k2d: Convert Kernel Matrix to Distance Matrix

Description

k2d provides various methods for converting kernel matrix to distance matrix and vice versa.

Usage

k2d(Mat, direction = "k2d", method = "norm", scale = 1, pos = TRUE)

Arguments

Mat

matrix, either kernel matrix or distance matrix to be converted.

direction

a character string "k2d" or "d2k". The latter interpret the Mat as a distance matrix and convert it to a kernel matrix. Default "k2d".

method

a character string to specify how the matrix is converted. "Default "norm".

scale

a numeric parameter used to scale the matrix. Typically used in exp( - d(x,y)/scale).

pos

logical. If TRUE when direction="d2k", negative eigenvalues are round to zero to obtain positive semidefinite kernel matrix. Default TRUE.

Details

There are various ways to convert kernel function values to distance between two points. Normal-distance (when method="norm") means a conversion d_ND(x,y) = sqrt k(x,x) - 2k(x,y) + k(y,y).

Cauchy-Schwarz-type conversion (method="CS") is more principled way: d_CS(x,y) = arccos k^2(x,y)/k(x,x)k(y,y).

Other two simple ways are d_exp(x,y) = exp(- k(x,y)/scale), which is an exponential-type distance (method="exp"), and d_n(x,y) = 1 - k(x,y)/sqrt k(x,x)k(y,y), which we call naive (method="naive").

For converting distance to kernel (direction="d2k"), it should be noted that we usually have distance between pairs of points only, and distances from "origin" are unknown. Double-centering (method="DC") is the most popular and simple way to convert distance to kernel. However, it does not make positive definite kernel in general, and it sometimes require post-processing, e.g., cutting off negative eigenvalues (pos=TRUE). Another simple way is exponential map (method="exp"), i.e., k(x,y) = exp( - d(x,y)/scale).