CCA_deflation: The Deflation Scheme for CCA

In DataSlingers/MoMA: MoMA - Modern Multivariate Analysis in R

Description

In MoMA one deflation scheme is provided for CCA.

Details

Let X,Y be two data matrices (properly scaled and centered) of the same number of rows. Each row represents a sample. The penalized CCA problem is formulated as

\min_{u,v} \, u^T X^T Y v + λ_u P_u(u) + λ_v P_v(v)

\text{s.t. } \| u \|_{I+α_u Ω_u} ≤q 1, \| v \|_{I + α_v Ω_v} ≤q 1.

In the discussion below, let u,v be the solution to the above problem. Let c_x = Xu, c_y = Yv. The deflation scheme is as follow:

X ≤ftarrow { X } - { c_x } ≤ft( { c_x } ^ { T } { c_x } \right) ^ { - 1 } { c_x } ^ { T } { X } = ( I - { c_x } ≤ft( { c_x } ^ { T } { c_x } \right) ^ { - 1 } { c_x } ^ { T } )X,

Y ≤ftarrow { Y } - { c_y } ≤ft( { c_y } ^ { T } { c_y } \right) ^ { - 1 } { c_y } ^ { T } { Y } = (I - { c_y } ≤ft( { c_y } ^ { T } { c_y } \right) ^ { - 1 } { c_y } ^ { T } ) Y.

References

De Bie T., Cristianini N., Rosipal R. (2005) Eigenproblems in Pattern Recognition. In: Handbook of Geometric Computing. Springer, Berlin, Heidelberg

DataSlingers/MoMA documentation built on Oct. 30, 2019, 5:55 a.m.