Any Sasakian structure can be closely mimicked by embeddings into weighted spheres.
problem Approximating Sasakian structures on closed manifolds.
method Using CR embeddings into weighted Sasakian spheres and strengthening previous approximation results.
result Sasakian structures can be approximated in the Cq-norm by embeddings into weighted Sasakian spheres. Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
We derive caplet volatilities for quadratic models, providing an asymptotic approximation.
problem Calculating caplet volatilities for quadratic term-structure models.
method Asymptotic approximation for caplet volatilities under quadratic models.
result Asymptotic accuracy of the derived caplet volatilities.
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…
Develops methods for structured variational inference with star-structured models.
problem Inference in models with interdependent variables.
method Star-structured variational inference, existence, uniqueness, self-consistency proofs, approximation error bounds, gradient-based algorithm.
result First results for existence, uniqueness, and self-consistency of variational approximations in star-structured models.
Paper introduces deep structured mixtures of Gaussian processes for scalable GP approximations.
problem Scalability issues with Gaussian Processes (GPs).
method Deep structured mixtures of GP experts for scalable approximate inference.
result Deep structured mixtures provide better predictive uncertainties and competitive performance.
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…
Improved sparse Gaussian processes using structured scaling matrices and Power-EP framework.
problem Scaling Gaussian processes for large datasets.
method Structured diagonal scaling matrix and Power-EP framework.
result Structured approximations improve performance without increasing computational cost.
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with ε-estimates. result Similarity structures converge to classical algebraic objects as εightarrow0. Continuous-time Bayesian networks (CTBNs) constitute a general and powerful framework for modeling continuous-time stochastic processes on networks. This makes them particularly attractive for learning the directed structures among interacting entities. However, if the available data is incomplete, one needs to simulat…
Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…
Neural and numerical methods approximate G2-structures on Calabi-Yau manifolds.
problem Approximating G2-structures on Calabi-Yau manifolds.
method Three stages: Ricci-flat metric computation, numerical approximations, and neural architecture training.
result Validated neural architecture for learning G2-structures and their metrics.
Study approximates probability measures using structured classes of functions.
problem Approximating probability measures in Wasserstein-p distance. method Structured classes of approximators for functions in Lp(Ω), transferring to measures in Wp(Ω). result Linear rate approximation for measures with densities bounded away from zero.
The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximat…
We simplify inference for TPP models with latent structures.
problem Intractable marginalization in TPP models with latent structures.
method Approximate inference over latent variables using a tight upper bound on the approximation gap.
result Improved results for models like Survival Analysis.
Theory for deep neural network approximation of score function and its derivatives.
problem Handling data distributions with low-dimensional structure and unbounded support.
method Simultaneous approximation of the score function and its derivatives using deep neural networks.
result Approximation error bounds match literature but relax bounded support requirement.
The geometry of supermanifolds provided with Q-structure (i.e. with odd vector field Q satisfying {Q,Q}=0), P-structure (odd symplectic structure ) and S-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of Batalin-Vilkovisky ap…
This work presents novel algorithms for learning Bayesian network structures with bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed-integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in uniformly sa…
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…
Constructs stable Hilbert bundles on curves using Diophantine approximation.
problem Constructing stable Hilbert bundles on complex projective curves.
method Investigating arithmetic properties of the upper half plane and applying Diophantine approximation to bound Hermitian-Einstein metrics.
result Constructs Hilbert bundles with Hermitian-Einstein metrics on curves of positive genus.
The paper analyzes deep ReLU CNNs' approximation properties in 2D space.
problem Establishing L2 approximation properties for deep ReLU CNNs. method Analysis based on decomposition theorem for convolutional kernels, properties of ReLU activation, and connections with one-hidden-layer ReLU NNs.
result Universal approximation theorem for deep ReLU CNNs with classic structure.
Structured Nonparametric Variational Inference for Dependent Latent Modeling
problem Approximating posterior distributions with complex dependencies among latent variables
method Structured Nonparametric Variational Inference (SN-VI)
result Flexible and accurate posterior approximation with arbitrary shapes
In this paper, using Donaldson's heat flow, we show that the semi-stability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian-Einstein structure on the Higgs bundle.
Compact Gaussian model approximates deep ensemble predictions.
problem Efficiently approximating deep ensemble models for image prediction.
method Sparse-structured multivariate Gaussian with Cholesky parameterization trained to match pre-trained ensemble outputs.
result Compact representation captures uncertainty and structured correlations explicitly.
Over the past years Robust PCA has been established as a standard tool for reliable low-rank approximation of matrices in the presence of outliers. Recently, the Robust PCA approach via nuclear norm minimization has been extended to matrices with linear structures which appear in applications such as system identificat…
Innovative PGMs match neural networks, revealing precise approximations during forward propagation.
problem Lack of precise semantics and probabilistic interpretation in neural networks.
method Constructing infinite tree-structured PGMs that correspond to neural networks.
result DNNs perform precise approximations of PGM inference during forward propagation.
This paper introduces a spline-based method for nonparametric ADVI that handles complex posterior distributions.
problem Learning complex posterior distributions with skewness, multimodality, and bounded support.
method Develops a spline-based nonparametric approximation approach for ADVI.
result Establishes the asymptotic consistency of the derived lower bound for importance weighted autoencoder.
New findings on how convolutional architectures approximate time series data.
problem Understanding the approximation properties of convolutional architectures in time series modeling.
method Mathematical analysis of convolutional architectures applied to time series modeling.
result A new definition of spectrum-based regularity for measuring temporal relationships under convolutional approximation.
Paper learns meaningful state and action representations from MDP trajectories.
problem Learning good state and action representations from MDP trajectories.
method Tensor decomposition, kernelization, importance sampling, low-Tucker-rank approximation.
result The learned state/action abstractions provide accurate approximations to latent block structures.
Low-rank approximation is an effective model compression technique to not only reduce parameter storage requirements, but to also reduce computations. For convolutional neural networks (CNNs), however, well-known low-rank approximation methods, such as Tucker or CP decomposition, result in degraded model accuracy becau…
Improves Bayesian neural networks inference efficiency and accuracy.
problem Inflexibility of factorized structure in Dropout posterior.
method Introduces Variational Structured Dropout (VSD) with orthogonal transformation.
result VSD induces adaptive regularization and better generalization.
Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.
Gaussian graphical models are relevant tools to learn conditional independence structure between variables. In this class of models, Bayesian structure learning is often done by search algorithms over the graph space. The conjugate prior for the precision matrix satisfying graphical constraints is the well-known G-Wish…
New model OPSS allows constant approximation for maximum coverage problem.
problem Optimizing coverage functions from samples is hard.
method Proposed OPSS model with structured samples.
result Achieved constant approximation for maximum coverage problem.
NODEs can approximate a wide range of diffeomorphisms with strong guarantees.
problem The approximation power of NODEs under certain conditions.
method Leveraging a structure theorem of the diffeomorphism group.
result NODEs can approximate a large class of diffeomorphisms with a stronger guarantee.
Bayesian structure learning improved using GFlowNets.
problem Inferring Bayesian network structure from data.
method Using Generative Flow Networks (GFlowNets) for approximating posterior DAG distributions.
result DAG-GFlowNet provides an accurate approximation of the posterior over DAGs.
Study shows smooth holomorphic structures can be approximated from weak connections.
problem Approximating smooth holomorphic structures from weak connections.
method Proves connections with specific properties can be approximated in Sobolev norms.
result Strong approximations of smooth holomorphic structures from weak connections.
Generative model captures repetitive industrial processes with varying durations and dynamics.
problem Capturing repetitive industrial processes with varying durations and dynamics using Gaussian Processes.
method Posterior-weighted Gaussian Process with a novel kernel to decouple intra-repetition and inter-repetition variability.
result Generative model produces realistic synthetic trajectories from toy datasets.
We simplify SSL by approximating redundant structural components with low-rank factorization.
problem Improving self-supervised learning performance with limited labeled data.
method Low-rank approximation of structural redundancy, introducing ε_s to measure approximation quality.
result The proposed method enhances SSL performance, as shown by theoretical and experimental validations.
Paper proposes Walsh-Hadamard Variational Inference for efficient approximate inference in large models.
problem Over-regularization in variational inference for large models.
method Walsh-Hadamard factorization strategies to reduce parameterization, accelerate computations, and increase posterior expressiveness.
result Efficient approximate inference achieved in over-parameterized models.
We simplify a complex volatility model to make it easier to price options.
problem The rough Bergomi model's non-Markovian nature complicates option pricing.
method We approximate the rBergomi model with a Bergomi model that is Markovian.
result The rBergomi model can be effectively approximated by a Markovian model.
Library learns Bayesian networks from mixed data without discretization.
problem Learning Bayesian networks from mixed data (discrete and continuous variables).
method Proposes an algorithm for structural and parameter learning of Bayesian networks from mixed data using a mixed MI score function and Gaussian approximation. Offers two graph structure enumeration algorithms.
result Advantages in solving approximation and gap recovery problems on synthetic and real datasets.
JacNet learns Jacobians to enforce structure on derivatives for invertibility and Lipschitz functions.
problem Enforcing structure on derivatives of neural network mappings.
method Proposes using a neural network to directly learn the Jacobian of the input-output function, allowing control over derivative structure.
result Demonstrates learning invertible approximations to simple and 1-Lipschitz functions.
Many predicted structured objects (e.g., sequences, matchings, trees) are evaluated using the F-score, alignment error rate (AER), or other multivariate performance measures. Since inductively optimizing these measures using training data is typically computationally difficult, empirical risk minimization of surrogate …
Dynamic trees are mixtures of tree structured belief networks. They solve some of the problems of fixed tree networks at the cost of making exact inference intractable. For this reason approximate methods such as sampling or mean field approaches have been used. However, mean field approximations assume a factorized di…
Bottom-Up Hidden Tree Markov Model is a highly expressive model for tree-structured data. Unfortunately, it cannot be used in practice due to the intractable size of its state-transition matrix. We propose a new approximation which lies on the Tucker factorisation of tensors. The probabilistic interpretation of such ap…
Proposes a balanced multi-component and multi-layer neural network for efficient function approximation.
problem Accurately and efficiently approximating complex functions with high degrees of freedom and computational cost.
method Inspired by a multi-component approach, MMNN combines single-layer networks with a multi-layer decomposition strategy.
result Significant reduction in training parameters, more efficient training process, and improved accuracy compared to FCNNs or MLPs.
The paper simplifies Bayesian posterior using clustering to make inference more manageable.
problem Handling large-scale, redundant datasets in Bayesian learning.
method Construct an approximate posterior by replacing data points in the same cluster with the centroid.
result The approximate posterior is close to the exact posterior and easier to sample from.