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Canonical Correlation

Canonical correlation is a statistical measure for expressing the relationship between two sets of variables. Formally, given two random vectors x ∈ Rdx and y ∈ Rdy with some joint (unknown) distribution D, the canonical correlation analysis (CCA) seeks vectors u ∈ Rdx and v ∈ Rdy, such that the random vectors when projected along these directions, that is, variables u > x and v > y, are maximally correlated. Equivalently, we can write CCA as the following optimization problem: find u ∈ Rdx, v ∈ Rdy that:

Maximizedxdyρ(u>x,v>y),uR,vR

where the correlation, ρ(u > x, v > y), between two random variables, is defined as ρ(u>x,v>y)=cov(u>x,v>y). Assuming that vectors x and y are 0 mean, we can write CCA as the problem var(u > x) var(u > ...

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