Variational inference is an umbrella term for algorithms which cast Bayesian inference as optimization. Classically, variational inference uses the Kullback-Leibler divergence to define the optimization. Though this divergence has been widely used, the resultant posterior approximation can suffer from undesirable stati…
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48 results for “variational operators”
Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equationsmath.DG
Characterizes symplectic and variational operators for scalar evolution equations.
problem Understanding the cohomology spaces and operators for scalar evolution equations.
method Analyzes cohomology spaces and uses isomorphisms to characterize operators.
result Cohomology spaces and operator spaces are isomorphic for certain scalar evolution equations.
We derive the first and second variation formula for the Green's function pole's value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of confor…
VANO uses neural operators for unsupervised learning of functional data.
problem Learning operators between infinite dimensional spaces for functional data.
method Variational Autoencoding Neural Operators (VANO) approach.
result VANO can learn and reconstruct functional data without supervision.
Study efficient neural operator learning using variation spaces.
problem Operator learning using encoder-decoder neural networks.
method Introduce variation space for nonlinear operators, establish approximation bounds.
result Algebraic approximation and learning rates for polynomially decaying input and output encoding errors.
The extragradient method fails for hypomonotone variational inequalities.
problem The convergence of the extragradient method for hypomonotone variational inequalities.
method Application of the extragradient method to hypomonotone linear operators.
result The extragradient method diverges for hypomonotone variational inequalities.
New method identifies vanishing arcs for curve singularities.
problem Characterizing arcs sent to geometric vanishing cycles.
method Introducing geometric variation operator and vanishing arcsets.
result Existence of topological exceptional collections of arcsets.
We establish higher-order weighted Sobolev and Holder regularity for solutions to variational equations defined by the elliptic Heston operator, a linear second-order degenerate-elliptic operator arising in mathematical finance. Furthermore, given -smooth data, we prove -regularity of solutions up t…
Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.
problem Eigenvalue problem for complex Hessian operator on pseudoconvex manifolds.
method Established -regularity and uniqueness of the first eigenfunction, derived variational formula for the first eigenvalue.
result Derivation of a bifurcation-type theorem and geometric bounds for the eigenvalue.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
Study fourth-order geometric flow of shape operator for co-dimension one immersions.
problem Analyzing the geometry of isometric immersions in Riemannian manifolds.
method Introduce a moduli flow to decrease curvature variation energy.
result The flow decreases a natural energy measuring curvature variation.
Enhanced DeepONet framework with uncertainty quantification for complex operators.
problem Learning complex operators with uncertainty quantification.
method Generalised variational inference (GVI) using Rényi's α-divergence.
result Superior predictive accuracy and uncertainty quantification.
Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.
problem Deriving formulas for eigenvalues of elliptic operators on compact manifolds.
method Variational methods applied to elliptic operators on compact Riemannian manifolds.
result Generic subsets of metrics yield simple spectra of elliptic operators.
In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's -entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly …
PDERL improves evolutionary reinforcement learning by using learning-based variation operators.
problem Scalability issue in Genetic Algorithms when combined with Deep Neural Networks.
method Integrates evolutionary and reinforcement learning through a hierarchical approach with learning-based variation operators.
result PDERL outperforms traditional evolutionary and reinforcement learning methods in robot locomotion tasks.
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…
New method accelerates energetic variational inference using particle dynamics.
problem Efficiently solving variational inference problems with reduced computational cost.
method Particle-based variational inference with implicit scheme, inspired by energy quadratization and operator splitting.
result Significantly reduces computational cost compared to existing methods.
Constructs Lepage equivalents for arbitrary-order Lagrangians.
problem Creating Lepage equivalents for complex Lagrangians.
method Uses variational bicomplex and symmetric linear connections to construct Lepage equivalents satisfying the closure property.
result Shows how to extend global Lepage equivalents to ones satisfying the closure property.
A new method solves variational inequality problems with multiple constraints without needing optimal Lagrange multipliers.
problem Solving variational inequality problems with multiple functional constraints efficiently.
method Constrained Gradient Method (CGM) for Minty variational inequality problems.
result The Constrained Gradient Method achieves complexity similar to projection-based methods but with cheaper oracles.
We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise -j…
Develops inference combinators for probabilistic programs using neural networks.
problem Creating efficient proposals for probabilistic program inference.
method Inference combinators using neural network parameterization of proposals.
result Correct by construction variational methods tailored to specific models.
New Stein operator improves robustness in model inference.
problem Improving robustness in inference for unnormalized models.
method Density-power weighted Stein operator (-Stein operator).
result Robust methods for goodness-of-fit testing and posterior approximation.
Defines a filtration on variational bicomplex for concise functional form conditions.
problem Expressing functional form vanishing conditions concisely.
method Introduces a filtration on the variational bicomplex and studies its properties.
result Graded components of the filtration inherit module structures, simplifying functional form conditions.
Improves gradient estimation for discrete distributions with variance reduction techniques.
problem Excessive variance in gradient estimation for discrete distributions.
method Stein operators for discrete distributions and control variates.
result Substantially lower variance in gradient estimation.
A new framework for recycling Gaussian process approximations.
problem Efficiently combining multiple Gaussian process approximations.
method Construct variational ensembles using a dictionary of fitted Gaussian processes.
result Framework allows for various tasks and scalability.
We discuss some consequences Fintushel-Stern `knot surgery' operation on 4-manifolds coming from its handlebody description. We give some generalizations of this operation and give a counterexample to their conjecture.
Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.
problem Understanding the relationship between the Schwarzian derivative and variational equations.
method Analyzing the Schwarzian derivative as a first integral and Euler-Lagrange operator for specific variations.
result The Schwarzian derivative is both a first integral and the Euler-Lagrange operator for a certain class of variations.
Decoupled Networkscs.CV
Inner product-based convolution has been a central component of convolutional neural networks (CNNs) and the key to learning visual representations. Inspired by the observation that CNN-learned features are naturally decoupled with the norm of features corresponding to the intra-class variation and the angle correspond…
We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) suc…
We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the as…
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
VB-DeepONet uses Bayesian inference to improve DeepONet's predictions and uncertainty quantification.
problem Overfitting and lack of uncertainty quantification in DeepONet.
method Variational Bayes approach to approximate posterior distribution, reducing computational cost.
result VB-DeepONet alleviates DeepONet's limitations and provides uncertainty quantification.
First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept o…
Reparameterization of variational auto-encoders with continuous random variables is an effective method for reducing the variance of their gradient estimates. In the discrete case, one can perform reparametrization using the Gumbel-Max trick, but the resulting objective relies on an operation and is non-dif…
We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation o…
New samplers minimize KL divergence for constrained and non-Euclidean geometries.
problem Efficient sampling from constrained and non-Euclidean distributions.
method Stein Variational Mirror Descent and Mirrored Stein Variational Gradient Descent.
result New samplers converge more rapidly and accurately than prior methods.
Learning Model Reparametrizations: Implicit Variational Inference by Fitting MCMC distributionsstat.ML
We introduce a new algorithm for approximate inference that combines reparametrization, Markov chain Monte Carlo and variational methods. We construct a very flexible implicit variational distribution synthesized by an arbitrary Markov chain Monte Carlo operation and a deterministic transformation that can be optimized…
We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As part of this formalism we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e. SL(2,C)) 2…
We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operat…
Researchers create new operators from Riemannian invariants.
problem Developing new mathematical tools for Riemannian geometry.
method Introducing formally self-adjoint conformally covariant polydifferential operators.
result Found a fourth-order, conformally covariant tridifferential operator.
VINNAS uses variational inference to avoid mode collapse in neural architecture search.
problem Mode collapse in gradient-based NAS methods, leading to suboptimal architectures.
method Differentiable variational inference with variational dropout and automatic relevance determination.
result State-of-the-art accuracy with up to twice fewer non-zero parameters.
Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.
problem Extending variational formula for Bismut-Cheeger eta form without kernel bundle assumption.
method Twisting spin Dirac operators by isomorphic vector bundles, proving -graded additivity.
result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in K-theory.
The Bialynicki-Birula decomposition of the space of lambda-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomor…
The flat trace of geodesic Koopman operators varies with negatively curved surfaces.
problem Understanding how the flat trace of geodesic Koopman operators changes with variations of negatively curved surfaces.
method Computing the first variation of the flat trace as a distribution and analyzing its leading singularity.
result The leading singularity coefficient is a linear functional of length variations, forcing marked lengths to be locally constant.
A new variational method for SSMs improves inference efficiency.
problem Hard variational inference for state space models.
method Proposes variational marginal particle filter (VMPF) based on Rao-Blackwellization.
result VMPF provides tighter variational bounds and sometimes benefits from unbiased reparameterization.
For For a given PDE system, or an exterior differential system possessing a Lie group of internal symmetries the orbit reduction procedure is introduced. It is proved that the solutions of the reduced exterior differential system are in one-to-one correspondence with the moduli space of regular solutions of the prolong…
Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators. At present two examples of generalized Yang-Baxter operators are known and recently three types of variations for one of these were discovered. We present the definition of enha…
We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze vario…