Study various numerical methods for expectation propagation in machine learning.
problem Improving the convergence and stability of the expectation propagation algorithm for large-scale learning tasks.
method Numerical approximation strategies including Laplace method, Gaussian quadrature, and variational sampling.
result Variational sampling yields the best convergence for the expectation propagation algorithm in training linear binary classifiers.
Two active learning algorithms improve HSI classification using Fermat distances and harmonic label propagation.
problem Semi-supervised hyperspectral image classification with limited labeled data.
method Combines Fermat distances with Poisson-reweighted harmonic label propagation for active point selection.
result FALL and A-FALL algorithms enhance labeling accuracy and scalability for large HSI scenes.
CEP improves inference efficiency and accuracy by conditional moment matching.
problem Intractable moment matching in EP.
method Conditional expectation propagation (CEP) performs conditional moment matching and expectation.
result CEP achieves better inference quality and efficiency.
EWC uses quadratic penalties that may double-count earlier task data.
problem Catastrophic forgetting in neural networks.
method Extended derivation of EWC with multiple tasks.
result Quadratic penalties in EWC might double-count earlier task data.
Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interprete…
EP approximates Bayesian posterior distributions using gradient descent.
problem Bayesian inference's uncomputable posterior distributions.
method Relates EP to gradient descent on a smoothed energy landscape.
result EP is equivalent to gradient descent on a smoothed energy landscape.
The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several B…
Paper combines deterministic and stochastic inference methods for PGMs.
problem Combining biases from deterministic methods and high costs from Monte Carlo.
method Sequential Monte Carlo algorithm that uses output from deterministic approximations.
result Improves upon deterministic methods and Monte Carlo by reducing biases and computational costs.
Let M be a compact C∞-smooth Riemannian manifold of dimension n, n≥3, and let φλ:ΔMφλ+λφλ=0 denote the Laplace eigenfunction on M corresponding to the eigenvalue λ. We show that Hn−1({φλ=0})≤Cλα, where α>1/2 is a constant, whi…
Study the geometry of gas giant planets to infer their internal structure.
problem Determine the interior structure of gas giant planets using boundary data.
method Geometric analysis of Riemannian manifolds with conformal blow-up at the boundary.
result The interior structure of a gas giant is uniquely determined by different types of boundary data.
Efficient GP models with non-Gaussian likelihoods using state space methods.
problem Modeling non-Gaussian likelihoods in Gaussian Process (GP) regression.
method State space formulation for efficient GP models, combining LA, VB, ADF, and EP schemes.
result Efficient inference methods for non-Gaussian likelihoods in GP models.
FAB-COST improves cold-start recommendation accuracy with less data.
problem Cold-start problem in recommendation systems.
method Contextual bandit algorithm using Expectation Propagation and Assumed Density Filtering.
result FAB-COST outperforms Laplace approximation on real data.
Improves hyperparameter learning in GP models with non-conjugate likelihoods.
problem Hyperparameter learning entangled with approximate inference in GP models.
method Hybrid training procedure combining VI for inference and EP-like marginal likelihood approximation for hyperparameter learning.
result Empirically demonstrates the effectiveness of the proposed training procedure across various data sets.
While learning the maximum likelihood value of parameters of an undirected graphical model is hard, modelling the posterior distribution over parameters given data is harder. Yet, undirected models are ubiquitous in computer vision and text modelling (e.g. conditional random fields). But where Bayesian approaches for d…
We consider probabilistic multinomial probit classification using Gaussian process (GP) priors. The challenges with the multiclass GP classification are the integration over the non-Gaussian posterior distribution, and the increase of the number of unknown latent variables as the number of target classes grows. Expecta…
A new neural network layer integrates graph learning into classification tasks.
problem Lack of relational information in standard deep learning architectures for label predictions.
method Derives backpropagation equations for a differentiable graph learning layer.
result Smooth label transitions, improved generalization, and robustness to adversarial attacks.
We propose a novel approach for nonlinear regression using a two-layer neural network (NN) model structure with sparsity-favoring hierarchical priors on the network weights. We present an expectation propagation (EP) approach for approximate integration over the posterior distribution of the weights, the hierarchical s…
Study traveling waves in hyperbolic space for Fisher-KPP equations.
problem Understanding wave behavior in hyperbolic space for Fisher-KPP equations.
method Analyzes the Cauchy problem in hyperbolic space for heat equation with Fisher-KPP forcing term.
result Proves new results on the dichotomy of solution propagation or vanishing based on diffusion and reaction strength.
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.
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.
The study improves bounds for Laplace eigenvalues in Kaehler manifolds.
problem Improving bounds for Laplace eigenvalues in Kaehler manifolds.
method Generalizing classical inequalities to higher eigenvalues and applying to analytic varieties.
result Proves inequalities for Laplace eigenvalues of Kaehler manifolds and analytic varieties.
The paper defines Laplace operators for algebroid spaces.
problem Developing mathematical tools for algebroid spaces.
method Introducing Laplace-type operators for functions and forms on algebroid prolongations.
result Locally expressed Laplace operators for algebroid spaces.
Revisits online Laplace methods for neural networks, showing they are sound under certain conditions.
problem Online Laplace methods violate the Laplace approximation's critical assumption.
method Re-derives online Laplace methods, showing they target a variational bound on a mode-corrected variant of the Laplace evidence.
result Online Laplace and its mode-corrected counterpart share stationary points that satisfy the Laplace method's assumption.
Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
Standard Laplace operator extends Hodge and Casimir operators to broader geometric contexts.
problem Extending Laplace operator to vector bundles and Riemannian manifolds.
method Functorial approach, showing commutation with homomorphisms and differential operators.
result Standard Laplace operator commutes with a wide range of differential operators.
Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
New methods improve Laplace approximations for deep neural networks by selecting key parameters.
problem Improving uncertainty quantification in deep neural networks using computationally feasible approximations.
method Gradient-Laplace and Greedy-Laplace methods for selecting parameters in sub-network Laplace approximations.
result Gradient-Laplace method outperforms existing heuristic approaches and provides formal optimality guarantees.
A new method combines Laplace and Variational Bayes for scalable inference.
problem Complex models and large datasets make exact inference infeasible.
method Low-Rank Variational Bayes Correction (VBC) using Laplace method and Variational Bayes correction in a lower dimension.
result The method ensures scalability in both model complexity and data size.
The paper constructs Laplace-Beltrami operators on noncommutative tori.
problem Developing Laplace-Beltrami operators for noncommutative tori.
method Construction of Laplace-Beltrami operators with consideration of non-trivial modular automorphisms.
result Laplace-Beltrami operators on noncommutative tori have properties similar to those on ordinary Riemannian manifolds.
The paper studies graph Laplace operator behavior near isolated singularities.
problem Investigating asymptotics of graph Laplace operator near isolated singularities.
method Analyzing curvature growth and conformal modifications to understand operator behavior.
result The graph Laplace operator converges to a weighted Laplace-Beltrami operator as bandwidth decreases, or behaves like \(O(\frac{1}{\sqrt{t}})\) if curvature grows too fast.
QLA improves Bayesian uncertainty estimation for DNNs without increasing computational cost.
problem Overconfident out-of-distribution predictions from DNNs.
method Proposes Quadratic Laplace Approximation (QLA) to improve Bayesian uncertainty quantification.
result QLA yields modest yet consistent uncertainty estimation improvements over Linearized Laplace Approximation (LLA) on five regression datasets.
Geometrically reformulates the Laplace method for optimal transport.
problem Approximating integrals using the Laplace method without geometric interpretation.
method Introduces the Kim-McCann Riemannian metric to give a geometric formulation of the Laplace method.
result Expresses the first-order term of the Laplace method using geometric objects.
Fundamental solutions found for p-Laplace equations in Heisenberg and Grushin spaces.
problem Finding solutions to p-Laplace equations with drift terms in specific geometric spaces.
method Analyzing fundamental solutions in the Heisenberg group and Grushin-type planes.
result Natural generalizations of Beals, Gaveau, and Greiner's solutions for the Laplace equation with drift term.
Study finds metrics maximizing one Laplace eigenvalue on 3D and higher manifolds.
problem Maximizing a Laplace eigenvalue on n-dimensional manifolds.
method Existence and regularity results for metrics of same volume in a conformal class.
result Existence and regularity of metrics maximizing the Laplace eigenvalue.
Neural Laplace models diverse DEs in the Laplace domain for better dynamics.
problem Inadequate ODEs for long-range dependencies and discontinuities.
method Unified framework in Laplace domain, using stereographic map for smoothness.
result Superior performance in diverse DEs, including complex history dependency and abrupt changes.
Bayesian method models spatiotemporal seizure dynamics.
problem Understanding complex seizure dynamics in space and time.
method Bayesian belief updating using variational Laplace in DCM framework.
result Framework assimilates spatial and temporal seizure dynamics.
Paper proves rigidity for certain PDEs on compact manifolds.
problem Proving rigidity for p-Laplace and n-Laplace equations. method Nonlinear flow and carré du champ methods.
result Rigidity means only constant solutions for certain parameters.
Graph Laplace operators uniquely identify metrics and densities on manifolds.
problem Identifying Riemannian metrics and sampling densities from graph Laplace operators.
method Analyzing intrinsic and extrinsic graph Laplace operators on compact Riemannian manifolds.
result Graph Laplace operators uniquely determine metrics and densities under certain conditions.
The paper decomposes and analyzes the higher spin Laplace operator.
problem Understanding the properties and solutions of the higher spin Laplace operator.
method Decomposition into Rrita-Schwinger operators, proving conformal invariance, establishing integral formulas.
result Established a Borel-Pompeiu type formula and a Green type integral formula for the higher spin Laplace operator.
Spatio-temporal point process models play a central role in the analysis of spatially distributed systems in several disciplines. Yet, scalable inference remains computa- tionally challenging both due to the high resolution modelling generally required and the analytically intractable likelihood function. Here, we expl…
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
Improved bounds on Laplace-Beltrami operator eigenvalues on real projective plane.
problem Improving upper bounds for Laplace-Beltrami operator eigenvalues.
method Analyzing eigenvalues with even indexes and providing bounds for Dirichlet, Neumann, and Steklov eigenvalues.
result Enhanced upper bounds for Laplace-Beltrami operator eigenvalues on the real projective plane.
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space R3. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve…
Variational Laplace improves Bayesian neural networks performance.
problem Improving Bayesian neural networks performance.
method Develops variational Laplace for BNNs, exploiting curvature of likelihood.
result Variational Laplace outperforms other inference methods.
The 1993 Laplace transform approach of Geman and Yor is a celebrated advance in valuing Asian options. Its insights are fundamental from both a mathematical and a financial perspective. In this paper, we discuss two observations regarding the financial relevance of its results. First, we show that the Geman and Yor Lap…