We inductively describe an embedding of a complete ternary tree Th of height h into a hypercube Q of dimension at most d(1.6)he + 1 with load 1, dilation 2, node congestion 2 and edge congestion 2. This is an improvement over the known embedding of Th into Q. And it is very close to a conjectured embedding of Havel [3] which states that there exists an embedding of Th into its optimal hypercube...

In this short paper we introduce a proper method to perform Korcak-analysis and obtain the correct Korcak-exponent on a set of patches, embedded into two-dimensions. Both artificial and natural data sets are used for the demonstration. The independence of the Korcak-exponent from the classical (Hausdorff) fractal dimension is also demonstrated. 2012 Elsevier B.V. All rights reserved.

We give a complete characterization of receptive field codes realizable by connected receptive fields and their minimal embedding dimensions. In particular, we show that all connected codes are realizable in dimension at most three. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known.

Many manifold learning algorithms aim to create embeddings with low or no distortion (isometric). If the data has intrinsic dimension d, it is often impossible to obtain an isometric embedding in d dimensions, but possible in s > d dimensions. Yet, most geometry preserving algorithms cannot do the latter. This paper proposes an embedding algorithm to overcome this. The algorithm accepts as inpu...

The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common pro...

In the analysis of chaotic time series, a standard technique is to reconstruct an image of the original dynamical system using delay coordinates. If the original dynamical system has an attractor, then the correlation dimension D2 of its image in the reconstruction can be estimated using the Grassberger–Procaccia algorithm. The quality of the reconstruction can be probed by measuring the length...

Any closed, connected Riemannian manifold M can be smoothly embedded by its Laplacian eigenfunction maps into Rm for some m. We call the smallest such m the maximal embedding dimension of M. We show that the maximal embedding dimension of M is bounded from above by a constant depending only on the dimension of M, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a vol...

Embedding a partially ordered set into a product of chains is a classical way to encode it. Such encodings have been used in various fields such as object oriented programming or distributed computing. The embedding associates with each element a sequence of integers which is used to perform comparisons between elements. A critical measure is the space required by the encoding, and several auth...

Phase space reconstruction of financial time series has become an effective approach for financial system analysis. Since the phase space obtained by the traditional method had the large embedding dimension and was susceptible to noise, this paper proposed a novel phase space reconstruction approach based on manifold learning, called manifold learning-based phase space reconstruction (MLPSR), w...