I want to understand how Schnass † arrived at the maximising objective as described below min D , A ‖ X − D A ‖ F 2 = min D ∑ n min | I | ≤ S ‖ x n − D I D I † x n ‖ 2 2 = ‖ X ‖ F − max D ∑ n max | I | ≤ S ‖ D I D I † x n ‖ 2 2 where † is the pseudo-inverse, S denotes the cardinality of column vectors an of matrix A.

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I want to understand how Schnass   † arrived at the maximising objective as described below                                          min                          D              ,              A                                ‖          X          −          D          A                      ‖            F            2                                              =                  min          D                          ∑          n                          min                                    |                        I                          |                        ≤            S                          ‖                  x          n                −                  D          I                          D          I                      †                                    x          n                          ‖          2          2                                                  =        ‖        X                  ‖          F                −                  max          D                          ∑          n                          max                                    |                        I                          |                        ≤            S                          ‖                  D          I                          D          I                      †                                    x          n                          ‖          2          2                    where   † is the pseudo-inverse,   S denotes the cardinality of column vectors an of matrix   A.

treskjeqlalo · Accepted Answer

First, decompose  ‖      x    n    −  A      x    n        ‖    2    2    =  ‖      x    n        ‖    2    2    +  ‖  A      x    n        ‖    2    2    −  2  ⟨      x    n    ,  A      x    n    ⟩In your case,   A  =      D    I        D    I          †      . By the properties of the Moore-Penrose pseudoinverse, we know that       D    I        D    I          †        =      D    I        D    I          †            D    I        D    I          †      . Therefore,  ‖      x    n    −      D    I        D    I          †            x    n        ‖    2    2    =  ‖      x    n        ‖    2    2    +  ‖      D    I        D    I          †            x    n        ‖    2    2    −  2  ⟨      x    n    ,      D    I        D    I          †            D    I        D    I          †            x    n    ⟩  =  ‖      x    n        ‖    2    2    −  ‖      D    I        D    I          †            x    n        ‖    2    2    .The rest follows from the fact that ∑n∥xn∥22=∥X∥2F and the property       min          z        x  −  f  (  z  )  =  x  −      max          z        f  (  z  ).Another proof of the first decomposition:       D    I        D    I          †       is a projection, so by the Pythagorean theorem you have  ‖      x    n        ‖    2    2    =  ‖      D    I        D    I          †            x    n        ‖    2    2    +  ‖  (  I  −      D    I        D    I          †        )      x    n        ‖    2    2    ⇒  ‖      x    n    −      D    I        D    I          †            x    n        ‖    2    2    =  ‖      x    n        ‖    2    2    −  ‖      D    I        D    I          †            x    n        ‖    2    2    .

I want to understand how Schnass † arrived at the maximising objective as des

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