Variational approach to basic manifold structures.
problem Understanding basic differential geometric structures.
method Variational description of geometric structures.
result Variational formulation of manifold structures.
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48 results for “variational structure”
The study characterizes complex structures using calculus of variations.
problem Variational characterization of complex structures.
method Calculus of variations for real vector bundle valued differential forms.
result Obtains variational characterization of complex structures.
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.
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
Characterizes obstacles to variational formulation of Dirac dynamics.
problem Obstacles to variational formulation of Dirac dynamics.
method Characterizes using basic cohomology for Dirac structures and Lie algebroids.
result Characterizes the obstruction to variational formulation.
The paper introduces structured variational families to improve scalability in black-box variational inference.
problem Scalability issues in black-box variational inference, especially for large datasets and hierarchical models.
method Developed structured variational families that achieve better iteration complexity of O(N) compared to full-rank families.
result Structured variational families can achieve better scaling with respect to dataset size N, improving iteration complexity from O(N^2) to O(N).
We give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan's -minimal models of certain differential graded algebras associated with real variations of Hodge structures.
Sheng and Zuo's characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross's canonical variations of Hodge structure of Calabi-Yau type over (Hermitian symmetric) tube domains.
The paper calculates variations of Einstein-Hilbert action on CR manifolds.
problem Variation of the Einstein-Hilbert action in pseudohermitian geometry.
method Computed first and second variations on CR manifolds, characterized critical points as pseudo-Einstein structures, and analyzed second variation on standard spheres.
result In three dimensions, the second variation of the Einstein-Hilbert action on CR structures differs from the Riemannian case due to embeddability.
Recent efforts on combining deep models with probabilistic graphical models are promising in providing flexible models that are also easy to interpret. We propose a variational message-passing algorithm for variational inference in such models. We make three contributions. First, we propose structured inference network…
A new EVI framework improves ParVI methods by maintaining variational structure and reducing KL-divergence.
problem Improving variational inference methods for better approximation of target distributions.
method EVI framework that minimizes the VI objective function based on an energy-dissipation law, including a new 'Approximation-then-Variation' scheme.
result The new scheme significantly decreases KL-divergence and outperforms existing ParVI methods in fidelity.
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.
ASVI automates variational inference for complex models.
problem Efficient variational inference for complex probabilistic models.
method Automatic structured variational inference (ASVI) using convex updates.
result ASVI outperforms other methods on a wide range of problems.
Improved Bayesian uncertainty quantification using variational bagging.
problem Inefficient and underestimating uncertainty in mean-field variational Bayes.
method Integrates bagging with variational Bayes for improved inference.
result Bagged variational posterior provides proper uncertainty quantification.
tvGP-VAE models tensor-valued latent variables with Gaussian processes for better data structure representation.
problem Agnostic latent variables in VAEs ignore data structure correlations.
method Proposes tensor-variate Gaussian process prior for variational autoencoder.
result Explicitly modeling correlation structures improves model performance in reconstruction.
Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Cascading flows improve variational inference in structured programs.
problem Challenges in variational inference for complex probabilistic programs.
method Integrates normalizing flows and ASVI to create cascading flows, which embed the forward-pass of probabilistic programs.
result Cascading flows outperform normalizing flows and ASVI in structured inference problems.
After a brief introduction to several variational problems in the study of shapes of thin thickness structures, we deal with variational problems on 2-dimensional surface in 3-dimensional Euclidian space by using exterior differential forms. The morphological problems of lipid bilayers and stabilities of cell membranes…
We show a very simple and general total second variation formula for Perelman's -functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for P…
Variational Auto-Encoders (VAEs) have been widely applied for learning compact, low-dimensional latent representations of high-dimensional data. When the correlation structure among data points is available, previous work proposed Correlated Variational Auto-Encoders (CVAEs), which employ a structured mixture model as …
The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.
Study variational properties of cone structures with infinitesimal symmetry.
problem Variational properties of cone structures with infinitesimal symmetry.
method Establishing a correspondence between cone structures and geometric structures via symmetry reduction and quasi-contactification.
result Invariant conditions for specific properties of cone structures.
Liouville structuresmath-ph
A 'Liouville structure' is a structure isomorphic to a cotangent vector fibration. A Liouville structure is an essential ingredient of every variational formulation of a physical theory. For reasons of interpretation the Liouville structure can not be replaced by the corresponding cotangent fibration. We give a precise…
We develop a framework for incorporating structured graphical models in the \emph{encoders} of variational autoencoders (VAEs) that allows us to induce interpretable representations through approximate variational inference. This allows us to both perform reasoning (e.g. classification) under the structural constraints…
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
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.
Beta process is the standard nonparametric Bayesian prior for latent factor model. In this paper, we derive a structured mean-field variational inference algorithm for a beta process non-negative matrix factorization (NMF) model with Poisson likelihood. Unlike the linear Gaussian model, which is well-studied in the non…
Paper uses non-Euclidean analysis to classify brain structure variations.
problem Classifying joint variations in multi-object brain structures.
method Combines non-Euclidean statistics and non-parametric integrative analysis.
result Effective, robust, and interpretable joint structure found.
The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the ra…
In this paper, we propose a novel structure for a cross-modal data association, which is inspired by the recent research on the associative learning structure of the brain. We formulate the cross-modal association in Bayesian inference framework realized by a deep neural network with multiple variational auto-encoders …
Study on complex variation of Hodge structures for non-Kähler manifolds.
problem Understanding complex variation of Hodge structures for non-Kähler manifolds.
method Analyzes holomorphic families of compact complex manifolds with specific cohomology properties.
result Period map is holomorphic and transversal under given conditions.
Variational Causal Networks approximate Bayesian inference over causal structures.
problem Quantifying uncertainty in causal structure inference from finite data.
method Parametric variational family over DAGs, using Evidence Lower Bound (ELBO) for tractable learning.
result Approximation of the true posterior over DAGs is demonstrated to be good.
We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.
New federated learning method for structured models.
problem Limited federated learning for structured probabilistic models.
method Structured variational inference adapted for federated learning.
result Effective algorithms compared to hierarchical Bayesian models.
VFG model embeds flow-based models with hierarchical structures using variational inference.
problem Flow-based models struggle with high-dimensional latent spaces and lack of tractable inference for graphical structures.
method Integrates flow-based functions through variational inference with aggregation nodes for hierarchical information integration.
result VFG models achieve improved ELBO and likelihood values on multiple datasets.
The cohomology of a variation of polarized Hodge structures over a quasi-compact Kähler manifoldmath.AG
A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact Kähler manifolds. Several isomorphisms between different cohomologies given.
Generating graph structures is a challenging problem due to the diverse representations and complex dependencies among nodes. In this paper, we introduce Graph Variational Recurrent Neural Network (GraphVRNN), a probabilistic autoregressive model for graph generation. Through modeling the latent variables of graph data…
Learning in the latent variable model is challenging in the presence of the complex data structure or the intractable latent variable. Previous variational autoencoders can be low effective due to the straightforward encoder-decoder structure. In this paper, we propose a variational composite autoencoder to sidestep th…
Continuous latent time series models are prevalent in Bayesian modeling; examples include the Kalman filter, dynamic collaborative filtering, or dynamic topic models. These models often benefit from structured, non mean field variational approximations that capture correlations between time steps. Black box variational…
Study functionals on almost complex structures for Yau's Challenge.
problem Yau's Challenge on compact -manifolds.
method Variational properties of functionals on almost complex structures.
result Variational properties could be used to tackle Yau's Challenge.
Study on -structures manifold geometry.
problem Understanding the space of closed -structures.
method Developed Sobolev-type metrics, Levi-Civita connections, geodesic equations.
result Formulated variational structures of torsion-free -structures.
We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidea…
The paper shows that almost every path structure is not variational.
problem Determining if a path structure is variational.
method Generalized Douglas's result to higher dimensions and analyzed path geometries with infinitesimal symmetries.
result Almost every path structure is not variational.
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.
Construct quaternionic-Kähler metrics from special Kähler manifolds with specific BPS structure variations.
problem Construct quaternionic-Kähler metrics from special Kähler manifolds with mutually local variations of BPS structures.
method Construct quaternionic-Kähler metrics from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. Provide global and local explicit formulas for the quaternionic-Kähler metric.
result Construct quaternionic-Kähler metrics that are positive-definite and deform the 1-loop corrected Ferrara-Sabharval metric.
There are a number of examples of variations of Hodge structure of maximum dimension. However, to our knowledge, those that are global on the level of the period domain are totally geodesic subspaces that arise from an orbit of a subgroup of the group of the period domain. That is, they are defined by Lie theory rather…
We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.
The study finds variations in ownership structure and efficiency across sectors in Malaysia.
problem Investigate variations in ownership structure and firm efficiency across sectors in Malaysia.
method Frequency distributions of ownership structure, DEA under CRS and VRS, stratified random sampling.
result There are variations in firm ownership structure and efficiency across sectors in Malaysia.