Related papers: The trouble with deautonomising higher order maps

This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient…

The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its…

In an earlier article, we presented a method to obtain integrals of motion and polynomial algebras for a class of two-dimensional superintegrable systems from creation and annihilation operators. We discuss the general case and present its…

Positive maps that are not decomposable are a key resource in entanglement theory because they can detect bound entangled states, yet systematic methods for constructing them remain limited. We introduce an optimization framework based on…

As gradient descent method in deep learning causes a series of questions, this paper proposes a novel gradient-free deep learning structure. By adding a new module into traditional Self-Organizing Map and introducing residual into the map,…

We study the local behavior of the closed-open discrete maps of Orlich--Sobolev classes in ${\Bbb R}^n,$ $n\geqslant 3.$ It was found that these mappings $f$ have continuous extension in isolated point $x_0$ in $D\setminus\{x_0 \},$ as soon…

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…

The goal of this thesis is to improve our understanding of the internal mechanisms by which deep artificial neural networks create meaningful representations and are able to generalize. We focus on the challenge of characterizing the…

In this paper, we introduce a graph matching method that can account for constraints of arbitrary order, with arbitrary potential functions. Unlike previous decomposition approaches that rely on the graph structures, we introduce a…

We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a…

Underpinning the success of deep learning is effective regularizations that allow a variety of priors in data to be modeled. For example, robustness to adversarial perturbations, and correlations between multiple modalities. However, most…

We define the notion of higher-order colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. When $M$ and $N$ are endowed with Riemannian metrics, $p\ge 1$ and $k\ge 2$, this allows us to define the intrinsic…

Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric…

The information processing abilities of a multilayer neural network with a number of hidden units scaling as the input dimension are studied using statistical mechanics methods. The mapping from the input layer to the hidden units is…

In this paper we study the dichotomy of Poincarce map. We give a relation between the dichotomy of the Poincarce map and boundedness of solutions of certain periodic Cauchy problems

We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional…

We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then…

Embedding discrete solvers as differentiable layers has given modern deep learning architectures combinatorial expressivity and discrete reasoning capabilities. The derivative of these solvers is zero or undefined, therefore a meaningful…

One of the major problems in modeling natural signals is that signals with very similar structure may locally have completely different measurements, e.g., images taken under different illumination conditions, or the speech signal captured…

In earlier works on Shape Dynamics (SD), a linear method of solving a particular set of Lichnerowicz-type equations through the implicit function theorem was developed in order to implicitly construct SD's global Hamiltonian and eliminate…