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.
The paper optimizes hyperplanes for binary classification in high-dimensional data with latent Gaussian mixtures.
problem Binary classification in high-dimensional data with latent Gaussian mixtures.
method Generalized least squares estimator for estimating the direction of the optimal separating hyperplane. Simple correction for intercept estimation.
result The procedure is minimax optimal in many scenarios and can retain the interpolation property.
Let M be an open Riemann surface and n≥3 be an integer. We prove that on any closed discrete subset of M one can prescribe the values of a conformal minimal immersion M→Rn. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the…
The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.
problem Extending the Mittag-Leffler theorem to meromorphic curves and minimal surfaces.
method Established a Mittag-Leffler-type theorem for meromorphic curves and minimal immersions, including interpolation and approximation.
result Complete minimal ends in R^5 are generically embedded, and open Riemann surfaces are characterized for minimal surfaces.
DSoftKI scales GP regression with full derivative observations.
problem Efficiently fitting and predicting full derivative observations in Gaussian Processes.
method Extends SoftKI by using local temperature vectors for interpolation, enabling encoding of local directional sensitivity.
result DSoftKI achieves accurate predictions and scales to larger datasets with full derivative observations.
This paper constructs PH spline curves with prescribed arc lengths.
problem Interpolating points, tangent directions, and curvatures with prescribed arc-length.
method Local construction of G2 planar PH biarc curves of degree 7. result Prescribed arc-length can be satisfied for any data and any chosen ratio between boundary tangents.
Lower bounds show OLS outperforms basis pursuit in overparameterized linear regression.
problem Excess risk of sparse interpolating procedures in overparameterized linear regression.
method Proved lower bounds on excess risk for OLS and basis pursuit.
result Excess risk of basis pursuit can converge at an exponentially slower rate than OLS.
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.
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.
Two-layer ReLU networks often converge to simpler solutions, improving generalization.
problem Understanding generalization in overparametrized neural networks, especially for complex tasks.
method Theoretical analysis of two-layer ReLU networks, focusing on the early alignment phase.
result Two-layer ReLU networks often converge to simpler solutions rather than interpolating the training data, leading to better generalization.
The paper connects flatness to generalization in learning multi-index models with neural networks.
problem Understanding the generalization of non-convex neural networks using flatness measures.
method Analyzes 2-layer non-convex homogeneous neural networks and their connection to multi-index models.
result Flattest interpolators achieve small population loss and generalize well, establishing a direct link between flatness and generalization.
Interpolating label noise makes models vulnerable to adversarial attacks.
problem Adversarial vulnerability of models trained on noisy labels.
method Theoretical analysis of label noise and adversarial risk relationship.
result Uniform label noise induces adversarial risk similar to worst-case poisoning.
This paper analyzes the interpolation error of nonlinear Attention compared to linear regression.
problem Understanding the interpolation error of nonlinear Attention in high-dimensional settings.
method Derives explicit expressions for mean-squared interpolation error using signal-plus-noise model and random matrix theory.
result Nonlinear Attention generally incurs a larger interpolation error than linear regression, but this gap can be reversed with structured signals.
Understanding the underlying mechanisms that enable the empirical successes of deep neural networks is essential for further improving their performance and explaining such networks. Towards this goal, a specific question is how to explain the "surprising" behavior of the same over-parametrized deep neural networks tha…
Dynamical system models (including RNNs) often lack the ability to adapt the sequence generation or prediction to a given context, limiting their real-world application. In this paper we show that hierarchical multi-task dynamical systems (MTDSs) provide direct user control over sequence generation, via use of a latent…
A common strategy to train deep neural networks (DNNs) is to use very large architectures and to train them until they (almost) achieve zero training error. Empirically observed good generalization performance on test data, even in the presence of lots of label noise, corroborate such a procedure. On the other hand, in…
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.
Navigated 2D multi-slice dynamic Magnetic Resonance (MR) imaging enables high contrast 4D MR imaging during free breathing and provides in-vivo observations for treatment planning and guidance. Navigator slices are vital for retrospective stacking of 2D data slices in this method. However, they also prolong the acquisi…
ε-Consistent Mixup improves semi-supervised classification accuracy.
problem Improving semi-supervised classification accuracy with limited labeled data.
method Combines Mixup's linear interpolation with consistency regularization, using an adaptive tradeoff between the two.
result ε-Consistent Mixup yields the largest gains in low label-availability scenarios.
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…
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.
A semi-supervised framework using stochastic interpolation and latent representations.
problem Challenges in conditional generative modeling with scarce labeled data.
method Combines conditional stochastic interpolation with low-dimensional latent representations.
result Significantly improves sample complexity and achieves faster convergence rate.
LSI enables joint learning of latent variables and generative models.
problem Joint optimization of latent variables and generative models.
method Developed a principled ELBO objective in continuous time for joint learning.
result LSI learns effective latent representations and generative transformations.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate O(N−1/2). 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.
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.
Proposes AEGAN for stable GAN training.
problem Training instability in GANs.
method Four-network model with adversarial and reconstruction losses.
result Stabilizes GAN training and prevents mode-collapse.
In this paper, we consider derivative free optimization problems, where the objective function is smooth but is computed with some amount of noise, the function evaluations are expensive and no derivative information is available. We are motivated by policy optimization problems in reinforcement learning that have rece…
This paper provides a practical method to extract caplet volatilities from quoted data.
problem Extracting caplet volatilities from quoted data is complex and not straightforward.
method The paper presents a constructive algorithm based on criteria and robust outlier detection. It includes direct interpolation, bootstrap methods, and global search methods.
result The paper introduces methods to extract caplet volatilities that are arbitrage-free and consistent with quoted data.
Bidirectional VAE reduces parameters and improves image tasks.
problem Improving image reconstruction, classification, interpolation, and generation.
method Uses a single neural network for both encoding and decoding in both forward and backward directions.
result Bidirectional VAEs reduce parameters by almost 50% and slightly outperform unidirectional VAEs.
There is a large literature explaining why AdaBoost is a successful classifier. The literature on AdaBoost focuses on classifier margins and boosting's interpretation as the optimization of an exponential likelihood function. These existing explanations, however, have been pointed out to be incomplete. A random forest …
New method uses neural networks to interpolate stellar atmospheres with high precision.
problem Recover precise stellar model atmospheres from grids of models.
method Deep neural network with 1D convolutional auto-encoder for feature extraction.
result Higher precision compared to traditional methods.
High-probability bound for distributed stochastic approximation tracking error.
problem Analyzing the convergence of distributed stochastic approximation schemes.
method Analysis using ODE approach to stochastic approximation.
result High probability bound for tracking error between iterates and limiting differential equation.
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.
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 neural network enforces mass conservation for better ice flow predictions.
problem Reliably project future sea level rise by improving ice sheet model inputs.
method Proposes divergence-free neural networks (dfNNs) enforcing local mass conservation.
result dfNNs yield more reliable ice flux estimates compared to other models.
Deep reinforcement learning (DRL) on Markov decision processes (MDPs) with continuous action spaces is often approached by directly training parametric policies along the direction of estimated policy gradients (PGs). Previous research revealed that the performance of these PG algorithms depends heavily on the bias-var…
Local Gradient Descent with local steps converges to the centralized model in the interpolation regime.
problem Understanding the implicit bias of Local Gradient Descent in the interpolation regime.
method Analyzing the implicit bias of Local Gradient Descent for classification tasks with linearly separable data.
result The aggregated global model from Local-GD converges exactly to the centralized model in the interpolation regime.
LFM learns a sequence of smaller models to generate data from noise.
problem Learning continuous, invertible flows between distributions.
method Stepwise Local Flow Matching (LFM) model, matching diffusion processes up to time-step size.
result LFM achieves competitive generative performance compared to Flow Matching.
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.
New method accelerates neural network training by focusing on flat directions.
problem Improving neural network training speed and stability.
method Bulk-SGD, interpolated gradient methods.
result Updates along the Dominant subspace can accelerate convergence but compromise stability.
Unified ODE model explains residual and non-residual networks.
problem Unclear relationship between residual and non-residual networks.
method Introducing a damping term in an ODE model to interpolate between ResNet and CNN architectures.
result Unified framework for understanding residual and non-residual networks.
New insights into optimization and generalization for linear models.
problem Understanding the implicit regularization of optimization methods for linear models.
method Investigating the norms minimized by interpolating solutions and using projections to move between solutions.
result Proving that for over-parameterized linear classification, projections onto the data-span enable the use of under-parameterized techniques.
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.
M2M tackles zero-shot structured noise suppression in images.
problem Structured noise with strong anisotropic correlations in real-world images.
method M2M introduces a novel sampling strategy that generates pseudo-independent sub-image pairs from a single noisy input, using directional interpolation and generalized median filtering.
result M2M consistently outperforms state-of-the-art zero-shot methods under correlated noise.
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-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.