We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain an example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.
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48 results for “Interpolation Spaces”
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.
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…
Interpolates curves using maximal and minimal surfaces in different spaces.
problem Interpolating curves in spacelike and Euclidean spaces.
method Using maximal and minimal surfaces, interpolates curves based on the Björling problem.
result Constructs surfaces to interpolate curves in a specified manner.
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.
Enhances interpolation paths in latent space using particle filters.
problem Generating meaningful interpolations between data points in latent space.
method Introduces a discriminator network to guide particle filter sampling of interpolation paths.
result Improved variability and stronger drift towards high data density areas.
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.
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.
This work bridges stochastic interpolants to infinite-dimensional Hilbert spaces.
problem Limited flexibility in generating arbitrary distributions for function-valued data.
method Establishes a rigorous framework for stochastic interpolants in infinite-dimensional Hilbert spaces.
result Achieves state-of-the-art results in conditional generation for complex PDE-based benchmarks.
In this article, a proof of the interpolation inequality along geodesics in -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…
The paper proves approximation and interpolation theorems for maxfaces with singularities.
problem Proving approximation and interpolation theorems for maxfaces with singularities.
method Surveying and applying Enneper--Weierstrass representation formula methods to maxfaces, incorporating singularity criteria.
result Existence of maxfaces with prescribed singularities and maxfaces with dense image singular set.
We present a neural network architecture based upon the Autoencoder (AE) and Generative Adversarial Network (GAN) that promotes a convex latent distribution by training adversarially on latent space interpolations. By using an AE as both the generator and discriminator of a GAN, we pass a pixel-wise error function acro…
Deep ReLU networks need Ω(N) parameters to interpolate at irregularly spaced points.
problem Interpolating at irregularly spaced data points with deep ReLU networks.
method Analyzing the number of parameters required for interpolation.
result Ω(N) parameters are necessary for interpolation when δ is exponentially small in N.
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.
Gradient flow in parameters equals linear interpolation in outputs.
problem Understanding and optimizing training algorithms in deep learning.
method Proving equivalence between gradient flow in parameter space and linear interpolation in output space, and deriving formulas for global minima.
result Gradient flow in parameters can be transformed into linear interpolation in outputs, leading to global minima.
The study identifies conditions under which algorithmic stability explains generalization in interpolating learning systems.
problem Understanding when algorithmic stability explains generalization in interpolating learning systems.
method Modeling training as a function-space trajectory and measuring sensitivity to single-sample perturbations.
result There exist interpolating regimes with small risk where contractive sensitivity cannot hold, showing that stability is not a universal explanation.
Prediction and interpolation for long-range video data involves the complex task of modeling motion trajectories for each visible object, occlusions and dis-occlusions, as well as appearance changes due to viewpoint and lighting. Optical flow based techniques generalize but are suitable only for short temporal ranges. …
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.
In implicit models, one often interpolates between sampled points in latent space. As we show in this paper, care needs to be taken to match-up the distributional assumptions on code vectors with the geometry of the interpolating paths. Otherwise, typical assumptions about the quality and semantics of in-between points…
Interpolates Sol geometry to Hyperbolic Space with a parameter.
problem Analyzing curvature and geometry of Lie groups.
method One-parameter family of solvable Lie groups with canonical metrics.
result Characterization of cut locus maximizing scalar curvature.
New approach uses interpolation models and error bounds for verifiable scientific machine learning.
problem Challenges in verifying and validating modern scientific machine learning workflows.
method Combines multiple standard interpolation techniques with error bounds for efficient computation and comparative performance analysis.
result Error bounds for interpolation techniques can be computed or estimated efficiently, aiding in validation goals.
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.
Paper investigates conditions for independence of weak gradients on metric spaces.
problem Dependence of weak gradients on in arbitrary metric measure spaces.
method Investigates the Bounded Interpolation Property to ensure independence of weak gradients.
result Bounded Interpolation Property guarantees independence of weak gradients.
The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
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.
Veronese webs appear as the natural way of passing to the quotient of curves in the projective space. In thi paper, we give the link between classical multidimensionnal webs and veronse webs by mean of interpolation.
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.
New method efficiently interpolates nonparametric density estimators.
problem Efficient evaluation of nonparametric density estimators.
method Piecewise multivariate polynomial interpolation scheme.
result New estimator with low space requirements and efficient querying.
Deep networks can interpolate noisy data without losing generalization.
problem Characterizing the relationship between interpolation and generalization in overparameterized deep networks.
method Analyzing the loss landscape of neural network functions over volumes around training data points, varying model parameters and training epochs.
result Loss sharpness in the input space follows a double descent, with large models predicting noisy targets over larger volumes around training data points.
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.
A physics-based method improves data interpolators and regression tasks.
problem Improving accuracy and efficiency in function learning.
method Inspired by statistical mechanics, introduces corrections to minimize energy.
result Improves performance in interpolation and regression tasks, especially in high-dimensional spaces.
Piecewise polynomial interpolation-based gradient descent reduces oracle complexity for smooth loss functions.
problem Optimizing empirical risk minimization loss functions
method Piecewise polynomial interpolation-based gradient descent
result Oracle complexity is reduced for smooth loss functions
We show that minimum-norm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected based on data. The result supports the empirical observation that minimum-norm …
Paper shows SVMs can interpolate data in various settings.
problem Understanding SVM performance and generalization.
method Flexible analysis framework for proving SVM interpolation in diverse settings.
result Support vector machines can interpolate data in many cases not previously covered.
The abstract proves the existence of CMC-1 surfaces with any complex structure in hyperbolic space.
problem Proving the existence of CMC-1 surfaces with arbitrary complex structures in hyperbolic space.
method Using a jet interpolation theorem and a uniform approximation theorem for holomorphic null curves.
result Existence of complete densely immersed CMC-1 surfaces in hyperbolic space with arbitrary complex structure.
Randomly sampled interpolators achieve zero generalization error with enough data.
problem Understanding the high generalization ability of machine learning models.
method Algebraic geometry tools to prove zero generalization error for random interpolators.
result Generalization error of randomly sampled interpolators becomes zero once the number of training samples exceeds a geometric threshold.
Constructive Approximation of Random Process via Stochastic Interpolation Neural Network Operatorsstat.ML
Paper develops SINNOs for approximating stochastic processes.
problem Approximating stochastic processes with neural networks.
method Developed stochastic interpolation neural network operators (SINNOs) with random coefficients.
result Established boundedness, interpolation accuracy, and approximation capabilities of SINNOs.
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.
On the Multiple Descent of Minimum-Norm Interpolants and Restricted Lower Isometry of Kernelsmath.ST
We study the risk of minimum-norm interpolants of data in Reproducing Kernel Hilbert Spaces. Our upper bounds on the risk are of a multiple-descent shape for the various scalings of , , for the input dimension and sample size . Empirical evidence supports our finding that minimum-norm interpo…
We investigate the properties of multidimensional probability distributions in the context of latent space prior distributions of implicit generative models. Our work revolves around the phenomena arising while decoding linear interpolations between two random latent vectors -- regions of latent space in close proximit…
Paper analyzes mistake and generalization of MNIC classifiers.
problem Understanding the performance of interpolating classifiers.
method Elementary analyses of MNIC's regret and generalization.
result MNIC generalizes with a rate proportional to the norm of the interpolating solution and inversely proportional to the number of data points.
New method interpolates high-dimensional scattered data using kernel theory.
problem Scattered data in high-dimensional spaces defy traditional distributional assumptions.
method Kernel interpolation framework based on integral operator theory.
result Spectra of kernel matrices predict performance of interpolation methods.
In order to generate novel 3D shapes with machine learning, one must allow for interpolation. The typical approach for incorporating this creative process is to interpolate in a learned latent space so as to avoid the problem of generating unrealistic instances by exploiting the model's learned structure. The process o…
In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the p…
New method certifies generative models' robustness.
problem Certifying generative models' robustness is challenging due to non-convex sets.
method ApproxLine, a scalable certification method capturing infinite sets or distributions over them.
result ApproxLine provides sound deterministic and probabilistic guarantees.
Proposes CLSM for better subsequence generation in music sequences.
problem Editing subsequences in music sequences without losing context.
method Context-informed prior and decoder for generative model, context position-informed encoder for inference.
result Contextual latent space is smoother in interpolation and generates higher quality samples.
The paper explains how certain neural network models can still perform well even when they fit training data perfectly.
problem Understanding how overparametrized models can generalize well despite fitting training data perfectly.
method Develops a framework to upper bound regression and classification risk in a reproducing kernel Hilbert space, providing conditions for harmless interpolation.
result Shows that harmless interpolation can occur in more general settings like bounded orthonormal systems, not just independent features.
This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in complex n-space. A fair amount of background…