Control on the Manifolds of Mappings with a View to the Deep Learning

Deep learning of the artificial neural networks (ANN) can be treated as a particular class of interpolation problems. The goal is to find a neural network whose input-output map approximates well the desired map on a finite or an infinite training set. Our idea consists of taking as an approximant the input-output map, which arises from a nonlinear continuous-time control system. In the limit such control system can be seen as a network with a continuum of layers, each one labelled by the time variable. The values of the controls at each instant of time are the parameters of the layer.

On the Terracini Locus of Projective Varieties

We introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite sets $$S \subset X$$ S ⊂ X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the study of interpolation problems over double points in special position, but they also enter naturally in the study of special loci contained in secant varieties to projective varieties.We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples in which the locus has codimension 1 in the symmetric product of X.

Local competing in interpolation problems

Простой пример иллюстрирует недостаточность известных подходов к интерполяции в задаче восстановления функции по немногим заданным отчётливо передающим форму частным значениям. Известные подходы дополняет локальный выбор между полиномиальной и рациональной локальными интерполянтами, минимизирующий ошибки локальной интерполянты в ближайших внешних узлах c одной или разных сторон. Новый подход сочетает предельную вычислительную простоту локальных интерполянт с тщательностью их подбора. Принципы построения алгоритма сформулированы в общем виде для отображений метрических пространств. Они обеспечивают точное (за редкими исключениями) восстановление отображений, локально совпадающих с какими-то из заданных возможных интерполянт. В одномерном случае двухэтапный алгоритм гарантирует непрерывность интерполянты и точное восстановление одновременно полиномов малой степени, несложных рациональных функций с линейным знаменателем и ломаных из длинных звеньев с узлами на концах в типичных ситуациях, когда эти требования не противоречивы. Дополнительный параметр позволяет заменить точное восстановление ломаных требуемой гладкостью интерполяции.

Local competing in interpolation problems

A simple example illustrates the insufficiency of the known approaches to interpolation in the problem of recovering a function from a few given specific values that clearly convey the form. A local choice between polynomial and rational local interpolants, which minimizes the local interpolant’s errors at the nearest external nodes from one or different sides, complements the known approaches. It combines the extreme computational simplicity of local interpolants with the thorought selection of them. The principles of constructing the algorithm are formulated in general terms for mappings of metric spaces. They provide accurate (with rare exceptions) reconstruction of mappings that locally coincide with some of the given possible interpolants. In the one-dimensional case, the two-stage algorithm guarantees the continuity of the interpolant and accurately reconstructs polynomials of small degree, simple rational functions with a linear denominator, and broken lines of long links with knots at the ends when these requirements do not contradict each other. An additional parameter allows you to replace the exact restoration of polylines with the required smoothness of interpolation.