arXiv research

Necessary and sufficient condition is given for a set $A\subset R^1$ to be a subset of the critical values set for a $C^k$ function $f:R^m \to R^1$.

Identifies minimal feature subsets for black-box model decisions.

In this article we extend a euclidean result of David and Semmes to the Heisenberg group by giving a sufficient condition for a $k$-Ahlfors-regular subset to have big pieces of bilipschitz images of subsets of $\R^k$. This Carleson type condition measures how well the set can be approximated by the Heisenberg $k$-plane…

P-SE explains model decisions with minimal feature subsets and fast estimators.

Consider a generic $r$-dimensional subspace of $\mathbb{R}^d$, $r<d$, and suppose that we are only given projections of this subspace onto small subsets of the canonical coordinates. The paper establishes necessary and sufficient deterministic conditions on the subsets for subspace identifiability.

For a Euclidean building $X$ of type $A_{2}$, we classify the 0-dimensional subbuildings $A$ of $\partial_{T}X$ that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of $A$) is (essentially) sufficient. To prove this, we construct n…

Researchers expand on best subset selection theory, identifying key complexities.

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on…

In Heisenberg groups, rectifiability is studied for subsets using $C^{1,α}$-regular surfaces.

Necessary and/or sufficient conditions are studied for the existence, uniqueness and holonomicity of bases in which on sufficiently general subsets of a differentiable manifold the components of derivations of the tensor algebra over it vanish. The linear connections and the equivalence principle are considered form th…

CNNs overinterpret inputs, leading to high accuracy without meaningful features.

New proof shows random neural networks contain sparse subnetworks.

Conditions for torsion-free connections with specific curvature maps are derived.

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $F\subset X$ of maximal dimension whose boundary sphere intersects every minimal $G$-invariant subset of $\partial_\infty X$. As a result we derive a necessary …

Paper tackles confounded ANMs, estimating ACEs with minimal interventions.

We investigate manifolds obtained as a quotient of a doubly warped product. We show that they are always covered by the product of two suitable leaves. This allows us to prove, under regularity hypothesis, that these manifolds are a doubly warped product up to a zero measure subset formed by an union of leaves. We also…

Proves weakly non-collapsed RCD spaces are strongly non-collapsed.

In this paper, for an immersion $f$ of an $n$-dimensional Riemannian manifold $M$ into $(n+d)$-Euclidean space we give a sufficient condition on $f$ so that, in case $d\leq 5$, any immersion $g$ of $M$ into $(n+d+1)$-Euclidean space that induces on $M$ a metric that is conformal to the metric induced by $f$ is locally …

Geometric proof shows regularity of anisotropic minimal surfaces in 2D.

An $n$ dimensional minimal submanifold $Σ$ of $\R^{n+m}$ is called non-parametric if $Σ$ can be represented as the graph of a vector-valued function $f:D\subset \R^n \mapsto \R^m$. This note provides a sufficient condition for the stability of such $Σ$ in terms of the norm of the differential $df$.

Subset pretraining speeds up neural network training.

We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of least areas circumscribed by simple closed curves $γ\subset \partial M$ uniquely determines the metric $g$, under some additional geometric assumptions. These are that $g$ …

Study efficient algorithms for identifying minimum interventional sets to learn causal relationships.

A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that any Lipschitz function on a positively curved homogeneous space is almost consta…

Inspired by work of Borzellino and Brunsden, we generalize the notion of a submanifold identifying a natural and sufficiently general condition which guarantees that a subset of an (effective) orbifold carries itself a canonical induced orbifold structure. We illustrate the strength of this approach generalizing typica…

Let $E$ be an arbitrary subset of $\mathbb{R}^n$ (not necessarily bounded), and $f:E\to\mathbb{R}$, $G:E\to\mathbb{R}^n$ be functions. We provide necessary and sufficient conditions for the $1$-jet $(f,G)$ to have an extension $(F, \nabla F)$ with $F:\mathbb{R}^n\to\mathbb{R}$ convex and of class $C^{1}$. Besides, if $…

We show that the maximal orbit dimension of a simultaneous Lie group action on n copies of a manifold does not pseudo-stabilize when n increases. We also show that if a Lie group action is (locally) effective on subsets of a manifold, then the induced Cartesian action is locally free on an open subset of a sufficiently…

Study geodesics on Grassmann manifold for functions vanishing on subsets of a set X.

Extends Eliashberg's principle to maps with specific singularities.

New rectifiability criteria for finite-perimeter sets in Carnot groups.

In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinct constant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$ denote the sphere and hyperbolic space of dimension $n$, respectively. We prove…

We consider a distributed learning approach in supervised learning for a large class of spectral regularization methods in an RKHS framework. The data set of size n is partitioned into $m=O(n^α)$ disjoint subsets. On each subset, some spectral regularization method (belonging to a large class, including in particular K…

Examines algebraic conditions for positive sectional curvature in 4D and higher.

Study on hyperbolic groups, focusing on separability and splittings.

New subsets without interior support Poincaré inequalities, expanding previous results.

Methods of transfer learning try to combine knowledge from several related tasks (or domains) to improve performance on a test task. Inspired by causal methodology, we relax the usual covariate shift assumption and assume that it holds true for a subset of predictor variables: the conditional distribution of the target…

Locally $C^{1,1}$ convex extensions for 1-jets defined on subsets of $R^n$.

A challenge in training discriminative models like neural networks is obtaining enough labeled training data. Recent approaches use generative models to combine weak supervision sources, like user-defined heuristics or knowledge bases, to label training data. Prior work has explored learning accuracies for these source…

Let $π:E\to M$ be a vector bundle over a simply connected manifold and $\nabla$ a linear connection in $π$. Let $σ: U \rightarrow E$ be a $\nabla$-parallel section of $π$ defined on a connected open subset $U$ of $M$. We give sufficient conditions on $U$ in order to extend $σ$ to the whole $M$. We mainly concentrate to…

New framework learns sufficient invariant features robustly across distribution shifts.

We describe Information Forests, an approach to classification that generalizes Random Forests by replacing the splitting criterion of non-leaf nodes from a discriminative one -- based on the entropy of the label distribution -- to a generative one -- based on maximizing the information divergence between the class-con…

A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. The…

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations a…

New construction shows VMRTs of unbendable curves can be Legendrian.

Meta-learning improves support recovery in high-dimensional PCA.

New algorithm learns causal graph to minimize regret in bandits without full structure.

Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in ρ_{2} ( TH^{4}(M;\Z) ) \}.$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset ρ_{2}( H^{2}(M; \Z) )$. Then the necessary and sufficient conditions for $M$ to admit a stabl…

We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_ξ(S)$. The definition of $L^2_ξ(S)$ involves the choice of a $p$-dimensional time-like subbundle $ξ\subset TM$. We establish a sufficient criterion for …