Two new regularizers leverage class information to improve deep network performance.
problem Improving deep network performance and feature independence for classification tasks.
method Class-wise Covariance Regularizer (cw-CR) and Variance Regularizer (cw-VR) designed to manipulate statistical characteristics per class.
result Significant improvements in classification performance for 21 out of 22 tasks.
Flexible per-class regularization improves binary classifiers.
problem Improving binary classifiers by addressing outliers and class imbalance.
method Graph-based adaptive regularization with flexible per-class thresholds.
result Flexible thresholds improve classifier performance and address class imbalance.
Regularization and data augmentation can be class-dependent, leading to poor performance on some classes.
problem Class-dependent effects of regularization and data augmentation.
method Evaluation of regularization and data augmentation techniques on Imagenet and INaturalist datasets.
result Regularization and data augmentation can lead to significant performance drops on some classes.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
Proposes regularization for robust image models using Wasserstein geometry.
problem Robustness to in-class variations in input data.
method Wasserstein-2 geometry, Tikhonov-type regularizer, data augmentation.
result Improves generalization under adversarial perturbations and large variations.
Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.
problem Understanding the Atiyah and Todd classes of Lie algebroids.
method Analyzing the Atiyah sequence of Lie algebroids and proving class restrictions.
result Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence.
Develops a new OT framework for class-based data with improved robustness.
problem Understand and recover class structure in optimal transport schemes.
method Proposes a convex OT program with sum-of-norms regularization and an accelerated proximal algorithm.
result The new regularizer preserves class structure better and is more robust to data geometry.
We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to a robust estimator of the class probability P(y∣x). The regularization term meas…
We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.
Improved covariance matrix estimation for multiple classes with limited data.
problem Estimating covariance matrices for multiple classes with scarce data.
method Coupled regularized sample covariance matrix estimator (RSCM) that combines pooled SCM and scaled identity matrix for regularization.
result The coupled RSCM estimators outperform cross-validation in classification tasks with comparable accuracy but faster computation.
A fast method for discrete OT with group-sparse regularization for class label preservation.
problem Efficiently measuring the distance between two discrete distributions with class labels.
method Fast discrete OT with group-sparse regularizers using gradient-based algorithms.
result Up to 8.6 times faster than original method without degrading accuracy.
The aim of the paper is to construct some Godbillon-Vey classes of a family of regular foliations, defined in the paper. These classes are cohomology classes on the manifold or on suitable open subsets. Some examples are also considered.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
Determines regular homotopy classes for link immersions of simple singularities.
problem Classifying immersions of link singularities.
method Computing complete invariants of immersions and comparing with Dynkin diagrams.
result Inclusion map of link into 5-sphere is regularly homotopic to immersion associated with Dynkin diagram.
Study on 3-manifolds finds regular conformal metrics for rough metrics.
problem Characterize conformal metrics for rough Riemannian metrics on 3-manifolds.
method Analogous to the Yamabe problem, study conformal classes and regularity.
result Characterize when a more regular representative exists in the conformal class.
For any regular Courant algebroid, we construct a characteristic class a la Chern-Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Severa class (in H^3_{DR}(M)). On the other hand, when the Courant algebroid is a …
New algorithms for latent class analysis using regularized spectral clustering.
problem Identifying latent classes within populations from categorical data.
method Developed two new algorithms using a regularized Laplacian matrix to estimate latent classes.
result Our algorithms provide consistent latent class analysis under mild conditions and can accurately infer the number of latent classes.
A simple regularization method improves model generalization.
problem Overfitting due to lack of labeled data in machine learning models.
method Density-fixing regularization method based on class prior distribution.
result Improves model generalization performance by approximating class prior distribution.
New findings on mapping class group actions on the circle, improving critical regularity.
problem Improving understanding of mapping class group actions on the circle.
method Analyzing actions of non-solvable groups and finite index subgroups of mapping class groups.
result Critical regularity of mapping class groups is at most one for surfaces of complexity at least three.
Regularizing for or against class selectivity in DNNs improves test accuracy.
problem The necessity and sufficiency of class selectivity in DNNs.
method Direct regularization of class selectivity in convolutional neural networks.
result Reducing class selectivity improves test accuracy, while increasing it decreases it.
Proves C1 regularity for abnormal minimizers in rank 2 sub-Riemannian structures.
problem Regularity of abnormal minimizers in sub-Riemannian structures.
method Proves C1 regularity using length-minimizers in rank 2 sub-Riemannian structures. result All length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class C1. Study slice-regular polynomial functions via twistor space group actions.
problem Characterize slice-regular functions and their polynomial subclasses.
method Employ the twistor construction and group actions of PGL(2,H). result Characterize slice-regular functions with planar twistor lifts and normal classes of polynomials.
Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.
problem Optimizing boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group.
method Analyzes critical sets of the prescribed mean curvature functional in the Heisenberg group.
result Characteristic curves of critical sets are C2-regular, optimal in the Heisenberg group. Classifies isotopy classes of links from Thompson's group F and its subgroup.
problem Classify isotopy classes of links from Thompson's group F and its subgroup.
method Introduced a method to produce links from elements of Thompson's group F and its subgroup.
result Classified isotopy classes of links from Thompson's group F and its subgroup.
Deep networks adapt to function regularity and data distribution.
problem Understanding deep learning's adaptability to function regularity and data distribution.
method Developed nonparametric approximation and estimation theories for a broad class of functions using deep ReLU networks.
result Deep neural networks are adaptive to different regularity of functions and nonuniform data distributions.
Deep neural networks achieve optimal learning rates for high-dimensional classification.
problem Learning classification functions from noisy data with smooth boundaries.
method Empirical risk minimization over deep neural networks for locally Barron-regular decision boundaries.
result Optimal estimation rates are independent of dimension and can be achieved by deep neural networks.
We show that a compact orientable 4-manifold M has a CR regular immersion into C3 if and only if both its first Pontryagin class and its Euler characteristic vanish, and has a CR regular embedding into C3 if and only if in addition the second Stiefel-Whitney class of M vanishes.
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. New method improves neural network performance with few samples.
problem Training neural networks with limited data.
method Proposes a regularization term for class-wise invariant representations.
result Improves neural network generalization with few samples.
Proves C1,1 regularity for complex Monge-Ampère equations and geodesic rays.
problem Complex Monge-Ampère equations on compact Kähler manifolds with degenerate cohomology.
method Proves C1,1 estimate for solutions. result Local C1,1 regularity of geodesic rays and quasi-psh envelopes. This paper extends foliation concepts to singular foliations using Lie ∞-algebroids.
problem Cohomological obstruction to volume forms in singular foliations.
method Replacing singular foliations with universal Lie ∞-algebroids to define modular class. result Geometric meaning of modular class as an obstruction to universal Lie ∞-algebroids. Deep neural solvers can approximate harmonic functions with low error.
problem Harmonic functions in Barron space are not regular.
method Elliptic regularity theory applied to Barron functions.
result Approximation of harmonic functions by Barron functions with low error.
Study MinMax methods for optimization problems, including optimal transport.
problem Optimization problems, especially optimal transport.
method MinMax framework, regularization, neural networks, approximation theorems.
result Justification of neural networks for solving optimization problems.
Functorial properties of knot invariants under specific concordances.
problem Computing obstructions to regular Lagrangian concordances.
method Functoriality of EH class and LOSS invariant under Lagrangian concordances in Weinstein cobordisms.
result Computable obstructions to regular Lagrangian concordances.
New method regularizes deep networks by distilling self-knowledge.
problem Overfitting in deep neural networks.
method Self-knowledge distillation to regularize class-wise predictions.
result Significant improvement in generalization and calibration.
We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…
Random feature approximation speeds up spectral methods and improves learning rates.
problem Improving the efficiency and generalization of spectral methods in large-scale algorithms.
method Combining random feature approximation with spectral regularization methods.
result Optimal learning rates for estimators over various regularity classes, including those not in the RKHS.
Implicit Policy simplifies complex reinforcement learning policies.
problem Complex action distributions in reinforcement learning.
method Rich policy class with entropy regularization.
result Entropy regularization with rich policy class achieves desirable properties.
Regularization fails in continual learning tasks.
problem Catastrophic forgetting in continual learning.
method Study of regularization approaches in continual learning.
result Regularization fails to learn class discrimination across tasks.
New regularizer improves DNN robustness against adversarial attacks.
problem Improving robustness of deep neural networks against adversarial attacks.
method Laplacian of similarity graphs to penalize large changes in class boundaries.
result Improves robustness of DNNs on vision datasets.
The study connects group structure to smooth actions on one-manifolds.
problem Understanding how group actions affect the smoothness of manifolds.
method Analyzes the relationship between group algebraic structure and smoothness of group actions on one-dimensional manifolds.
result Uniform construction of groups acting on compact interval and circle with prescribed regularity.
Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.
problem Characterizing Vaisman manifolds with vanishing first Chern class.
method Categorization into three types based on Bott-Chern class sign, showing canonical metrics, quasi-regularity, stability, and automorphism group behavior.
result Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics and are stable under deformations.
We construct the equivalent of the Godbillon-Vey class and its generalizations for regular foliations on super-manifolds. We interpret these classes as classes of foliated flat connections.
We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.
A new algorithm for faster model selection in twin multi-class SVM.
problem Challenges in effective solution of multi-classification and fast model selection in twin multi-class SVM.
method Sample data set partition strategy, Lagrangian multipliers, piecewise linear update, initialization algorithm, and event-based iteration.
result Comparable classification performance achieved without solving quadratic programming problems.
WAR method improves classifier robustness in noisy label datasets.
problem Learning robust classifiers in presence of noisy labels.
method Adversarial regularization based on Wasserstein distance.
result WAR method outperforms state-of-the-art competitors on noisy label datasets.
Researchers propose and solve a class of pseudo-Finslerian metrics with angle-separation.
problem Characterizing and solving pseudo-Finslerian metrics with specific angle-separation conditions.
method Derived complete algebraic and differential equations, solved the set for angle-regular solutions.
result Found and described an angle-regular solution for the Finsleroid-in-pseudo-Finsleroid type.