objective.fun {matchFeat}

Calculate Cost of Multidimensional Assignment

Description

Calculates the objective value in the multidimensional assignment problem with decomposable costs (MDADC). The dissimilarity function used in this problem is the squared Euclidean distance.

Usage

objective.fun(x, sigma = NULL, unit = NULL, w = NULL)

Arguments

x

data: matrix of dimensions (mn,p) or 3D array of dimensions (p,m,n) with m = number of labels/classes, n = number of sample units, and p = number of variables)

sigma

permutations: matrix of dimensions (m,n)

unit

integer (=number of units) or vector mapping rows of x to sample units (length mn). Must be specified only if x is a matrix.

w

weights for loss function: single positive number, p-vector of length, or (p,p) positive definite matrix

Details

Given n datasets having each m vectors of same size, say {x_{11},...,x_{1m}},...,x_{n1},...,x_{nm}, and permutations \sigma_1,...,\sigma_n of {1,...,m}, the function calculates 1/(n(n-1)) sum_{i,j} sum_{k} || x_{i,sigma_i(k)- x_{j,\sigma_j(k) \|^2}} where i and n run from 1 to n and k runs from 1 to m. This is the objective value (1) of Degras (2021), up to the factor 1/(n(n-1)).

Value

Objective value

References

Degras (2022) "Scalable feature matching across large data collections." doi:10.1080/10618600.2022.2074429

See Also

objective.gen.fun

Examples

data(optdigits)
m <- 10
n <- 100
sigma <- matrix(1:m,m,n) # identity permutations
objective.fun(optdigits$x, sigma, optdigits$unit)