more coherently: given a pair of vectors x and y representing our input and target sets (encoded as distributions) find a function f from some space that best

more coherently: given a pair of vectors x and y representing our input and target sets (encoded as distributions) find a function f from some space that best solves: f(x) = y if we require x, y, f(x) to be distributions, we might restate this as a minimisation problem: given input x and target y s.t.: 0 <= x <= 1 and sum(x) = 1 0 <= y <= 1 and sum(y) = 1 constraints: for any z s.t. 0 <= z <= 1 and sum(z) = 1 we require 0 <= f(z) <= 1 and sum(f(z)) = 1 minimise d( f(x), y ) where d( . , . ) is some appropriate metric. such as KL divergence or inner product or so on. in the simple case where we additionally require that f be a linear operator A, and pick the L^2 norm as our loss function, we get a constrained minimisation problem to solve for a matrix A that minimises ||Ax - y||^2 subject to all our constraints.