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Register a new userReconstruction of manifold embeddings into Euclidean spaces via intrinsic distances
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We consider one of the classical manifold learning problems, that of reconstructing up to an almost isometry an embedding of a compact connected Riemannian manifold in a Euclidean space given the information on intrinsic distances between points from its almost dense subset. It will be shown that the most popular methods in data science to deal with such a problem, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU) actually miss the point and may provide results very far from an isometry (and even may give no biLipshitz embedding). We will then provide an easy variational formulation of this problem which leads to an algorithm always providing an almost isometric imbedding with given controlled small distortion of original distances.
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Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-$c_0$ spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on $k$-subsets of $mathbb{N}$. We apply this characterization to show that the class of separable, reflexive, and asymptotic-$c_0$ Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-$c_0$ property, called the asymptotic-subsequential-$c_0$ property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming graphs. We present examples of spaces that are asymptotic-subsequential-$c_0$. In particular $T^*(T^*)$ is asymptotic-subsequential-$c_0$ where $T^*$ is Tsirelsons original space.
For controller design for systems on manifolds embedded in Euclidean space, it is convenient to utilize a theory that requires a single global coordinate system on the ambient Euclidean space rather than multiple local charts on the manifold or coordinate-free tools from differential geometry. In this article, we apply such a theory to design model predictive tracking controllers for systems whose dynamics evolve on manifolds and illustrate its efficacy with the fully actuated rigid body attitude control system.
We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on the influence of $A$ on the embedding behaviour. For a large range of parameters, we derive sharp characterizations of embeddings.
For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the 2n-dimensional Euclidean space. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding of L into the (2n-1)-dimensional sphere, where L is the join of n copies of a 4-point set, has a pair of linked (n-1)-dimensional spheres.
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $mathbb{H}^n$ and spherical $mathbb{S}^n$ spaces, for $ngeq 2$. For that purpose, the kissing number is replaced by the kissing function $kappa_H(n, r)$, resp. $kappa_S(n, r)$, which depends on the dimension $n$ and the radius $r$. After we obtain some theoretical upper and lower bounds for $kappa_H(n, r)$, we study their asymptotic behaviour and show, in particular, that $kappa_H(n,r) sim (n-1) cdot d_{n-1} cdot B(frac{n-1}{2}, frac{1}{2}) cdot e^{(n-1) r}$, where $d_n$ is the sphere packing density in $mathbb{R}^n$, and $B$ is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of $kappa_S(n, r)$, for $n= 3,, 4$, over subintervals in $[0, pi]$ with relatively high accuracy.
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