Dirichlet pruning compresses neural networks by removing unimportant units.
problem Compressing large neural network models without sacrificing performance.
method Assigns Dirichlet distribution over network layers' units and uses variational inference to estimate parameters.
result Achieves state-of-the-art compression performance on larger architectures like VGG and ResNet.
TPBS models improve robustness to overfitting with localized Dirichlet energy regularization.
problem Global Dirichlet energy-based regularization fails for TPBS models due to perfect interpolation.
method Propose local Dirichlet energy regularization and two inference estimators.
result TPBS models outperform neural networks in overfitting regimes and maintain competitive performance otherwise.
Enhances uncertainty estimation in neural networks using Dirichlet-based MC Dropout.
problem Deterministic predictions without uncertainty estimates in neural networks.
method Integrates Dirichlet-based framework within Monte Carlo Dropout.
result Improves quality of uncertainty estimates in deep learning models.
Efficiently approximates uncertainty in classification models using Dirichlet distributions.
problem Inefficient computation of uncertainty estimates in Bayesian deep learning.
method Revised Laplace Bridge method to construct a Dirichlet approximation of softmax output distributions.
result The Dirichlet approximation leads to more efficient computation and better uncertainty estimates.
New method improves AI system's uncertainty estimation.
problem Precise uncertainty estimation in AI predictions.
method Information Aware Max-Norm Dirichlet Networks.
result Outperforms state-of-the-art neural networks for uncertainty estimation.
Neural networks solve the Dirichlet problem for Monge-Ampère equations.
problem Solving the Dirichlet problem for the Monge-Ampère equation.
method Using deep input convex neural networks to find the unique convex solution.
result Deep input convex neural networks can solve the Monge-Ampère Dirichlet problem.
PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.
problem Challenges in enforcing Dirichlet boundary conditions in PINNs.
method Hybrid approach combining PINNs and FEM for strong boundary condition enforcement.
result PINN-FEM outperforms standard PINN models in accuracy and robustness.
New neural network approach solves Poisson equations efficiently.
problem Approximating solutions to Poisson equations with Dirichlet boundary conditions.
method Using shallow ReLUα-networks to solve Laplace operator equations. result Neural networks can approximate solutions to the Laplace operator with Dirichlet boundary conditions efficiently.
New method samples manifolds efficiently using Dirichlet distribution.
problem Sampling on complex manifolds efficiently.
method Data-driven Dirichlet sampling on manifolds.
result Efficient sampling respects manifold structure with low computational effort.
Bayesian neural networks improve with summary information and Dirichlet process.
problem Lack of prior knowledge in BNNs for complex architectures.
method Incorporates external summary information about predicted probabilities using a Dirichlet process.
result Improves model accuracy, uncertainty calibration, and robustness.
Graph neural networks over-smooth when layers increase, reducing discriminative power.
problem Over-smoothing in graph neural networks reduces model performance as the number of layers increases.
method Analyzed over-smoothing in general graph neural network architecture using Dirichlet energy.
result The Dirichlet energy of embeddings converges to zero, leading to loss of discriminative power.
Geometric Occam's Razor shapes deep learning solutions.
problem Understanding the regularization in over-parameterized neural networks.
method Analyzing the geometric model complexity and Dirichlet energy in neural networks.
result Over-parameterized neural networks are implicitly regularized by geometric model complexity.
This research improves neural network uncertainty estimates and reliability.
problem Lack of inherent uncertainty estimates and variability in softmax scores.
method Ensemble-based Dirichlet modeling with method of moments estimator.
result Improved stability and predictive uncertainty estimates.
New PINNs method improves accuracy in computing Mean Escape Time from bounded domains.
problem Computing Mean Escape Time from bounded domains with high accuracy.
method Boundary-adapted Physics-Informed Neural Networks (PINNs) with exact Dirichlet boundary enforcement.
result Derivation of H2(Ω) a priori error bounds for PINNs with normalized distance approximations. We study the problem of multimodal generative modelling of images based on generative adversarial networks (GANs). Despite the success of existing methods, they often ignore the underlying structure of vision data or its multimodal generation characteristics. To address this problem, we introduce the Dirichlet prior fo…
Recurrent-DBN models dynamic relational data with interpretable latent structures.
problem Interpreting dynamic relational data with hidden structures.
method Recurrent Dirichlet Belief Network framework with hierarchical latent structures and efficient inference strategy.
result Recurrent-DBN discovers interpretable latent structures and improves link prediction.
Proposes a new method to improve multiclass probability calibration.
problem Uncalibrated class probabilities in multiclass classifiers leading to over-confidence.
method Dirichlet calibration method applicable to any model class, derived from Dirichlet distributions.
result Improved probabilistic predictions across various datasets and classifiers.
In the Bayesian approach to structure learning of graphical models, the equivalent sample size (ESS) in the Dirichlet prior over the model parameters was recently shown to have an important effect on the maximum-a-posteriori estimate of the Bayesian network structure. In our first contribution, we theoretically analyze…
Novel simplex-valued distribution improves on existing models.
problem Limitations of existing simplex-valued distributions like Dirichlet.
method Introducing continuous categorical distribution.
result Continuous categorical resolves limitations of Dirichlet.
The paper calculates Morse indices and nullities for embedded networks on spheres.
problem Computing Morse indices and nullities for embedded networks on spheres.
method Using the Dirichlet-to-Neumann map and properties of eigenvalues and eigenfunctions.
result For all stationary triple junction networks in S2, there is only one eigenvalue -1. The class of chain event graph models is a generalisation of the class of discrete Bayesian networks, retaining most of the structural advantages of the Bayesian network for model interrogation, propagation and learning, while more naturally encoding asymmetric state spaces and the order in which events happen. In this…
Proposes DSM priors for Bayesian neural networks to improve interpretability and robustness.
problem Bayesian neural networks struggle with interpretability, overconfidence, and adversarial attacks.
method Introduces Dirichlet scale mixture (DSM) priors to address these issues.
result DSM priors lead to sparse networks, robustness against adversarial attacks, and competitive predictive performance.
Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.
problem Solving partial differential equations on complex geometries with physics constraints.
method Combines admissible NURBS parametrizations and PINN solver for arbitrary geometries.
result High convergence rate and accuracy for most PDEs using Deep NURBS.
nnLDA combines neural and probabilistic methods for better topic modeling with side information.
problem Lack of integration of auxiliary information in traditional topic models.
method nnLDA integrates side information through a neural prior mechanism, optimizing both neural and probabilistic components.
result nnLDA outperforms traditional models in topic coherence, perplexity, and classification.
Enhances neural network solvers for PDEs with complex boundary conditions.
problem Challenges in solving PDEs with high accuracy and complex boundary conditions.
method Integrates natural gradient optimization with numerical time-stepping schemes to enforce Dirichlet boundary conditions.
result Superior accuracy and computational efficiency of the proposed methods for solving PDEs.
A classic approach for learning Bayesian networks from data is to identify a maximum a posteriori (MAP) network structure. In the case of discrete Bayesian networks, MAP networks are selected by maximising one of several possible Bayesian Dirichlet (BD) scores; the most famous is the Bayesian Dirichlet equivalent unifo…
Proposes a new Bayesian score for learning network structure from related datasets.
problem Learning network structure from heterogeneous related data sets.
method Bayesian Hierarchical Dirichlet (BHD) score based on a hierarchical model.
result BHD outperforms BDeu in reconstruction accuracy and sparsity for related datasets.
Privacy preserving networks can be modelled as decentralized networks (e.g., sensors, connected objects, smartphones), where communication between nodes of the network is not controlled by an all-knowing, central node. For this type of networks, the main issue is to gather/learn global information on the network (e.g.,…
Method learns Dirichlet-to-Neumann maps on graphs using Gaussian processes.
problem Coupling multiphysics simulations on graphs with conservation constraints.
method Gaussian processes combined with discrete exterior calculus and maximum likelihood estimation.
result Data-driven predictions with uncertainty quantification on entire graph.
BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.
problem Enforcing boundary conditions in neural networks for PDE solutions.
method Boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (BEKAN) with radial basis functions (RBFs). Incorporates Dirichlet, periodic, and Neumann conditions.
result BEKAN outperforms MLP and B-splines KAN in solving PDEs with boundary conditions.
New metrics reveal oversmoothing in GNNs more accurately than traditional methods.
problem Oversmoothing in graph neural networks reduces model performance.
method Rank-based metrics to measure oversmoothing in GNNs.
result Rank-based metrics consistently capture oversmoothing, while energy-based metrics often fail.
Proposes a new model for clustering multiplex networks with compositional data.
problem Clustering multiplex networks with multiple types of relations and compositional data.
method Multiplex Dirichlet stochastic block model for compositional networks.
result Validated through simulation and applied to international export data.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
A new model clusters network nodes based on relative edge weights.
problem Clustering networks ignores node capacities, leading to biased results.
method Proposes a Dirichlet stochastic block model for composition-weighted networks.
result Validated on simulated and real-world networks, showing improved clustering accuracy.
Being among the easiest ways to find meaningful structure from discrete data, Latent Dirichlet Allocation (LDA) and related component models have been applied widely. They are simple, computationally fast and scalable, interpretable, and admit nonparametric priors. In the currently popular field of network modeling, re…
Bayesian network structure learning is often performed in a Bayesian setting, by evaluating candidate structures using their posterior probabilities for a given data set. Score-based algorithms then use those posterior probabilities as an objective function and return the maximum a posteriori network as the learned mod…
Solves a specific Dirichlet problem on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds.
method Solves the Dirichlet problem for Monge-Ampère type equations for (n−1)-plurisubharmonic functions on Hermitian manifolds. result Solves a specific Dirichlet problem on Hermitian manifolds.
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.
Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
LDTA expands LDA's topic modeling capacity with tree-structured priors.
problem Limited expressiveness of Dirichlet priors in LDA for complex topic relationships.
method Introduces Latent Dirichlet-Tree Allocation (LDTA) with Dirichlet-Tree (DT) priors, and develops universal mean-field variational inference and Expectation Propagation.
result LDTA enables expressive, tree-structured priors over topic proportions, expanding modeling capacity of LDA.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
With the widespread success of deep neural networks in science and technology, it is becoming increasingly important to quantify the uncertainty of the predictions produced by deep learning. In this paper, we introduce a new method that attaches an explicit uncertainty statement to the probabilities of classification u…
Solves a specific Dirichlet problem on Riemannian manifolds.
problem Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds.
method Derives existence of C1,1-solutions under appropriate assumptions. result Existence of C1,1-solutions. Feature extraction has gained increasing attention in the field of machine learning, as in order to detect patterns, extract information, or predict future observations from big data, the urge of informative features is crucial. The process of extracting features is highly linked to dimensionality reduction as it impli…
Proposes a hierarchical model for learning discrete Bayesian networks with shrinkage.
problem Learning discrete Bayesian networks with high-order interactions and cell probabilities.
method Hierarchical Dirichlet shrinkage model with Metropolis-adjusted Langevin algorithm for sampling.
result Efficiently learns graph structure and selects between DAGs from sparse count data.
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.
Paper improves Bayesian network learning from related data sets.
problem Learning from heterogeneous data sets with different probabilistic structures.
method Mixed-effects models to pool information across related data sets.
result Mixed-effects models outperform traditional methods in accuracy.