In MoMA one deflation scheme is provided for CCA.
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
\( \min_{u,v} \, u^T X^T Y v + \lambda_u P_u(u) + \lambda_v P_v(v) \)
\( \text{s.t. } \| u \|_{I+\alpha_u \Omega_u} \leq 1, \| v \|_{I + \alpha_v \Omega_v} \leq 1. \)
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:
\(X \leftarrow { X } - { c_x } \left( { c_x } ^ { T } { c_x } \right) ^ { - 1 } { c_x } ^ { T } { X } = ( I - { c_x } \left( { c_x } ^ { T } { c_x } \right) ^ { - 1 } { c_x } ^ { T } )X,\)
\( Y \leftarrow { Y } - { c_y } \left( { c_y } ^ { T } { c_y } \right) ^ { - 1 } { c_y } ^ { T } { Y } = (I - { c_y } \left( { c_y } ^ { T } { c_y } \right) ^ { - 1 } { c_y } ^ { T } ) Y\).
De Bie T., Cristianini N., Rosipal R. (2005) Eigenproblems in Pattern Recognition. In: Handbook of Geometric Computing. Springer, Berlin, Heidelberg