arXiv research

A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler c…

It is well known that a random vector with given marginal distributions is comonotonic if and only if it has the largest sum with respect to the convex order [ Kaas, Dhaene, Vyncke, Goovaerts, Denuit (2002), A simple geometric proof that comonotonic risks have the convex-largest sum, ASTIN Bulletin 32, 71-80. Cheung (2…

This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.

We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new …

Characterizes symmetric Bernoulli distributions with minimal convex sums.

In this paper we prove local analytic hypoellipticity for a degenerate sum of squares of complex vector fields generalizing those of Kohn in "Hypoellipticity and Loss of Derivatives". Kohn's article is to appear in the Annals of Mathematics with an appendix by Derridj and Tartakoff proving local analyticity in that cas…

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR he…

Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.

New insights into tail behavior of heavy-tailed random vectors and processes.

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = τ\cdot v_0^{\otimes 3} + A$, where $τ\geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover th…

We study a variation of Bagchi and Datta's $σ$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated…

This note quantifies, via a sharp inequality, an interplay between (a) the characteristic rank of a vector bundle over a topological space X, (b) the Z/2Z-Betti numbers of X, and (c) sums of the numbers of certain partitions of integers. In a particular context, (c) is transformed into a sum of the readily calculable B…

Study Eisenstein metrics on modular group representations.

The Courant bracket defined originally on the sections of the vector bundle $TM \oplus T^*M \to M$ is extended to the direct sum of the 1-jet vector bundle and its dual. The extended bracket allows to interpret many structures encountered in differential geometry in terms of Dirac structures. We give here a new approac…

New dHYM connections found on complex vector bundles.

Log-concavity proven for multinomial likelihoods under specific constraints.

Consider a family $Z=\{\boldsymbol{x_{i}},y_{i}$,$1\leq i\leq N\}$ of $N$ pairs of vectors $\boldsymbol{x_{i}} \in \mathbb{R}^d$ and scalars $y_{i}$ that we aim to predict for a new sample vector $\mathbf{x}_0$. Kriging models $y$ as a sum of a deterministic function $m$, a drift which depends on the point $\boldsymbol…

New method preserves privacy by aggregating feature-vectors with weighted sums, ensuring label differential privacy.

We prove the existence of extremal, non-csc, Kähler metrics on certain unstable projectivised vector bundles $¶(E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic Kähler classes. In particular, the vector bundles $E \to M$ under consideration are assumed to split as a …

The study extends Hano's theorem to semi-Riemannian product manifolds with specific conditions.

We consider the problem of minimizing the composition of a smooth (nonconvex) function and a smooth vector mapping, where the inner mapping is in the form of an expectation over some random variable or a finite sum. We propose a stochastic composite gradient method that employs an incremental variance-reduced estimator…

Estimates box dimension of fractal interpolation surfaces using oscillation vectors.

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models …

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain dir…

By drawing on ideas from optimisation theory, artificial neural networks (ANN), graph embeddings and sparse representations, I develop a novel technique, termed SENNS (Sparse Extraction Neural NetworkS), aimed at addressing the feature extraction problem. The proposed method uses (preferably deep) ANNs for projecting i…

Flow of curves with curvature and forcing vector field exists.

New index formula for hypoelliptic operators on manifolds.

For a complete minimal surface in the Euclidean 3-space, the so-called flux vector corresponds to each end. The flux vectors are balanced, i.e., the sum of those over all ends are zero. Consider the following inverse problem: For each balanced n vectors, find an n-end catenoid which attains given vectors as flux. Here,…

We propose an approach to the aggregation of risks which is based on estimation of simple quantities (such as covariances) associated to a vector of dependent random variables, and which avoids the use of parametric families of copulae. Our main result demonstrates that the method leads to bounds on the worst case Valu…

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras $\mathfrak{cga}_\ell(d,{\mathbb C})$ with $d=1$ for any integer value $\ell \in \mathbb{N}$. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type, and id…

The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimens…

The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.

Constructs complete metrics and solitons on complex vector bundles.

Study $Sp(n)$-orbits in complex and $Σ$-complex subspaces of Hermitian quaternionic vector spaces.

We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands.

From the paper "Formality Conjecture" (Ascona 1996): "I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with $4$ vertices $($and $6$ edges$)$. This class gives a remarkable vector field on the space of bi-vector fields on $\mathbb{R}^{d}$. The evolution with respect to the t…

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum $E\oplus C$ where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive …

The paper bounds the mean absolute error in DNN vector-to-vector regression.

New tree-structured Markov fields with Poisson marginals for counting variables.

In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of sufficiently large rank, a metric with nonnegative sectional curvature. We also examine the case of complex vector bundles over other manifolds, and give…

One of the ubiquitous representation of long DNA sequence is dividing it into shorter k-mer components. Unfortunately, the straightforward vector encoding of k-mer as a one-hot vector is vulnerable to the curse of dimensionality. Worse yet, the distance between any pair of one-hot vectors is equidistant. This is partic…

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big(M^n,\,g,\,X,\,λ\big)$ with constant scalar curvature is isometric to a Euclidean sphere $\Bbb{S}^{n}$. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreo…

Introduces a variational framework for indefinite Lagrangians with specific symmetries.

A new framework for fair representation learning using correction vectors.

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a covariant functor from the category of finite-dimensional diffeological vector …