$\max_\limits{S} \sum_{j=1}^kS_j^TKS_j $ $s.t\ S^TS = I$ where $S = (S_1, .., S_k),\ S_i\ (nx1)$
Not sure how to solve this problem since it cannot form a lagrangian equation due to conformability issues.
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what are the properties of $K$? – user619894 Feb 10 '20 at 06:30
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symmetric and psd. – John Karasev Feb 10 '20 at 08:49
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If you treat $S^TS=I$ as an orthonormality constraint, and try to optimize $\sum_{j=1}^kS_j^TKS_j +\lambda(S^TS-I)$ for S, you will find that the $S$ are eigenvectors of $K$. – user619894 Feb 10 '20 at 09:29
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check the dimensions of the objective, that won't work. Maybe breaking down the $S^TS$ into two summations one along diagonal other off-diagonal. – John Karasev Feb 10 '20 at 21:59
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Yes in the end it is spectral decomp of K taking the first highest e-values. Just wanted to figure if there is a nice way to put that constraint into the lagrangian. – John Karasev Feb 10 '20 at 22:00