This research solves Hermite interpolation on manifolds using retractions.
problem Interpolating data on non-Euclidean spaces with matching derivatives.
method Proposes a novel procedure using retractions for Hermite interpolation on various manifolds.
result Establishes the well-posedness of the method and extends Hermite interpolation results to manifolds.
In this paper, we propose a method to learn a minimizing geodesic within a data manifold. Along the learned geodesic, our method can generate high-quality interpolations between two given data samples. Specifically, we use an autoencoder network to map data samples into latent space and perform interpolation via an int…
We improve autoencoder image interpolation by shaping latent space.
problem Incongruities in autoencoder interpolation leading to artifacts or unrealistic results.
method Propose a regularization technique to shape latent space to follow a smooth, locally convex manifold consistent with training images.
result Faithful interpolation between data points achieved.
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.
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.
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.
Exploiting the deep generative model's remarkable ability of learning the data-manifold structure, some recent researches proposed a geometric data interpolation method based on the geodesic curves on the learned data-manifold. However, this interpolation method often gives poor results due to a topological difference …
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.
GPMI method interpolates uncertain atrial conduction velocity on non-Euclidean manifolds.
problem Uncertainty in atrial conduction velocity calculations.
method Gaussian Process Manifold Interpolation (GPMI) on human atrial manifolds.
result GPMI accounts for atrial topology and calculates CV uncertainty.
Two methods for interpolating manifold-valued data are presented.
problem Interpolating manifold-valued functions with derivative constraints.
method Two approaches: weighted Riemannian barycenters and tangent space interpolation.
result Both methods are valid and perform well with numerical examples.
Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In …
New method interpolates training data and is consistent for various data distributions.
problem Establishing generalization guarantees for ensemble methods in the interpolating regime.
method Developed manifold-Hilbert kernel for Riemannian manifolds and used it in ensemble classification.
result Consistent ensemble classification method for broad data distributions.
Generative models learn manifold structure; new approach uses atlas and geodesic interpolation.
problem Challenges in representing manifolds with topology different from Euclidean space.
method Atlas Generative Models (AGMs) with hybrid latent spaces and geodesic interpolation.
result Geodesic interpolation can be extended to AGMs, improving manifold representation.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
This work proposes a novel method for interpolating ROMs without solving FEM models.
problem Interpolating ROMs for unseen parameter values without solving FEM models.
method Non-intrusive Space-Time POD interpolation on compact Stiefel manifolds.
result Robust ROMs derived for unseen parameter values with strong correlations to high-fidelity simulations.
Improves latent space structure for better data representation.
problem Limited ability of conventional priors to encode data manifold structure.
method Introduces an Encoded Prior Sliced Wasserstein AutoEncoder with iterative training and geodesic interpolation.
result Learned manifold encoding preserves topological and geometric properties of data.
MFM improves generative model interpolations by learning approximate geodesics on data manifolds.
problem Straight interpolations fail to capture dynamics on data manifolds.
method Metric Flow Matching (MFM) learns approximate geodesics by minimizing kinetic energy of a data-induced Riemannian metric.
result MFM outperforms Euclidean baselines, achieving SOTA on single-cell trajectory prediction.
We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midp…
In this article we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between the functionals for harmonic and biharmonic maps. In the case of …
This paper presents an efficient algorithm for evolving point cloud data on smooth manifolds using B-Splines.
problem Evolution of point cloud data on smooth manifolds in higher dimensions.
method Lagrangian approach using adaptive B-Spline interpolation.
result Demonstrates the convergence of geometric quantities and the effectiveness of the approach.
We extend to any simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in C and in the unit disk given by Berndtsson, Ortega-Cerdà and Seip. The main tool is a comparison theorem for the Hessian in Kähler geometry due to Greene, Wu and Siu, Yau.
This work ensures stability in POD basis interpolation for pMOR in hyperelasticity.
problem Stability of POD basis interpolation on Grassmann manifolds for pMOR in hyperelasticity.
method Stability conditions derived from Grassmannian Exponential map and principal angles.
result Explicit stability conditions for practical pMOR applications and non-monotonic error behavior.
RNGI model bridges two probability densities on Riemannian manifolds efficiently.
problem Limited applicability of Euclidean stochastic interpolants to Riemannian manifolds.
method Introduces RNGI model interpolating between Riemannian manifold probability densities along geodesics.
result Proves temporal marginal density solves transport equation on Riemannian manifold.
Geometric families of low-rank covariances improve flexibility and tractability in high dimensions.
problem Interpolating and identifying covariance matrices in high dimensions with limited data.
method Differential geometric construction of low-rank covariance families, interpolation on manifolds, and distance minimization for identification.
result Differential geometric covariance families offer significant flexibility and computational tractability.
Interpolating between points is a problem connected simultaneously with finding geodesics and study of generative models. In the case of geodesics, we search for the curves with the shortest length, while in the case of generative models we typically apply linear interpolation in the latent space. However, this interpo…
A new tradeoff between regularization and sharpness improves model performance in overparameterized settings.
problem Improving model performance in overparameterized settings with minimum-norm interpolators.
method Proposes a regularization-sharpness tradeoff for overparameterized linear regression with an ℓ^p penalty.
result Empirical validation shows the tradeoff terms can distinguish performant linear interpolators.
The paper proposes a method to learn 3D object pose manifolds using GANs and elasticae.
problem Learning image manifolds of 3D objects with limited data.
method Geom-SGAN and elasticae for geometry-preserving image interpolation.
result The method outperforms state-of-the-art GANs and VAEs in learning rotation paths.
On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them in…
Deep neural networks excel at learning the training data, but often provide incorrect and confident predictions when evaluated on slightly different test examples. This includes distribution shifts, outliers, and adversarial examples. To address these issues, we propose Manifold Mixup, a simple regularizer that encoura…
Preserves positive intermediate curvature on manifolds.
problem Obstructs positive intermediate curvature on partial tori.
method Shows smooth interpolation of metrics with positive intermediate curvature.
result Proves non-existence of certain manifolds with positive intermediate curvature.
Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.
problem Approximation and interpolation for regular immersions directed by algebraically elliptic cones.
method Uses homotopy-theoretic necessary and sufficient conditions for approximation and interpolation.
result Homotopy-theoretic conditions for approximation and interpolation are satisfied in many cases of interest.
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 …
In this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature…
Exact universal interpolation property for landmark configurations in Euclidean space.
problem Representing and deforming landmark configurations through flows of vector fields.
method Explicitly describe vector fields for exact universal interpolation property in all dimensions.
result Achieve controllability by combining constant and polynomial vector fields.
Geometric theory connects machine learning classifiers to differential geometry.
problem Classifying data points in machine learning.
method Mapping binary classification to vector bundles and differential geometry.
result Harmonic interpolation solves RKHS interpolation problems.
In this paper, we firstly establish an Interpolating curvature invariance between the well known nonnegative and 2-non-negative curvature invariant along the Ricci flow. Then a related strong maximum principle for the (λ1,λ2)-nonnegativity is also derived along Ricci flow. Based on these, finally we obtain a rigid…
GAGA learns a warped metric for geometry-aware data generation and interpolation.
problem Challenges in generating data with meaningful geometry in high-dimensional datasets.
method Combines manifold learning with generative modeling to learn a warped Riemannian metric.
result GAGA improves trajectory inference by 30% in single-cell population-level data.
We replace the output layer of deep neural nets, typically the softmax function, by a novel interpolating function. And we propose end-to-end training and testing algorithms for this new architecture. Compared to classical neural nets with softmax function as output activation, the surrogate with interpolating function…
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
Ricci flow stabilizes hyperbolic 3-manifolds near the hyperbolic metric.
problem Stability of Ricci flow on hyperbolic 3-manifolds.
method Normalized Ricci-DeTurck flow with exponential convergence to the hyperbolic metric.
result Normalized Ricci-DeTurck flow converges exponentially to the hyperbolic metric.
The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.
problem Characterizing the null energy condition in Lorentzian manifolds.
method Characterization via convexity of the relative entropy along displacement interpolations on null hypersurfaces.
result The null energy condition is characterized in terms of convexity of the relative entropy.
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
New theorems compare Laplacian on Kähler manifolds.
problem Comparing Laplacian on Kähler manifolds.
method New curvature notions between Ricci and holomorphic bisectional curvatures.
result Established Laplacian comparison theorems and rigidity theorems.
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.
New method samples from multi-modal distributions on Riemannian manifolds without training.
problem Sampling from multi-modal distributions on Riemannian manifolds is challenging.
method Simulation of a non-equilibrium deterministic dynamics to transport noise toward target distributions.
result Method is entirely training-free and effective on various multi-modal problems.
GPS model predicts subspace-valued functions efficiently.
problem Accurate and efficient prediction of subspace-valued functions.
method Gaussian Process Subspace regression (GPS) model, using multivariate Gaussian distributions on Euclidean space.
result GPS provides accurate, smooth predictions with uncertainty quantification.
Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by constructing an interpolation function that maps from the input space to the low-dimen…
Upper bounds on constants for Brownian motion with sticky boundary.
problem Bounding constants for Brownian motion with sticky boundary.
method Interpolation approach based on energy interactions and Reilly formula.
result Upper bounds on Poincaré and Logarithmic Sobolev constants.