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.
A scalable method for Bayesian inference in large linear models.
problem High computational cost in Bayesian linear models for large networks.
method Sample-based inference and g-prior for hyperparameter selection.
result Linearised neural network inference on large datasets (ResNet-18, ResNet-50, U-Net).
Develops scalable Bayesian inference methods for neural networks.
problem Lack of model uncertainty in deep learning leading to overconfident predictions.
method Linearised Laplace approximation, conjugate Gaussian-linear models, stochastic gradient descent, sample-based EM algorithm.
result Equips neural networks with model uncertainty using scalable methods.
Bayesian neural networks struggle with uncertainty estimates between regions.
problem Limited expressiveness of predictive uncertainty estimates in between regions.
method Compared mean-field variational inference (MFVI) with linearised Laplace approximation.
result Linearised Laplace approximation handles 'in-between' uncertainty better.
Conditions found for linearizing divergence-free fields on invariant tori.
problem Linearizing divergence-free vector fields on invariant tori.
method Assuming a solution to the cohomological equation and using Bers' results in pseudo-analytic function theory.
result The field B on S is either identically zero or nowhere vanishing, with a linearizable form. Study linear perturbations in Schwarzschild black hole spacetime.
problem Linear perturbations of Schwarzschild black hole spacetime.
method Investigate linearised perturbation of constant mass aspect function foliation at null infinity.
result Linearised perturbations of Bondi energy and mass vanish, and all linear momentum can be achieved.
We present an elementary argument that one can shield linearised gravitational fields using linearised gravitational fields. This is done by using third-order potentials for the metric, which avoids the need to solve singular equations in shielding or gluing constructions for the linearised metric.
Study proves curvature estimates for Kerr spacetime's linearized perturbations.
problem Proving elliptic L2(S2)-estimates for linearised curvature quantities in Kerr spacetime. method Applies linearised system from doctoral thesis, covers full sub-extremal range of Kerr parameters.
result Elliptic L2(S2)-estimates for linearised curvature quantities in the full sub-extremal range of Kerr parameters. This paper proposes a new algorithm for Gaussian process classification based on posterior linearisation (PL). In PL, a Gaussian approximation to the posterior density is obtained iteratively using the best possible linearisation of the conditional mean of the labels and accounting for the linearisation error. PL has s…
Quaternion-Kaehler four-manifolds, or equivalently anti-self-dual Einstein manifolds, are locally determined by one scalar function subject to Przanowski's equation. Using twistorial methods we construct a Lax Pair for Przanowski's equation, confirming its integrability. The Lee form of a compatible local complex struc…
New method parameterizes solutions to linearized vacuum constraints on Einstein manifolds.
problem Parameterizing solutions to linearized vacuum constraints on Einstein manifolds.
method Parameterize solutions using unconstrained potentials and shield linearized gravitational fields.
result Showed how to shield linearized gravitational fields without TT gauge for any value of cosmological constant.
Scattering theory developed for linearised gravity near Schwarzschild black hole.
problem Linear stability of Schwarzschild spacetime and scattering of gravitational waves.
method Physical-space Chandrasekhar transformation and Teukolsky-Starobinsky correspondence.
result Construction of scattering theory for spin 2 Teukolsky equations.
We construct the full linearisation functor which takes a graded bundle of degree k (a particular kind of graded manifold) and produces a k-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of k-fold vector bundles consisting of symmetric $…
Linearized Einstein equations simplified via Calabi operator.
problem Linearizing Einstein equations for cosmological applications.
method Using the Calabi operator from projective differential geometry.
result Linearized Einstein equations simplified.
The paper extends gluing theorems for gravitational fields in higher dimensions.
problem Proving gluing theorems for linearised gravitational fields on characteristic hypersurfaces.
method Analyzing linearised vacuum gravitational fields in (n+1)-dimensional static spacetimes with cosmological constant. result Generalization of gluing theorems to higher dimensions, extending previous work on light cones.
Scattering theory for linearised gravity on Schwarzschild black hole exterior.
problem Constructing a scattering theory for linearised gravity equations on Schwarzschild background.
method Building on previous work, constructing Hilbert space-isomorphisms for finite energy initial data and scattering states.
result Past and future linear memories are related by an antipodal map for Bondi-normalised solutions.
Proposes an INLA-based method for state and parameter estimation in nonlinear systems.
problem Difficulty in learning parameters accurately in nonlinear dynamical systems.
method Iterated INLA for state and parameter estimation in nonlinear dynamical systems.
result Outperforms existing methods on data assimilation tasks.
The paper extends gluing theorems for linearized gravitational fields in static spacetimes with cosmological constant.
problem Establishing gluing theorems for linearized vacuum gravitational fields on characteristic surfaces.
method Analyzing linearised Einstein equations in Bondi gauge on static four-dimensional spacetimes with cosmological constant.
result Generalization and extension of gluing theorems to include cosmological constant and arbitrary topology.
New method controls sparse feature updates in deep networks.
problem Understanding sparse feature updates in deep networks during training.
method Iterative linearised training method to control sparse feature updates.
result Iterative linearised training surprisingly performs on par with standard training, requiring less frequent feature learning.
We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new in…
We calculate eigenvector overlaps between intersecting time periods of covariance matrices.
problem Analyzing overlapping time periods in covariance matrices.
method Girko linearisation and extended local laws.
result Computed eigenvector overlaps for intersecting time intervals.
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to…
Method estimates uncertainty in CT reconstructions.
problem Lack of accurate uncertainty estimates in deep-learning CT reconstructions.
method Linearised deep image prior with conjugate Gaussian-linear model error bars and Gaussian surrogate for TV regularisation.
result Method provides superior calibration of uncertainty estimates.
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely solvable up to gauge solutions, given initial data on a spacelike Cauchy hypersur…
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.
The Plebański complex is a differential operator that squares to the Laplacian and is composed of two Dirac operators.
problem The Plebański complex studies the linearization of equations for hyper-Kähler manifolds.
method Defined and studied properties of the Plebański complex, showing it fits into the elliptic complex framework.
result The Plebański complex is an elliptic differential operator that squares to the Laplacian and is composed of two Dirac operators.
Introduces Nijenhuis Geometry, a new field of endomorphisms.
problem Defines Nijenhuis operators and their properties.
method Develops new techniques and summarises basic facts.
result Proves new, not obvious, results in Nijenhuis Geometry.
Unified framework for efficient Gaussian process inference.
problem Efficient inference in non-conjugate Gaussian process models.
method Combines expectation propagation with linearization for improved efficiency.
result Unified view of various inference schemes, including classical smoothers and EP.
Defines formal exponentials for graded manifolds and linearizes QP-manifolds.
problem Formal exponentials and linearizations of QP-manifolds.
method Definition of formal exponential maps, Grothendieck connections, and connections on tangent bundles.
result Linearizes QP-manifolds at points, giving formal tangent spaces L∞-algebra structures. This research explores the dynamics of linearised neural nets, revealing distinct learning phases and layer growth rates.
problem Understanding the fundamental mechanics of neural nets and their learning dynamics.
method Derivation of properties of learning dynamics in general multi-layer linear neural nets, including orthogonal networks.
result Linear multi-layer neural nets exhibit distinct phases of learning with different layer growth rates, and nonlinearity affects these dynamics.
We explain how to encode the essential data of a PDE on jet bundle into a more intrinsic object called Pfaffian fibration. We provide motivations to study this new notion and show how prolongations, integrability and linearisations of PDEs generalise to this setting.
We show how one can handle the formalism developped by Yurii Vorobjev in order to give general results about the problems of linearisation and of normal form of a Poisson structure in the neighborhood of one of its symplectic leaves.
We review the concept of a graded bundle, which is a generalisation of a vector bundle, its linearisation, and a double structure of this kind. We then present applications of these structures in geometric mechanics including systems with higher order Lagrangian and the Plateau problem.
This work proposes a new method for simultaneous probabilistic identification and control of an observable, fully-actuated mechanical system. Identification is achieved by conditioning stochastic process priors on observations of configurations and noisy estimates of configuration derivatives. In contrast to previous w…
We consider faithful projective actions of a cocompact lattice of SL(2,R) on the projective plane, with the following property: there is a common fixed point, which is a saddle fixed point for every element of infinite order of the the group. Typical examples of such an action are linear actions, ie, when the action ar…
We prove that the TT-gauge-fixed linearised Einstein operator is non-degenerate for Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics with dimension- and topology-dependent ranges of mass parameter. We provide evidence that this remains true for all such metrics except the spherical ones with a critical mas…
We give a simple analytic criterion which characterizes linearizable 1-codimensional webs. Then we give an invariant geometrical interpretation of it, in term of projective connection. We explain then how our approach allows to study linearization of more general objects than 1-codimensional webs. By way of illustratio…
This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are i…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear in the existing literature. We start with a discussion of graded spaces, passing …
New framework for gravitational perturbations of Kerr spacetimes, focusing on stability.
problem Stability of Kerr spacetimes to gravitational perturbations.
method New geometric framework with tailored null frames and gauge, reformulating Einstein equations.
result Derivation of linearised vacuum Einstein equations in the new framework.
The paper shows instability in Minkowski spacetime for a quantum system.
problem Linear instability of the semiclassical Einstein-Klein-Gordon system in Minkowski spacetime.
method Formulated a forcing problem for metric and state perturbations, used tensor decomposition and quantum Møller operator.
result Metric perturbations grow exponentially, bounded by a universal scale H, indicating quantum backreaction.
Study on type-D Ricci-flat metrics with Killing spinors and Killing vectors.
problem Characterizing and solving metrics with specific properties.
method Using Killing spinors and Killing vectors, rederive results via Toda field equations and axisymmetric solutions.
result Field equations linearize for certain metrics, including axisymmetric solutions.
In a recent seminal paper \cite{D-H-R} of Dafermos, Holzegel and Rodnianski the linear stability of the Schwarzschild family of black hole solutions to the Einstein vacuum equations was established by imposing a double null gauge. In this paper we shall prove that the Schwarzschild family is linearly stable as solution…
Let x:M→Em be an isometric immersion of a Riemannian manifold M into a Euclidean m-space. Denote by Δ the Laplace operator of M. Then Δ gives rise to a differentiable map L:M→Em, called the Laplace map, defined by L(p)=(Δx)(p), p∈M. We call L(M) the Laplace image, and the transformat…
The paper examines special Q-nets that terminate after a finite number of Laplace steps.
problem Understanding the termination of Laplace sequences in Q-nets.
method Analyzing discrete Koenigs nets and their Laplace sequences.
result For certain Koenigs nets, Laplace sequences terminate after a finite number of steps.
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.
problem Estimating eigenvalues of poly-Laplace operator on subgraphs of lattice graphs.
method Introduced discrete poly-Laplace operator, derived upper and lower bounds for eigenvalues.
result Poly-Laplace eigenvalues are at least squares of lower-order poly-Laplace eigenvalues.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.