arXiv research

A method to improve few-shot learning using continual local replacement and pseudo labeling.

We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calcula…

We establish a functor $Kan$ from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.

New convergence bounds for shuffling-based SGD methods in distributed learning.

If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally …

Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, $K^{-1}=\i…

Unified proof of knot unknotting bounds using Ma-Qiu index.

SGD without replacement decouples into curvature-following and flatness-regularizing steps.

Improves arc separation result for homogeneous spaces.

Localizes smooth spaces to study their homotopy properties.

We propose a reduction for non-convex optimization that can (1) turn an stationary-point finding algorithm into an local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algor…

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the loca…

Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a…

This paper studies the convergence behaviour of dictionary learning via the Iterative Thresholding and K-residual Means (ITKrM) algorithm. On one hand it is proved that ITKrM is a contraction under much more relaxed conditions than previously necessary. On the other hand it is shown that there seem to exist stable fixe…

Large scale agglomerative clustering is hindered by computational burdens. We propose a novel scheme where exact inter-instance distance calculation is replaced by the Hamming distance between Kernelized Locality-Sensitive Hashing (KLSH) hashed values. This results in a method that drastically decreases computation tim…

Entropy data replaces classical charts for smooth manifolds.

New covering moves for 3-manifolds up to degree 4.

We show two novel concentration inequalities for suprema of empirical processes when sampling without replacement, which both take the variance of the functions into account. While these inequalities may potentially have broad applications in learning theory in general, we exemplify their significance by studying the t…

In this paper we prove a convergence result for sequences of Willmore immersions with simple minimal bubbles. To this end we replace the total curvature control in T. Rivière's proof of the $\varepsilon$-regularity for Willmore immersions by a control of the local Willmore energy.

Paper introduces P-sensitive functions and their applications in robust optimization and financial models.

Modified condition proves no positive scalar curvature for enlargeable manifolds.

The calibration of a local volatility models to a given set of option prices is a classical problem of mathematical finance. It was considered in multiple papers where various solutions were proposed. In this paper an extension of the approach proposed in LiptonSepp2011 is developed by i) replacing a piecewise constant…

Study shows how certain spaces can be mapped to R^n with specific properties.

Develops a functional generalization of Eldan's stochastic localization for optimization and privacy.

A new method simplifies HLLE for better robustness.

Learning long-term dependencies is a key long-standing challenge of recurrent neural networks (RNNs). Hierarchical recurrent neural networks (HRNNs) have been considered a promising approach as long-term dependencies are resolved through shortcuts up and down the hierarchy. Yet, the memory requirements of Truncated Bac…

Replacing MSE with f-divergence in diffusion models improves robustness under data contamination.

The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses we introduce variational characterizations that naturally suggest a two-step scheme for their optimizatio…

Proposes a new regression method using $L_p$-norms for non-Gaussian noise.

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition of the corresponding coboundary rel…

A $4$-move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant of knots based on the fundamental group $π_1(S^3\setminus K)$…

Spatial graphs study tangle replacement with equivalence classes.

New method reveals corners of drum shapes.

We introduce a new local sparse attention layer that preserves two-dimensional geometry and locality. We show that by just replacing the dense attention layer of SAGAN with our construction, we obtain very significant FID, Inception score and pure visual improvements. FID score is improved from $18.65$ to $15.94$ on Im…

Localized curvature bounds ensure harmonic maps are constant.

Improved Local SGD convergence for general convex objectives with bounded second-order heterogeneity.

We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define $k$-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic $k$ 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-r…

Study counts sub-chord diagrams to classify spherical curves.

For any virtual link $L = S \cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T \mapsto T'$ is a replacement of $T$ with the $n$-tangle $T'$ in a way that preserves the orientation of $L$. After developing a general decomposition for the Jones polynomial of …

Federated learning (FL) is a distributed learning approach where a set of end-user devices participate in the learning process by acting on their isolated local data sets. Here, we process local data sets of users where worst-case optimization theory is used to reformulate the FL problem where the impact of local data …

Efficiently samples sequences without replacement for machine learning models.

The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

Let M be an oriented compact 3-manifold and let T be a (loose) triangulation of M, with ideal vertices at the components of the boundary of M and possibly internal vertices. We show that any spin structure s on M can be encoded by extra combinatorial structures on T. We then analyze how to change these extra structures…

This paper addresses credit valuation adjustment with a new closeout convention.

Paper tightens statistical aggregation results using local complexity.

Compact ECS manifolds are proven to be bundles over the circle.

Stochastic LWTA networks resist adversarial attacks while maintaining accuracy.