Applications Taxicab geometry

1 applications

1.1 measures of distances in chess
1.2 compressed sensing
1.3 differences of frequency distributions

applications
measures of distances in chess

in chess, distance between squares on chessboard rooks measured in taxicab distance; kings , queens use chebyshev distance, , bishops use taxicab distance (between squares of same color) on chessboard rotated 45 degrees, i.e., diagonals coordinate axes. reach 1 square another, kings require number of moves equal distance; rooks, queens , bishops require 1 or 2 moves (on empty board, , assuming move possible @ in bishop s case).

compressed sensing

in solving underdetermined system of linear equations, regularisation term parameter vector expressed in terms of



ℓ
1


{\displaystyle \ell 1}

-norm (taxicab geometry) of vector. approach appears in signal recovery framework called compressed sensing.

differences of frequency distributions

taxicab geometry can used assess differences in discrete frequency distributions. example, in rna splicing positional distributions of hexamers, plot probability of each hexamer appearing @ each given nucleotide near splice site, can compared l1-distance. each position distribution can represented vector each entry represents likelihood of hexamer starting @ nucleotide. large l1-distance between 2 vectors indicates significant difference in nature of distributions while small distance denotes shaped distributions. equivalent measuring area between 2 distribution curves because area of each segment absolute difference between 2 curves likelihoods @ point. when summed segments, provides same measure l1-distance.