Abstract notes on generative modeling techniques.
problem Improving generative modeling techniques.
method Connections between optimal transport and Schrödinger bridge, flow matching.
result Showed connections between mathematical principles and generative modeling techniques.
Paper introduces a new generative learning model using Schrödinger bridge diffusion in latent space.
problem Learning distributions from divergent data distributions.
method Pre-training with large-scale models, Schrödinger bridge diffusion model in latent space.
result Effective control of second-order Wasserstein distance between generated and target distributions.
Unified framework for robust, stable, and efficient density ratio estimation.
problem Density-chasm and support-chasm problems in density ratio estimation.
method Dequantified diffusion-Schrödinger bridge (D3RE) framework with DDBI and DSBI.
result Offers uniform approximation and bounded time scores in theory and empirical performance.
CMCD sampler connects transport and variational inference for efficient sampling.
problem Efficient sampling and generative modeling in Bayesian computation.
method Developed a principled framework using divergences on path space, CMCD sampler with adaptive dynamics.
result CMCD sampler outperforms competing approaches across various experiments.
The study establishes bounds for Schrödinger operators on Riemannian manifolds.
problem Bounding Schrödinger operators on Riemannian manifolds.
method Utilizes weighted manifolds and Faber-Krahn inequalities to derive bounds.
result Establishes conditions for Schrödinger operators to be positive and for their spectra.
Generative model for time series using Schrödinger bridge.
problem Creating synthetic time series data with temporal dynamics.
method Schrödinger bridge approach for entropic interpolation via optimal transport.
result The method generates synthetic time series that respect temporal dynamics.
In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr o ¨ \ddot{o} o ¨ dinger equations on some Riemannian manifolds like the standard 2-sphere S 2 S^2 S 2 and the hyperbolic 2-space H 2 ( − 1 ) H^{2}(-1) H 2 ( − 1 ) . Using the similar idea, we establish such blow-up results on…
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V V V for bounded Schrödinger operator Δ − V Δ-V Δ − V . method Investigates weighted L 2 L^2 L 2 -boundedness of Hodge projector. result Characterizes function V V V for Schrödinger operator boundedness. Unified framework extends adjoint Schrödinger bridge sampler to discrete spaces.
problem Challenges in learning discrete neural samplers due to gradients and combinatorial complexity.
method Introduces discrete ASBS, a unified framework that extends adjoint Schrödinger bridge sampler to discrete spaces.
result Empirically, discrete ASBS achieves competitive sample quality with significant advantages in training efficiency and scalability.
ASBS improves sampling from Boltzmann distributions without importance weighting.
problem Sampling from Boltzmann distributions with known energies but unknown samples.
method Adjoint Schrödinger Bridge Sampler using kinetic-optimal transportation.
result ASBS achieves scalable and efficient sampling without importance weighting.
We give a new lower bound for the first gap λ 2 − λ 1 λ_2 - λ_1 λ 2 − λ 1 of the Dirichlet eigenvalues of the Schr{ö}dinger operator on a bounded convex domain Ω Ω Ω in R n ^n n or S n ^n n and greatly sharpens the previous estimates. The new bound is explicit and computable.
Suppose that G = ( V , E ) G=(V, E) G = ( V , E ) is a finite graph with the vertex set V V V and the edge set E E E . Let Δ Δ Δ be the usual graph Laplacian. Consider the following nonlinear Schr o ¨ \ddot{o} o ¨ dinger type equation of the form { − Δ u − α u = f ( x , u ) , u ∈ W 1 , 2 ( V ) , \left \{ \begin{array}{lcr} -Δu-αu=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right. { − Δ u − α u = f ( x , u ) , u ∈ W 1 , 2 ( V ) , on graph G G G , where $f(x…
Study shows observability for Schrödinger equations on product manifolds with specific conditions.
problem Observability of Schrödinger equations on product manifolds with product metrics.
method Proof of observability in finite time on open subsets satisfying Vertical Geometric Control Condition, under gap condition on spectrum of F(g).
result Observability on ω for the Schrödinger equation is strictly weaker than Geometric Control Condition on product of spheres.
For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs …
We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…
The goal of this article is twofold: in a first part, we prove Gaussian estimates for the heat kernel of Schr{ö}dinger operators delta + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz transform with potential d(delta+V) --1/2…
New inequalities for spectral zeta kernels on spheres and manifolds.
problem Establishing new inequalities for spectral zeta functions.
method Applying Kato's inequalities and majorisation techniques.
result Generalized Kato's comparison inequalities to higher dimensions.
Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.
problem Limited theory and algorithms for sampling infinite-dimensional function spaces.
method Adjoint Sampler for infinite-dimensional function spaces based on stochastic maximum principle.
result FAS achieves superior performance in synthetic and real systems.
Explicit formula for Reidemeister torsion of two-bridge knots.
problem Calculating Reidemeister torsion for two-bridge knots.
method Provided an explicit formula and proved vanishing identities.
result Adjoint Reidemeister torsion satisfies vanishing identities.
We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
Discrete diffusion samplers improve sampling from unnormalised densities.
problem Sampling from discrete unnormalised densities efficiently.
method Introduce off-policy training techniques and data-to-energy Schrödinger bridge training for discrete diffusion samplers.
result Improved performance on synthetic and new benchmarks.
Localized sampler tackles high-dimensional sampling with fewer samples.
problem Sampling from unknown distributions with limited data.
method Combining Schrödinger bridges and plug & play Langevin samplers with localization strategy.
result Localized sampler reduces dimensionality, making sampling more efficient.
Improved diffusion bridge sampling with rKL-LD loss.
problem Improving sampling from unnormalized distributions using diffusion bridges.
method Employing the rKL-LD loss instead of the Log Variance (LV) loss for diffusion bridges.
result rKL-LD consistently outperforms LV loss in diffusion bridges.
A new sampler for FLMs improves token-level decoding controls.
problem Sampling from FLMs using standard methods collapses marginals and produces invalid sequences.
method Samples clean one-hot endpoints from FLM token marginals and uses Ornstein-Uhlenbeck bridges conditioned on these endpoints.
result The method preserves token-wise posterior-predictive marginals and improves quality-diversity tradeoff.
Develops new samplers to approximate target distributions via modified Markov processes.
problem Approximating a target distribution with a modified Markov process.
method Iterative proportional fitting and Sinkhorn algorithm to modify transition kernels.
result Schrödinger bridge samplers can approximate target distributions and estimate their normalizing constants.
1-loop invariant equals torsion for 2-bridge knots.
problem Proving a conjecture about knot invariants.
method Combining Ohtsuki-Takata work with explicit computation.
result 1-loop term equals Reidemeister torsion for hyperbolic 2-bridge knots.
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…
New method speeds up diffusion models without requiring complex assumptions.
problem Slow sampling in diffusion models due to high computational cost.
method Training-free acceleration scheme under minimal assumptions.
result Provable acceleration within O ~ ( d 5 / 4 / ε ) \widetilde{O}(d^{5/4}/\sqrt{\varepsilon}) O ( d 5/4 / ε ) iterations. Paper reviews methods for conditional sampling in generative diffusion models.
problem Extending generative diffusion models to sample from conditional distributions.
method Review of existing computational approaches to conditional sampling.
result Highlight key methodologies for constructing conditional generative samplers.
Explicit relation found between knot torsion and TQFT signatures.
problem Relating Reidemeister torsion of two-bridge knots to TQFT signatures.
method Established an explicit relation using parabolic representations and SU 2 _2 2 -TQFT. result Inverse sum of torsions is constant and signatures have similar asymptotic behavior.
New method samples from time-integrated stochastic bridges using neural networks.
problem Sampling from time-integrated stochastic bridges with high accuracy and speed.
method Polynomial chaos expansion and artificial neural networks.
result Robust, data-driven Monte Carlo sampling with thousands of samples in milliseconds.
AdaPID optimizes diffusion-based samplers by dynamically adjusting schedules.
problem Optimizing the intermediate-time dynamics in diffusion-based samplers.
method Develops a time-varying stiffness schedule using Piece-Wise-Constant (PWC) parametrizations and a hierarchical refinement approach.
result QoS-driven PWC schedules consistently improve sampling fidelity and accuracy.
DDS samples from noisy data by reversing diffusion, providing theoretical guarantees.
problem Sampling from unnormalized densities.
method Denoising diffusion process, score matching, optimal control, Schrödinger bridges.
result DDS provides theoretical guarantees for sampling.
Method infers parameters in complex diffusion processes.
problem Parameter inference in high-dimensional, non-linear diffusion processes.
method Differentiable score matching to approximate diffusion bridges, used in an importance sampler.
result Numerically stable framework for parameter inference and diffusion mean estimation.
New method for efficient conditional sampling from diffusion models.
problem Efficient conditional simulation from diffusion models.
method Explicit forward-backward bridging to express conditional simulation as an inference problem.
result Principled particle Gibbs and pseudo-marginal samplers for conditional distribution.
DBIMs speed up DDBMs and improve image translation.
problem Efficiently sampling from DDBMs for image translation.
method Generalized diffusion bridges and booting noise.
result DBIMs are up to 25 i m e s imes im es faster and maintain generation diversity. Generative model uses Schrödinger bridges for stable sampling.
problem Sampling from unknown distributions with limited training samples.
method Combines Schrödinger bridges and Langevin dynamics.
result Effective stability and generation of samples within convex hull.
Study on heat flow across two half-lines with special boundary conditions.
problem Low energy mode of heat flow transmission across a Grushin-type cylinder.
method Analysis of heat equation with inverse-square potential and bridging boundary conditions.
result First insight into qualitative features of the heat flow solution at later times.
A new method samples from multi-modal distributions without hyperparameter tuning.
problem Sampling from multi-modal distributions is challenging and requires tuning hyperparameters.
method Learned Reference-based Diffusion Sampler (LRDS) that learns a reference model on high-density regions and uses it to train a diffusion-based sampler.
result LRDS best exploits prior knowledge on multi-modal distributions compared to competing algorithms.
In this paper, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of A…
Generative Bayesian Inference uses GANs for approximate posterior sampling.
problem Bayesian inference without explicit likelihoods.
method Develops Bayesian GAN (B-GAN) for posterior simulation.
result B-GAN achieves highly competitive performance in posterior sampling.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n n n -th order spectral variations of singular values. This work presents a partitioned solution procedure to compute shape gradients in fluid-structure interaction (FSI) using black-box adjoint solvers. Special attention is paid to project the gradients onto the undeformed configuration. This is due to the mixed Lagrangian-Eulerian formulation of large-displacement FSI in…
Derives adjoint polynomials of torus knots in explicit form.
problem Understanding adjoint invariants of torus knots.
method Closed-form double sum expression derivation.
result Explicit double sum form of adjoint polynomials.
Corrected samplers reduce discretization error in discrete flow models without additional computational cost.
problem Discretization error in samplers for discrete flow models.
method Established non-asymptotic error bounds for samplers, proposed time-corrected and location-corrected samplers.
result Location-corrected sampler has lower complexity and better generation quality.
Stochastic gradient Langevin dynamics (SGLD) is a computationally efficient sampler for Bayesian posterior inference given a large scale dataset. Although SGLD is designed for unbounded random variables, many practical models incorporate variables with boundaries such as non-negative ones or those in a finite interval.…
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
New samplers improve MCMC efficiency in high dimensions.
problem Efficient sampling in high-dimensional problems.
method Affine invariant ensemble samplers, including derivative-free and derivative-based HMC.
result Affine invariant ensemble HMC outperforms standard HMC in high dimensions.