Formulas for a characteristic of spheres and balls in binary high-dimensional spaces

2019 ◽  
Vol 29 (5) ◽  
pp. 311-319
Author(s):  
V. G. Mikhailov
Keyword(s):  
Linear Space ◽  
Special Function ◽  
Poisson Approximation ◽  
High Dimensional ◽  
Dimensional Vector ◽  
Number Of Solutions ◽  
Hamming Metric ◽  
Approximate Formulas

Abstract We consider a special function ρ(H) of the subset H of n-dimensional vector linear space over the field K. This function is used in the estimates of accuracy of the Poisson approximation for the distribution of the number of solutions of systems of random equations and random inclusions over K. For the case when K = GF(2) and H is a sphere or ball (in the Hamming metric) in {0, 1}n we obtain explicit and approximate formulas for ρ(H) for sufficiently large values of n.

scholarly journals Tropical Balls and Its Applications to K Nearest Neighbor over the Space of Phylogenetic Trees

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 779
Author(s):  
Ruriko Yoshida
Keyword(s):  
Supervised Learning ◽  
Phylogenetic Trees ◽  
Nearest Neighbor ◽  
Nearest Neighbors ◽  
High Dimensional ◽  
Learning Method ◽  
Dimensional Vector ◽  
K Nearest Neighbor ◽  
K Nearest Neighbors

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space.

scholarly journals Greedy measurement selection for state estimation

2018 ◽  
Author(s):  
J. Nichols ◽  
Albert Cohen ◽  
Peter Binev ◽  
Olga Mula
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
High Dimensional ◽  
Dimensional Vector ◽  
Physical Processes ◽  
Linear Measurements ◽  
Measurement Selection ◽  
Selection For ◽  
Unknown Solution

Parametric PDEs of the general form $$ \mathcal{P}(u,a)=0 $$ are commonly used to describe many physical processes, where $\mathcal{P}$ is a differential operator, a is a high-dimensional vector of parameters and u is the unknown solution belonging to some Hilbert space V. Typically one observes m linear measurements of u(a) of the form $\ell_i(u)=\langle w_i,u \rangle$, $i=1,\dots,m$, where $\ell_i\in V'$ and $w_i$ are the Riesz representers, and we write $W_m = \text{span}\{w_1,\ldots,w_m\}$. The goal is to recover an approximation $u^*$ of u from the measurements. The solutions u(a) lie in a manifold within V which we can approximate by a linear space $V_n$, where n is of moderate dimension. The structure of the PDE ensure that for any a the solution is never too far away from $V_n$, that is, $\text{dist}(u(a),V_n)\le \varepsilon$. In this setting, the observed measurements and $V_n$ can be combined to produce an approximation $u^*$ of u up to accuracy $$ \Vert u -u^*\Vert \leq \beta^{-1}(V_n,W_m) \, \varepsilon $$ where $$ \beta(V_n,W_m) := \inf_{v\in V_n} \frac{\Vert P_{W_m}v\Vert}{\Vert v \Vert} $$ plays the role of a stability constant. For a given $V_n$, one relevant objective is to guarantee that $\beta(V_n,W_m)\geq \gamma >0$ with a number of measurements $m\geq n$ as small as possible. We present results in this direction when the measurement functionals $\ell_i$ belong to a complete dictionary.

Classification of distance-transitive orbital graphs of overgroups of the Jevons group

2020 ◽  
Vol 30 (1) ◽  
pp. 7-22
Author(s):  
Boris A. Pogorelov ◽  
Marina A. Pudovkina
Keyword(s):  
Vector Space ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Isometry Group ◽  
Complete Bipartite Graph ◽  
Dimensional Vector ◽  
Translation Group ◽  
Dimensional Vector Space ◽  
Hamming Metric

AbstractThe Jevons group AS̃n is an isometry group of the Hamming metric on the n-dimensional vector space Vn over GF(2). It is generated by the group of all permutation (n × n)-matrices over GF(2) and the translation group on Vn. Earlier the authors of the present paper classified the submetrics of the Hamming metric on Vn for n ⩾ 4, and all overgroups of AS̃n which are isometry groups of these overmetrics. In turn, each overgroup of AS̃n is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group AS̃n. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph 2n, the complete bipartite graph K2n−1,2n−1, the halved (n + 1)-cube, the folded (n + 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.

CONSTRUCTION OF m-METRIC ARRAY CODES DETECTING AND CORRECTING CT-BURST ARRAY ERRORS

2008 ◽  
Vol 01 (04) ◽  
pp. 589-617 ◽  
Author(s):  
Sapna Jain ◽  
Seul Hee Choi
Keyword(s):  
Finite Field ◽  
Error Correction ◽  
Linear Space ◽  
Lower Bounds ◽  
Array Codes ◽  
Hamming Metric ◽  
Array Errors

In [11], the first author introduced the notion of CT bursts in the space Mat m×s(Fq), the linear space of all m × s matrices with entries from a finite field Fq, endowed with a non-Hamming metric [16] and obtained some lower bounds for CT burst array error correction. In this paper, we first obtain various construction bounds on the parameters of m-metric array codes [16] for the detection and correction of CT burst array errors and then construct the array codes meeting these bounds.

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