One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the …
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48 results for “interpolating Hamiltonians”
Proposes a geometry-aware VAE for better latent space modeling.
problem Lack of meaningful latent space structure in VAEs for small datasets.
method Introduces a Riemannian Hamiltonian VAE with a learned metric.
result Improves latent space structure leading to better interpolations and data generation.
We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, …
Introduces HGN for learning Hamiltonian dynamics from images.
problem Learning Hamiltonian dynamics from high-dimensional data.
method Hamiltonian Generative Network (HGN) and Neural Hamiltonian Flow (NHF).
result HGN can sample, roll out, and modify learned dynamics.
CHMC improves HMC efficiency for multimodal distributions.
problem Slow convergence of HMC in multimodal distributions.
method Integrates a counterdiabatic term to optimize Hamiltonian changes.
result CHMC achieves efficient sampling from challenging distributions.
Symplectic GP regression models Hamiltonian systems for particle tracing.
problem Efficiently modeling long-term Hamiltonian flow maps for charged particles.
method Multi-output Gaussian process regression with symplectic matrix-valued covariance function.
result Symplectic methods outperform existing approaches in learning Hamiltonian functions.
Develops a new deformation theory for Dirac structures.
problem Interpolating between twisted Dirac and Poisson geometries.
method Introduces a new deformation theory compatible with Dirac geometry operations.
result Uniform deformation theory recovering various special cases.
Investigates billiard dynamics on smooth curves in higher dimensions.
problem Extending conventional billiard dynamics to higher-dimensional smooth curves.
method Area-preserving twist map, KAM theory, Mather's converse KAM, interpolating Hamiltonians.
result Uniform distribution of impact points for small chords on nice wires.
Study on feature learning in Leaky ResNets, explaining bottleneck structure.
problem Understanding feature learning in deep neural networks.
method Lagrangian and Hamiltonian reformulation of representation geodesics.
result Emergence of a bottleneck structure in large effective depth networks.
New method learns population dynamics from snapshots, outperforming existing models.
problem Capturing periodic and other dynamical properties of population dynamics.
method Wasserstein Lagrangian Mechanics (WLM) for learning second-order dynamics from observed marginals.
result WLM outperforms existing methods across various dynamics, including vortex dynamics, embryonic development, and flocking.
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…
AIS uses a suboptimal extended target distribution, which this paper improves using SGM.
problem Improving the efficiency of Annealed Importance Sampling for marginal likelihood estimation.
method Leveraging score-based generative modeling to approximate the optimal extended target distribution.
result Demonstrated novel, differentiable AIS procedures on synthetic and real-world data.
The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant…
Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.
problem Limited coverage of O(n)-invariant metrics by kernel metrics.
method Characterization of O(n)-invariant metrics, intermediate classes construction.
result Introduction of cometric-stability as a key property for geodesics.
This paper shows how forward rate interpolations are equivalent to discount factor interpolations in yield curve construction.
problem The challenge of choosing between different interpolation methods for yield curve construction.
method Demonstrates the equivalence between forward rate interpolations and discount factor interpolations.
result Some popular interpolation methods on forward rates are equivalent to classical interpolation methods on discount factors.
We create a smooth manifold of triangular meshes with a geodesically complete metric.
problem Representing and manipulating 2D shapes as triangular meshes.
method Developed a geodesically complete Riemannian metric for triangular meshes.
result The metric preserves mesh connectivity and avoids mesh degradation.
The paper proposes a method for generating uniform interpolations on data manifolds.
problem Generating high-quality interpolations between data samples on complex manifolds.
method Autoencoder network with interpolation network, regularized by a Riemannian metric.
result The method generates interpolations that remain within the manifold's distribution.
Study finds conditions for Legendre curves to be interpolating sesqui-harmonic in Sasakian space forms.
problem Characterizing Legendre curves in Sasakian space forms.
method Analyzes necessary and sufficient conditions for Legendre curves to be interpolating sesqui-harmonic.
result Obtains an example of an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
New FX option interpolations impact implied volatilities.
problem Different interpolations of FX option quotes lead to varying implied volatilities.
method Analysis of various exact interpolations of broker quotes.
result Different interpolations result in different implied volatilities.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Bayesian interpolants explain neural network inferences concisely.
problem Understanding neural network inferences.
method Adapting Craig interpolants for neural networks.
result Produces precise, understandable explanations.
Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalizationstat.ML
Near-interpolating models grow norms quickly, affecting generalization.
problem Understanding the trade-off between interpolation and generalization in near-interpolating models.
method Random matrix theory and eigendecay analysis of data covariance matrix.
result Near-interpolating models exhibit rapid norm growth and worse generalization trade-offs.
The paper improves interpolation in generative models by using specific base distributions.
problem Unexpected side effects in linear interpolations of normalizing flows.
method Enforces a specific manifold using Dirichlet and von Mises-Fisher base distributions.
result Superior performance in terms of bits per dimension, FID, and KID scores for interpolation.
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
Paper presents a unified approach to interpolation and geodesics in latent spaces of generative models.
problem Finding geodesics and interpolating in latent spaces of non-Gaussian densities.
method General approach to interpolation and geodesics in latent space for arbitrary density.
result Maximizing quality measure of an interpolating curve is equivalent to finding geodesic.
Improved sparse-view CT images with deep learning sinogram interpolation.
problem Sparse-view CT images quality improvement with limited projection data.
method Combination of U-Net and residual learning for sinogram interpolation.
result Significantly improved CT image quality (RMSE and SSIM metrics) over standard methods.
Classifies Hamiltonian and quasi-Hamiltonian manifolds with specific group actions.
problem Classifying specific types of manifolds under group actions.
method General classification of multiplicity free manifolds, focusing on rank one.
result Obtained numerous new concrete examples of quasi-Hamiltonian manifolds.
Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.
problem Existence and properties of non-vertical fully-Hamiltonian vector fields in almost symplectic manifolds with Lagrangian fibrations.
method Investigates vector fields in 2n-dimensional almost symplectic manifolds with Lagrangian fibrations, focusing on partially-Hamiltonian and fully-Hamiltonian vector fields.
result Non-vertical fully-Hamiltonian vector fields exist under certain genericity conditions and can be reduced to families of symplectic-Hamiltonian vector fields.
Deep neural networks can interpolate any dataset in the overparametrized regime.
problem Interpolating any dataset with deep neural networks in the overparametrized regime.
method Proving universal approximations and interpolating any dataset with deep neural networks, considering specific conditions on activation functions.
result Interpolation of any dataset is possible in the overparametrized regime with deep neural networks.
Interpolation hurts robust generalization even without noise.
problem The challenge of robust generalization in the absence of noise.
method Avoiding interpolation through ridge regularization.
result Ridge regularization improves robust generalization.
Uniform convergence of interpolators proven for Gaussian data.
problem Interpolation learning in high-dimensional linear regression with Gaussian data.
method Generic uniform convergence guarantee in terms of Gaussian width.
result Consistency of interpolators for minimum-norm and near-minimal-norm cases.
Interpolation improves nearest neighbor algorithms' performance.
problem Improving nearest neighbor algorithms' performance.
method Considered a class of interpolated weighting schemes and characterized their asymptotic performances.
result Mild degree of data interpolation strictly improves prediction accuracy and statistical stability.
The paper characterizes vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
problem Characterizing vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
method Characterization theorem and critical point condition for interpolating sesqui-harmonic vector fields.
result Conditions for vector fields to be interpolating sesqui-harmonic maps on compact manifolds.
SoftKI combines SKI and variational methods for scalable GP regression.
problem Scalable Gaussian Process regression on high-dimensional datasets.
method SoftKI approximates kernel via softmax interpolation from a smaller number of learned points.
result SoftKI is competitive with other approximated GP methods for modest data dimensions.
Proves a new law of robustness for interpolating arbitrary data distributions.
problem Understanding robust interpolation for arbitrary data distributions.
method Proves a Lipschitzness lower bound for robust interpolation.
result Demonstrates a two-fold law of robustness for interpolating functions.
Develops Hamiltonian Score Matching and Generative Flows for machine learning.
problem Estimating score functions and designing generative models.
method Introduces Hamiltonian velocity predictors (HVPs) for score matching and generative flows.
result Hamiltonian Generative Flows (HGFs) rival leading generative modeling techniques.
Holographic energy equals Hamiltonian energy.
problem Equating holographic and Hamiltonian energies.
method Relative holographic and Hamiltonian energy comparison.
result Holographic energy is identical to Hamiltonian energy.
Sum of Hamiltonian manifoldsmath.SG
Summing Hamiltonian manifolds with a common submanifold.
problem Combining Hamiltonian manifolds with a shared submanifold.
method Establishing symplectic reduction and comparing Chern classes.
result Symplectic reduction of the sum agrees with the sum of reductions.
Interpolating estimators in nonparametric regression become suboptimal under adversarial attacks.
problem Adversarial robustness of interpolating estimators in nonparametric regression.
method Investigation of adversarial robustness of interpolating estimators in a nonparametric regression framework.
result Interpolating estimators must be suboptimal even under a subtle future -attack.
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
Paper investigates optimal interpolation methods in linear regression.
problem Understanding when interpolating methods generalize well in linear regression.
method Investigates optimal response-linear interpolators using functions linear in the response variable.
result Provides a closed-form expression for the optimal interpolator and shows it can be derived as the limit of gradient descent.
The paper studies co-Hamiltonian diffeomorphisms on compact cosymplectic manifolds.
problem Fix-point theory and co-Hamiltonian diffeomorphisms on compact cosymplectic manifolds.
method Fix-point theory, Arnold's conjecture, co-Hofer norms, topologies, approximations lemmas.
result Minimum number of fix points for co-Hamiltonian diffeomorphisms is at least 1.
Compactness proven for isospectral Birkhoff billiard tables.
problem Proving compactness of isospectral Birkhoff billiard tables.
method Derived a hierarchical structure for integral invariants and used interpolating Hamiltonian.
result Compactness of equivalence classes of marked length isospectral Birkhoff billiard tables.
Covariance shrinkage via stochastic interpolation
problem High-dimensional covariance estimation
method Recasting shrinkage as empirical risk minimization
result Reduces statistical risk through scheduling, flow maps, and early stopping
Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.
problem Understanding the phase diagram of kernel interpolation in large dimensions.
method Characterization of variance and bias under various source conditions.
result Determined the -phase diagram of large-dimensional kernel interpolation.
This work predicts and interpolates long-range videos using unsupervised landmarks.
problem Predicting and interpolating long-range video data with occlusions and appearance changes.
method Unsupervised latent structure inference followed by temporal prediction in a latent space.
result High-quality long-range video interpolation and extrapolation achieved through landmark representation.
Study on biharmonic and interpolating sesqui-harmonic vector fields on para-Kähler--Norden manifolds.
problem Investigating higher-order harmonicity in pseudo-Riemannian geometry.
method Deriving first variations of bienergy and interpolating sesqui-energy functionals, characterizing biharmonic and interpolating sesqui-harmonic vector fields.
result Explicit characterizations and examples of vector fields satisfying biharmonic and interpolating sesqui-harmonic conditions.
New algorithms improve MCMC efficiency for complex distributions.
problem High variance and low effective sample size in MCMC samplers.
method Antithetic Riemannian Manifold and Quantum-Inspired Hamiltonian Monte Carlo.
result Improved effective sample size and variance reduction.