The paper improves spectral clustering by analyzing the asymptotic normalised cut value.
problem No agreed method for tuning scaling parameter or automatically determining cluster number.
method Investigates asymptotic value of normalised cut for increasing samples.
result Provides recommendations for improving spectral clustering methodology.
New methods for scalable inference in modular models with misspecified sub-models.
problem Model misspecification in multi-modular models complicates evidence combination.
method Variational methods for approximating Cut and SMI posteriors, and Variational Meta-Posterior.
result Feasibility of analysis with multiple cuts using a single set of variational parameters.
SUQUAN optimizes quantile normalisation for better downstream analysis.
problem Optimizing target distribution for better downstream analysis.
method Optimizes target distribution jointly with other parameters in the analysis.
result SUQUAN outperforms standard quantile normalisation on various data types.
A new method normalizes EBM training by introducing a learnable parameter.
problem Training energy-based models with maximum likelihood is challenging due to intractable normalisation constants.
method Proposes a self-normalised log-likelihood (SNL) objective that introduces a learnable parameter representing the normalisation constant.
result The SNL objective is a lower bound of the log-likelihood and can be directly optimised using stochastic gradient techniques.
Batch normalisation doesn't affect variational inference but fails for larger batch sizes.
problem Failure of Monte Carlo Batch Normalisation (MCBN) for capturing epistemic uncertainty in larger batch sizes.
method Investigated MCBN as an approximate inference technique for Bayesian neural networks, showing its limitations and providing insights for improvement.
result For larger batch sizes, MCBN fails to capture epistemic uncertainty, requiring the batch size to be a variational parameter.
Proposes a method to apply conformal prediction to probabilistic time series forecasting models.
problem Obtaining accurate prediction regions for multi-step time series forecasting with probabilistic models.
method Conformalises conditional normalising flows to generate potentially disjoint prediction regions.
result Improves predictive efficiency in time series forecasting with multimodal distributions.
AON improves neural network generalization by making weights approximately orthogonal.
problem Improving generalization of deep neural networks.
method Approximated orthonormal normalisation (AON) technique to make weight vectors approximately orthogonal.
result AON yields promising validation performance compared to orthonormal regularisation.
Study of superintegrable systems linked to affine hypersurfaces.
problem Understanding superintegrable systems through geometric structures.
method Established a correspondence between superintegrable systems and affine hypersurfaces, defining conformal equivalence.
result Identified conformal classes of abundant manifolds with abundant hypersurface immersions.
Study of Coxeter diagrams and Artin-Tits groups, focusing on normalisers and wall intersections.
problem Understanding normalisers of parabolic subgroups in Artin-Tits groups and their connections to Coxeter diagrams.
method Analyzing hyperplane arrangements, Coxeter groups, and wall-and-chamber structures.
result Complexified hyperplane complement is a K(π,1) space for normalisers of parabolic subgroups in finite-type Coxeter diagrams.
Kernelised flows improve density estimation and generation with fewer parameters.
problem Limited expressiveness of flow-based models due to invertibility constraints.
method Integrates kernels into normalising flows to enhance expressiveness and efficiency.
result Kernelised flows outperform neural network-based flows in parameter efficiency and low-data scenarios.
CIFs replace single bijections with continuous families to avoid topological limitations.
problem Normalising flows struggle with targets with complex topologies.
method Propose Continuously Indexed Flows (CIFs) replacing single bijections with a continuous family.
result CIFs avoid topological limitations and perform better empirically.
Bayesian normalising flows improve conditional density estimation.
problem Complex conditional distributions in various settings.
method Bayesian normalising flows for flexible likelihood modeling.
result State-of-the-art performance on benchmark datasets.
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.
Method estimates bivariate causal models using normalising flows and variational Gaussian process regression.
problem Lack of explainability in AI models, especially in causal mechanisms.
method Combination of normalising flows for density estimation and variational Gaussian process regression for post-nonlinear models.
result Method better explains cause-effect pairs than simple additive noise models.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
New invariant dominates Jones, Kuperberg, and arrow polynomials.
problem Dominating known knot invariants.
method Picture formalism leading to a new invariant.
result New invariant outperforms Jones, Kuperberg, and arrow polynomials.
Score-based methods fail with isolated components and incorrect mixing proportions.
problem Score-based methods struggle with distributions having isolated components and incorrect mixing proportions.
method Score-based methods, including score matching, are used but fail in the presence of isolated components and incorrect mixing proportions.
result Score-based methods cannot discover isolated components or identify correct mixing proportions.
Adapts linearised Laplace method for deep learning models.
problem Incompatibility of linearised Laplace method with modern deep learning tools.
method Examines and adapts linearised Laplace method for model selection in deep learning.
result Recommendations for better adapting linearised Laplace method to modern deep learning.
RotRNN uses rotations to simplify long sequence modelling.
problem Complex initialisation and normalisation schemes in linear recurrent models.
method RotRNN employs rotation matrices to simplify and normalise linear recurrent models.
result RotRNN achieves competitive performance on long sequence modelling datasets.
Improved spectral convergence bounds for diffusion maps on tori.
problem Weak theoretical error bounds for diffusion maps.
method Spatial Hardy space estimates, PDE spectral stability, Sinkhorn weights.
result Matched pointwise error bounds for spectral data and operator convergence.
In quantitative finance, we often model asset prices as a noisy Ito semimartingale. As this model is not identifiable, approximating by a time-changed Levy process can be useful for generative modelling. We give a new estimate of the normalised volatility or time change in this model, which obtains minimax convergence …
Unique solutions found for a specific flow equation.
problem Finding unique solutions for a specific flow equation.
method Used a uniform bound for the Liouville energy and a natural space-time L2-bound for the time derivative of the solution. result Uniqueness of classical solutions for the normalised two-dimensional Hamilton-Ricci flow.
Bitcoin volatility shows multifractal structure, contradicting rough volatility models.
problem Applying rough volatility models to Bitcoin volatility data.
method Normalised p-variation framework, multifractal Detrended Fluctuation Analysis, log-log moment scaling, wavelet leaders.
result Bitcoin volatility exhibits multifractal structure, violating rough volatility model assumptions.
New algorithm STCV improves sparse model discovery from normalised data.
problem Distortion of sparse model discovery due to data scaling.
method STCV, a novel sparse regression algorithm robust to data scaling.
result STCV outperforms standard methods on normalised, noisy datasets.
NeuralCut learns to select cutting planes by looking ahead, outperforming traditional methods.
problem Selecting effective cutting planes for MILP optimization.
method Imitation learning on a lookahead expert to train a neural network for cut selection.
result NeuralCut outperforms standard baselines in cut selection for MILP benchmarks.
Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.
problem Solving parametric mixed-integer linear optimization problems with changing data.
method Introducing cutting-plane layers (CPLs) for differentiable cutting-plane generation.
result The algorithm computes solutions with low integrality gaps and generalizes to unseen instances.
NeVI-Cut uses neural networks to efficiently propagate uncertainty without feedback.
problem Efficiently propagating uncertainty in downstream Bayesian analysis without feedback.
method NeVI-Cut combines neural networks and normalizing flows for variational inference.
result NeVI-Cut achieves significant computational gains and higher accuracy than traditional methods.
A new approach to kernel adaptive filters reduces sparsity for monotonic signals.
problem Kernel adaptive filters struggle with trivial monotonic signals, leading to inaccurate predictions and high computational complexity.
method Proposes a unit-norm Gaussian kernel and sparsification criterion to compare new observations against dictionary samples.
result The method achieves more accurate predictions and smaller dictionary size compared to standard KAF.
Paper connects probability density cuts to graph theory eigenfunctions.
problem Developing sparse cuts for probability densities.
method Defines sparse cuts and principal eigenfunctions for probability densities, proving Cheeger and Buser inequalities.
result No such inequalities hold for prior definitions, proving new inequalities for probability densities.
The paper derives inequalities for Riemannian submersions and their applications.
problem Characterizing Casorati inequalities for Riemannian submersions.
method Algebraic and geometric analysis of Casorati inequalities for normalised scalar and Casorati curvatures.
result Characterization of equality cases for Casorati inequalities in Riemannian submersions.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Understanding Riemannian metrics on lens spaces and their geometric properties.
method Geometric control theory methods applied to axisymmetric metrics.
result Cut loci and cut times converge to sub-Riemannian structure's values.
Study Riemannian metrics on lens spaces, find cut loci and diameters.
problem Analyzing Riemannian metrics on lens spaces.
method Geometric control theory methods.
result Cut loci and cut times converge to sub-Riemannian structure's cut locus and time.
We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …
In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In …
Extends cutting and blowing up to nonrational symplectic settings.
problem Nonrational symplectic toric structures.
method Cutting and blowing up in nonrational directions.
result Extension to symplectic toric manifolds and orbifolds.
A new algorithm PD improves stock-correlation network clustering and robustness.
problem Improving clustering and robustness of stock-correlation networks.
method Proposes a new proportional degree algorithm to filter information on a complete graph of normalised mutual information.
result The PD algorithm produces a network with better homogeneity and robustness compared to PMFG.
Stability of cut locus under metric perturbations in compact Riemannian manifolds.
problem Stability of cut locus under C2-perturbations of the metric. method Proving stability with respect to the Hausdorff metric of the cut locus under C2 perturbation of the metric. result The Hausdorff distance between cut loci converges to zero as the metrics converge.
New equivalence relation for links using cut-diagrams.
problem Classical link concordance.
method Cut-diagrams and cut-concordance.
result Nilpotent peripheral system invariant of cut-concordance.
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
Gradient flow method solves isoperimetric inequality for maps.
problem Finding maps with optimal enclosed area.
method Sobolev gradient flow for area-normalised Dirichlet energy.
result Solutions converge to a circle as time goes to infinity.
Study geodesics and metrics on PSL(2) group for hyperbolic plane.
problem Geodesics and metrics on PSL(2) group for hyperbolic plane.
method Parametrization of geodesics, conjugate time, cut time, cut locus computation.
result Cut time and cut locus converge to sub-Riemannian problem as third eigenvalue tends to infinity.
Study shows convergence rates for Cheeger cuts on data clouds.
problem Optimizing graph cuts for clustering data sampled from a manifold.
method Analyzes statistical properties of Cheeger cuts on proximity graphs built from data.
result Obtains high probability convergence rates for Cheeger constant and cuts.
Unified framework for differentiable graph partitioning with probabilistic cuts.
problem Lack of general guarantees and principled gradients in prior probabilistic relaxations of graph cuts.
method Unified probabilistic framework covering a wide class of cuts, including Normalized Cut, with tight analytic upper bounds.
result Rigorous, numerically stable foundation for scalable, differentiable graph partitioning.
Study of cut locus in Carnot groups disproves conjectures and finds new sets.
problem Disprove conjectures on the shape of cut loci in Carnot groups.
method Analyzing left-invariant Carnot-Carathéodory metric and semi-algebraic sets.
result Found sets of cut points with codimension 2, disproving previous conjectures.
Combines neural networks with splitting-up method for filtering equations.
problem Approximating the solution of filtering equations for signal processes.
method Combines splitting-up method with neural networks.
result Produces an approximation of the unnormalised conditional distribution.
Contrary to standard statistical models, unnormalised statistical models only specify the likelihood function up to a constant. While such models are natural and popular, the lack of normalisation makes inference much more difficult. Here we show that inferring the parameters of a unnormalised model on a space Ω can …
Stochastic cutting planes improve data-driven optimization speed.
problem Data-driven Mixed-Integer Nonlinear Optimization problems.
method Stochastic version of cutting-plane method.
result Stochastic algorithm converges to ε-optimal solution with high probability.
Study of Randers metrics on spheres with simple cut loci.
problem Understanding Randers metrics on spheres and their cut loci.
method Analyzing geodesics, conjugate, and cut loci of Finsler metrics of Randers type.
result Found new families of Randers metrics with simple cut loci.