Cubic spline smoothing improves interpolation between irregularly sampled data.
problem Interpolation discontinuity in recurrent neural networks for irregularly sampled sequences.
method Cubic spline smoothing compensation module trained end-to-end with ODE-RNN.
result Improves interpolation between irregularly sampled data points.
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.
Study shows overparameterization helps in generalizing from smooth interpolants.
problem Understanding generalization in overparameterized linear models.
method Analysis of random Fourier series model with weighted trigonometric interpolation.
result Weighted trigonometric interpolation leads to lower generalization error in overparameterized scenarios.
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.
AutoEncoder smooths noisy sensor data and interpolates missing values.
problem Noisy sensor data and missing timepoints require interpolation.
method Uses AutoEncoder to denoise and interpolate data.
result AutoEncoder improves data quality and reveals dynamics.
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
Gradient descent with logistic loss can interpolate deep networks with smoothed ReLU activations under certain conditions.
problem Conditions for gradient descent to drive logistic loss to zero in deep networks with smoothed ReLU activations.
method Gradient descent applied to fixed-width deep networks with smoothed ReLU approximations (e.g., Swish, Huberized ReLU).
result Gradient descent can drive logistic loss to zero under specific conditions, providing bounds on convergence rate.
Diffusion models improve creativity by smoothing the score function, leading to interpolated data.
problem Improving creativity in diffusion models.
method Analyzing the effect of score smoothing on diffusion model dynamics.
result Score smoothing causes diffusion models to generate data that interpolate the training set.
This paper introduces a new bound to explain generalization in over-parameterized models.
problem Understanding why some over-parameterized models generalize well while others do not.
method PAC-Chernoff bounds and smoothness measures based on large deviation theory.
result Interpolators with smoother structures generalize better, according to the new theoretical framework.
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.
We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied…
The paper extends a variance gamma model to quadratic functions, reducing arbitrage and computational costs.
problem Creating an arbitrage-free interpolation for option pricing models.
method Generalizing the local variance gamma model to a piecewise quadratic local variance function.
result The quadratic model results in an arbitrage-free interpolation of class C3, reducing knots and computational cost.
New law explains why deep learning models often have more parameters than needed.
problem Why deep learning models often have more parameters than classical theory suggests.
method Proved a universal law of robustness for smooth interpolation.
result Smooth interpolation requires d times more parameters than mere interpolation.
New method certifies images against transformations like rotations and translations.
problem Certifying robustness of images against transformations like rotations and translations.
method Randomized smoothing with three different kinds of defenses.
result Individual certificates can be obtained via statistical error bounds or efficient online inverse computation.
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 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.
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.
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.
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.
SGD achieves near optimal convergence rate in smooth interpolation regime.
problem Optimization of smooth convex objectives with zero noise at optimum.
method Stochastic Gradient Descent (SGD) with large stepsize analysis.
result Last iterate of SGD achieves expected excess risk of O(1/T + σ* / √T) with optimal stepsize.
Estimates individual treatment effects using gradient interpolation and kernel smoothing.
problem Estimating individualized continuous treatment effects in observational data.
method Augment training data with independently sampled treatments and inferred counterfactual outcomes using gradient interpolation and kernel smoothing.
result Our method outperforms state-of-the-art methods on counterfactual estimation error.
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.
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.
Combines spline interpolation and ARIMA for stock market forecasting.
problem Limited predictive performance of ARIMA in noisy data.
method Integrates cubic spline interpolation and ARIMA for time series forecasting.
result Demonstrates guidance for short-term stock market forecasting.
Gradient descent with logistic loss can make two-layer networks interpolate binary classification data.
problem Training two-layer networks for binary classification.
method Gradient descent with logistic loss applied to two-layer networks.
result Gradient descent can drive training loss to zero under certain conditions.
A common approach for defining a reward function for Multi-objective Reinforcement Learning (MORL) problems is the weighted sum of the multiple objectives. The weights are then treated as design parameters dependent on the expertise (and preference) of the person performing the learning, with the typical result that a …
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…
New algorithms optimize spectral risk measures, improving interpolation between average and worst-case performance.
problem Optimizing spectral risk measures for learning systems.
method Developed stochastic algorithms to optimize spectral risk measures by characterizing their subdifferential and addressing challenges like biasedness of subgradient estimates and non-smoothness.
result Our approach outperforms out-of-the-box stochastic subgradient and dual averaging methods in optimizing spectral risk measures.
We introduce a new wavelet transform suitable for analyzing functions on point clouds and graphs. Our construction is based on a generalization of the average interpolating refinement scheme of Donoho. The most important ingredient of the original scheme that needs to be altered is the choice of the interpolant. Here, …
Classifies Real line bundles with Real connections on manifolds with involution.
problem Classifying Real line bundles with Real connections on manifolds with involution.
method Defines Real smooth Deligne cohomology to interpolate between equivariant sheaf cohomology and smooth imaginary-valued forms.
result Classifies Real line bundles with Real connections on manifolds with involution.
We carry out a Carlotto-Schoen-type gluing with interpolating scalar curvature on cone-like sets, or deformations thereof, in the category of smooth Riemannian asymptotically Euclidean metrics.
A comprehensive methodology is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers which are typical…
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
The paper establishes bounds on the smoothness parameter in Gaussian process interpolation.
problem Estimating the smoothness parameter in Gaussian process models.
method Approximation theory in Sobolev spaces and general theorems on parameter estimation.
result Maximum likelihood estimation recovers the true smoothness for certain classes of functions.
Optimal nonparametric regression estimator adapts to unknown smoothness.
problem Nonparametric regression with unknown smoothness.
method Constructs an interpolating estimator that adapts to unknown smoothness.
result Minimax optimal rates achieved on Hölder classes.
We discuss results for the Ribaucour transformation of curves or of higher dimensional smooth and discrete submanifolds. In particular, a result for the reduction of the ambient dimension of a submanifold is proved and the notion of Ribaucour coordinates is derived using a Bianchi permutability result. Further, we disc…
BWFlow improves graph generation by smoothly interpolating graph components.
problem Disjoint modeling of graph nodes and edges leads to irregular and non-smooth probability paths.
method Modeling graphs as MRFs and using optimal transport displacement for a smooth probability path.
result BWFlow achieves better training convergence and efficient sampling in graph generation.
ResNets promote smoother interpolations than MLPs, enhancing generalization.
problem Understanding the difference in smoothness between ResNets and MLPs.
method Neural Tangent Kernel (NTK) analysis during gradient descent training.
result ResNet's NTK results in smoother interpolations than MLPs.
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.
Study on learning properties of scale-dependent kernels controlling stability and error.
problem Understanding the learning properties of scale-dependent kernels in nonparametric ridge-less least squares.
method Combines probabilistic results with interpolation theory to analyze stability and error.
result Different regimes of learning error depending on sample size and data dimension.
The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.
problem Approximating \(J_b\)-holomorphic maps to Oka manifolds.
method Constructing continuous or smooth families of \(J_b\)-holomorphic maps to Oka manifolds with approximation on compact Runge sets.
result Runge and Mergelyan approximation theorems and Weierstrass interpolation theorem for families of open Riemann surfaces.
In this paper we introduce the "interpolation-degneration" strategy to study Kahler-Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By "interpolation" we show the angles in (0,2π] that admit a conical Kahler-Einstein metric…
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.
This overview article concerns the notion of fractional smoothness of random variables of the form g(XT), where X=(Xt)t∈[0,T] is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance main…
New insights into why neural networks can overfit without interpolating data.
problem Understanding why neural networks can overfit without interpolating data in fixed dimensions.
method Analyzing the smoothness of estimators and their derivatives.
result Benign overfitting is possible with estimators that have large enough derivatives, not just in high dimensions but also in fixed dimensions.
In this paper, we propose a nonlinear distance metric learning scheme based on the fusion of component linear metrics. Instead of merging displacements at each data point, our model calculates the velocities induced by the component transformations, via a geodesic interpolation on a Lie transfor- mation group. Such vel…
A new method evolves point clouds using B-splines for smooth surfaces.
problem Evolution of smooth surfaces from discrete point clouds.
method Adaptive Lagrangian B-spline framework for geometric evolution.
result Efficient and accurate reproduction of surface evolution phenomena.