Consider a data matrix $M_{n ~ x ~Features}$ and its top two principal components ($PC_1$ and $PC_2$). For any of submatrix $S_{n ~x ~F}$, where $F \subset Features$ (i.e. S contains all samples but a subset of Features), how to find principal components of S that are also orthogonal to $PC_1$ and $PC_2$?

A heuristic approach might be to find principal components of $S - W_1 * PC_1 - W_2 *PC_2 $, where $W_1$ and $W_2$ are linear regression coefficient. Is this correct or any other better/principled approach?

  • $\begingroup$ Please explain what sense of "independence" you are using: is this the probabilistic meaning, where the matrix is random, or perhaps do you mean "linearly independent"? And in that case, please explain how a PC of a submatrix can reasonably be compared to a PC of the main matrix. (This will depend on what you mean by "submatrix.") $\endgroup$ – whuber Jul 24 '19 at 13:11
  • $\begingroup$ @whuber Yes, I meant linearly independent (i.e. orthogonal). Also for the sake of simplicity lets assume M is normally distributed. I also clarified the submatrix S contains all samples but only a subset of features. $\endgroup$ – avi Jul 24 '19 at 17:41
  • $\begingroup$ Thank you. The answer to your modified question is "yes." $\endgroup$ – whuber Jul 24 '19 at 20:43

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