The notion of embedding a class of dichotomies in a class of linear half spaces is central to the support vector machines paradigm. We examine the question of determining the minimal Euclidean dimension and the maximal margin that can be obtained when the embedded class has a finite VC dimension. We show that an overwhelming majority of the family of finite concept classes of any constant VC di...

The rotational dimension is a minor-monotone graph invariant related to the of Euclidean space containing spectral embedding corresponding first nonzero eigenvalue Laplacian, which introduced by Göring, Helmberg and Wappler. In this paper, we study dimensions graphs contain large complete graphs. characterized its dimension. It will be obtained that chordal may made while keeping constant.

We analyze the tubular phase of self-avoiding anisotropic crystalline membranes. A careful analysis using renormalization group arguments together with symmetry requirements motivates the simplest form of the large-distance free energy describing fluctuations of tubular configurations. The non-self-avoiding limit of the model is shown to be exactly solvable. For the full self-avoiding model we ...

A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into H2 is achieved.

The analysis of observed time series from nonlinear systems is usually done by making a time-delay reconstruction to unfold the dynamics on a multi-dimensional state space. An important aspect of the analysis is the choice of the correct embedding dimension. The conventional procedure used for this is either the method of false nearest neighbors or the saturation of some invariant measure, such...

(d) Hm−∗(M,M \X), where M is a smooth oriented manifold of dimension m and there exists an embedding X ↪→M as before. (d’) Hm−∗(X) if X itself is a smooth oriented manifold of dimension m. (e) H∗(D•(X), d). Assume there exists an embedding X ↪→M to a smooth manifold M of dimension m. Then we define D•(X) be a chain complex of distributions supported on X. (A distributions of degree k is a conti...

In this paper we show that two dynamical invariants, the second order Renyi entropy and the correlation dimension, can be estimated from recurrence plots (RPs) with arbitrary embedding dimension and delay. This fact is interesting as these quantities are even invariant if no embedding is used. This is an important advantage of RPs compared to other techniques of nonlinear data analysis. These e...

The paper proposes a new method to estimate the distribution of the embedding dimension associated with a time series, using the Self Organizing Map decision taken in Output Space (SOMOS) dimensionality reduction neural network. It is shown that SOMOS, besides estimating the embedding dimension, it also provides an approximation of the overall distribution of such dimension for the set where th...

We study the problem of embedding compact subsets of Rn into C1 submanifolds of minimal dimension. In [4], we define a generalized tangent space TxA suitable for a general compact subset A of Rn and we prove that A may be locally embedded into a C1 manifold of dimension dim(TxA). This result leads naturally to the global conjecture that for a compact subset A of Rn, there exists a C1 manifold M...