performing predictors.
problem Explaining generalization in machine learning.
method Analyzing necessary conditions for PAC-Bayes bounds to provide meaningful generalization guarantees.
result Achieving a target generalisation level requires the prior to place sufficient mass on high-performing predictors.
We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Lapla…
New GLPs split Lévy bridges into non-overlapping subprocesses.
problem Creating multivariate stochastic processes with specific properties.
method Defining GLPs by splitting Lévy bridges and using time changes.
result GLPs have terminal values and increments with generalised multivariate Liouville distributions.
Abstract reviews distributions and subbundles in differential geometry.
problem Understanding distributions and subbundles in differential geometry.
method Systematic review of distributions and subbundles, including sheaves and differentiability cases.
result Detailed consideration of Orbit Theorem and its applications.
This work improves generalisation bounds using chaining and information theory.
problem Improving generalisation bounds for supervised learning algorithms.
method Developed a theoretical framework linking generalisation bounds to their chained counterparts, derived new bounds using Wasserstein distance.
result Chained generalisation bounds can be tighter than standard bounds, especially for concentrated hypothesis distributions.
CRL improves recommendation systems by reducing distribution shift.
problem Offline metrics fail to predict online performance due to distribution shift in recommender systems.
method Proposes an information-theoretic disentanglement criterion and a variational lower bound for better generalisation under distribution shift.
result CRL variants deliver substantial online gains in listener engagement compared to baseline models.
The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one …
New approach to compute generalization performance using known risk distribution.
problem Computing generalization performance in machine learning.
method Assumes known risk distribution ρ(r), computes expected error using empirical risk minimization, and considers power-law behavior of ρ(r). result Corrected typical behavior of generalization performance due to chance correlations in training set.
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…
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.
Generalizes Kauffman's clock theorem to surfaces.
problem Proving a lattice structure on graph states in various surfaces.
method Using matchings and graph orientations, extending Propp's results.
result Two generalizations of Kauffman's theorem for more surfaces.
Bayesian inference uses Stein discrepancy for robustness in intractable likelihoods.
problem Intractable likelihoods in Bayesian inference.
method Generalised Bayesian inference with Stein discrepancy as the loss function.
result Robust generalised posteriors with closed form or accessible using MCMC.
Proposes vMF distribution for skewed elliptical distributions.
problem Skewed distributions not adequately modeled by symmetric distributions.
method Introduces von-Mises-Fisher (vMF) distribution to represent skewed elliptical distributions.
result vMF distribution provides an explicit and simple probability representation of skewed elliptical distributions.
Generalised regularisation equals robustness for exotic function classes.
problem Adversarial examples highlight the need for robust models in exotic function classes.
method Equality result linking distributional robustness and Lipschitz regularisation.
result Certified robustness of Lipschitz-regularised models with mild assumptions.
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…
Regularizes ML algorithms for robust multivariate analysis against distribution shifts.
problem Ensuring robustness of multivariate analysis algorithms against distribution shifts.
method Integrates a causal regularisation term into the loss function of multivariate analysis algorithms.
result Demonstrates improved out-of-distribution generalisation with reduced-rank regression and partial least squares.
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.
Enhances robustness in experimental design through Generalised Bayesian inference.
problem Poor inference and estimates of information gain when statistical model is incorrectly specified.
method Generalised Bayesian (Gibbs) inference framework applied to experimental design.
result GBOED enhances robustness to outliers and incorrect assumptions about noise distribution.
Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First w…
Graph neural networks generalize well under certain conditions, explained by learning theory.
problem Understanding why graph neural networks generalize well in transductive inference.
method Analysis of transductive Rademacher complexity to explain generalization properties of graph convolutional networks.
result Transductive Rademacher complexity can explain the generalization of graph convolutional networks for node classification in stochastic block models.
The GARCH algorithm is the most renowned generalisation of Engle's original proposal for modelising {\it returns}, the ARCH process. Both cases are characterised by presenting a time dependent and correlated variance or {\it volatility}. Besides a memory parameter, b, (present in ARCH) and an independent and id…
We provide necessary and sufficient conditions on the derived type of a vector field distribution $\Cal V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from R to Rq, q≥1. This result fully genera…
We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.
Study reveals phase transition in neural networks near interpolation.
problem Understanding generalization and learning transitions in neural networks.
method Effective theory for approximating Bayes-optimal generalisation error.
result Unveils a discontinuous phase transition between universal and specialisation phases.
Improved normalising flows using Student's t-distribution for robust training.
problem Training deep probabilistic models with robust statistics.
method Propose Student's t-distribution as a robust alternative to Gaussian in normalising flows.
result Improved robustness and reduced generalization gap with Student's t-distribution.
Generative models converge to data distribution but not principal latent factors.
problem Understanding when generative models converge to the true data distribution.
method Analytical characterisation of transition from memorisation to generalisation in linear generative models.
result Convergence captures matching the bulk of the data distribution but not principal latent factors.
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.
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.
Recursive KalmanNet generalizes well in noisy, out-of-distribution scenarios.
problem Generalization in noisy, out-of-distribution scenarios.
method Recurrent neural network guided by a Kalman filter.
result Recursive KalmanNet performs well in scenarios with different temporal dynamics from training data.
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.
Researchers develop a generalised geometric Brownian motion for better asset pricing.
problem Irregularities in simple geometric Brownian motion for asset dynamics.
method Introduce a memory kernel to generalise GBM, derive moments and probability density functions.
result The performance of kernels in pricing options depends on option maturity and moneyness.
Unsupervised learning representations generalize better than supervised learning under distribution shifts.
problem Robustness of unsupervised representations to distribution shift.
method Extensive evaluation on synthetic and realistic datasets, including controllable domain generalization datasets.
result Unsupervised representations learned from SSL and AE generalize better than supervised learning under various distribution shifts.
Paper tackles long-tailed labels in classification problems.
problem Imbalanced or long-tailed label distribution in real-world classification problems.
method Logit adjustment applied post-hoc or during training to encourage a large relative margin between rare and dominant labels.
result Unified and generalised techniques for coping with long-tailed labels, improving generalisation and performance.
New measures generalize existing ones, linking information and risk.
problem Linking information measures and risk in statistical decision problems.
method Introducing new families of divergence measures and deriving an information processing equality.
result Extension of variational φ-divergence representation to multiple distributions. We revisit the Kolmogorov-Smirnov and Cramér-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not…
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.
Study analyzes household capital risk and poverty trapping, deriving a new function for capital deficit distribution.
problem Analyzing the risk of household capital falling into poverty.
method Introduced a new Gerber-Shiu function to model trapping time and capital deficit distribution.
result Derived a model for capital deficit distribution at trapping using GB distributions.
Study different masking schemes for a universal marginaliser.
problem Understand how well a neural approximator learns conditional distributions.
method Compare networks trained with various masking schemes.
result Neural approximators perform differently based on the masking scheme.
ARC algorithm optimizes dynamic pricing with correlated observations.
problem Optimizing dynamic pricing with correlated and generally distributed observations.
method Extends ARC algorithm to batched bandits with generalised linear model.
result ARC algorithm outperforms alternative approaches in dynamic pricing.
Defines T-duality and generalised Ricci flow relations using Courant algebroid relations.
problem Establishing compatibility between T-duality and generalised Ricci flow.
method Introducing Courant algebroid relations, invariant divergence operators, and generalised isometries.
result T-duality is compatible with generalised Ricci flow, and T-dual solutions are also solutions of generalised Ricci flow.
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…
New approach ties loss curvature to model performance in deep learning.
problem Understanding the relationship between loss curvature and model performance in deep learning.
method Empirical analysis of loss Hessians and theoretical results on input-output Jacobians.
result Novel generalization bound in terms of empirical Jacobian.
Bayesian framework tackles measurement error in covariates.
problem Misleading inference due to corrupted covariates.
method Bayesian Nonparametric Learning framework robust to misspecification.
result General framework for Classical and Berkson error models.