arXiv research

Statistical characteristics of deep network representations, such as sparsity and correlation, are known to be relevant to the performance and interpretability of deep learning. When a statistical characteristic is desired, often an adequate regularizer can be designed and applied during the training phase. Typically, …

Flexible per-class regularization improves binary classifiers.

Regularization and data augmentation can be class-dependent, leading to poor performance on some classes.

New class of complex manifolds defined, properties studied.

Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.

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.

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 $P(y|\pmb{x})$. The regularization term meas…

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.

We propose regularization strategies for learning discriminative models that are robust to in-class variations of the input data. We use the Wasserstein-2 geometry to capture semantically meaningful neighborhoods in the space of images, and define a corresponding input-dependent additive noise data augmentation model. …

Improved covariance matrix estimation for multiple classes with limited data.

A fast method for discrete OT with group-sparse regularization for class label preservation.

Study Godbillon-Vey class for regular Jacobi foliations.

Determines regular homotopy classes for link immersions of simple singularities.

Study on 3-manifolds finds regular conformal metrics for rough metrics.

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.

A simple regularization method improves model generalization.

New findings on mapping class group actions on the circle, improving critical regularity.

Regularizing for or against class selectivity in DNNs improves test accuracy.

Study slice-regular polynomial functions via twistor space group actions.

Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.

Classifies isotopy classes of links from Thompson's group F and its subgroup.

We develop a novel theoretical framework for understating OT schemes respecting a class structure. For this purpose, we propose a convex OT program with a sum-of-norms regularization term, which provably recovers the underlying class structure under geometric assumptions. Furthermore, we derive an accelerated proximal …

Deep networks adapt to function regularity and data distribution.

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.

Deep neural networks achieve optimal learning rates for high-dimensional classification.

We show that the EH class and the LOSS invariant of Legendrian knots in contact 3-manifolds are functorial under regular Lagrangian concordances in Weinstein cobordisms. This gives computable obstructions to the existence of regular Lagrangian concordances.

This paper extends foliation concepts to singular foliations using Lie $\infty$-algebroids.

Study MinMax methods for optimization problems, including optimal transport.

New method regularizes deep networks by distilling self-knowledge.

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.

Training deep neural networks is known to require a large number of training samples. However, in many applications only few training samples are available. In this work, we tackle the issue of training neural networks for classification task when few training samples are available. We attempt to solve this issue by pr…

We prove the $C^{1}$ regularity for a class of abnormal length-minimizers in rank $2$ sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank $2$ sub-Riemannian structures of step up to $4$ are of class $C^{1}$.

The study connects group structure to smooth actions on one-manifolds.

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.

Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.

We establish the equivalence between the family of closed uniformly regular Riemannian manifolds and the class of complete manifolds with bounded geometry.

In an earlier paper, I defined a new winding number of regular closed curves on complete euclidean/hyperbolic surfaces and showed that this winding number, together with the free homotopy class, determines the regular homotopy class. In this paper, I give a Whitney-type formula for the winding number of non-null-homoto…

Noisy labels often occur in vision datasets, especially when they are obtained from crowdsourcing or Web scraping. We propose a new regularization method, which enables learning robust classifiers in presence of noisy data. To achieve this goal, we propose a new adversarial regularization scheme based on the Wasserstei…

A new algorithm for faster model selection in twin multi-class SVM.

Generalizes Frobenius theorem to quasiconformal deformations.

This study connects Jacobian regularization to adversarial robustness and improves generalization.

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähle…

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…

We establish a general theorem improving regularity of solutions of elliptic pseudodifferential equations. It allows to resolve in a unified way the regularity issue for a broad class of nonlinear elliptic equations and systems appearing in different areas of geometry and analysis.