We consider stable signal recovery in `q quasi-norm for 0 < q ≤ 1. In this problem, given a measurement vector y = Ax for some unknown signal vector x ∈ R and a known matrix A ∈ Rm×n, we want to recover z ∈ R with ‖x − z‖q = O(‖x − x‖q) from a measurement vector, where x∗ is the s-sparse vector closest to x in `q quasi-norm. Although a small value of q is favorable for measuring the distance to...

Let L be the space of complex-valued functions f on the set of vertices T of an infinite tree rooted at o such that the difference of the values of f at neighboring vertices remains bounded throughout the tree, and let Lw be the set of functions f ∈ L such that |f v − f v− | O |v|−1 , where |v| is the distance between o and v and v− is the neighbor of v closest to o. In this paper, we character...

This paper aims to fuzzify the width problem of classical approximation theory. New concepts fuzzy Kolmogorov n-width and linear are introduced on basis α-fuzzy distance which is induced by norm. Furthermore, relationship between widths in normed space discussed. Finally, exact asymptotic orders corresponding a given norm finite-dimensional Sobolev estimated.

The success of many computer vision problems and machine learning algorithms critically depends on the quality of the chosen distance metrics or similarity functions. Due to the fact that the real-data at hand is inherently taskand data-dependent, learning an appropriate distance metric or similarity function from data for each specific task is usually superior to the default Euclidean distance...

The contour of a planar shape is essentially one-dimensional signal embedded in 2-D space; thus the orthogonal distance, which only considers 1-D (norm) deviation from suggested models, is not rich enough to characterize the description quality of arbitrary model/shape pairs. This paper suggests a generalized distance measure, called Transport Distance, for probabilistic shape modeling. B-Splin...

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specif...

A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel’s Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff distance is zero. We establish a completeness theorem. As appli...

Distance Geometry puts the concept of distance at its center. The basic problem in distance geometry could be described as drawing an edge-weighted undirected graph in R for some given K such that the positions for adjacent vertices have distance which is equal to the corresponding edge weight. As we are unaware of any work in this field using any other norm but `2, we move some first steps usi...

The modification of the Clarke generalized subdiNerentia1 due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Glteaux differentiability of any real function can be deduced from the GBteaux differentiability of the norm if the function has a directional derivative which attains a constant related...

FCM is used for image segmentation in some applications. It is based on a specific distance norm and does not use spatial information of the image, so it has some drawbacks. Various kinds of improvements have been developed to extend the adaptability, such as BFCM, SFCM and KFCM. These methods extend FCM from two aspects, one is replacing the Euclidean norm, and the other is considering the spa...