# Representation within a RKHS framework

Given a p.s.d kernel $Q$, can minimization/maximization of $Tr(X^TQX)$ over X be represented within a reproducing kernel Hilbert space (RKHS) framework? If there is a primary concern with the trace function being unbounded(for maximization) or trivial zero-solution(minimization); you may consider constraints over $X$; for example-say $X$ being orthogonal. Am trying to see if I can fit this in an RKHS framework.

Also, a few thoughts around this function using hilbert-schmidt norms are as follows: I do see that $TrX^TQX$ can be represented as $Tr[(SX)^T(SX)]=||SX||^2_{HS}$, using the hilbert schmidt norm where $S$ is the p.s.d square root of Q. (Ex: $S=U\lambda^{1/2}$, where $Q=U\lambda U^T$ is the eigen-decomposition of $Q$).

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May you help us to understand your original problem first, without introducing a RKHS? –  Zen Sep 2 '12 at 3:42
You have a symmetric positive semidefinite $n\times n$ matrix $Q$, which we may decompose as $Q=S S^\top$, and you want to minimize $\textrm{tr} (X^\top Q X) = ||S X||^2_{HS}$, where $X$ is an ortogonal $n\times n$ matrix. Is this description correct? –  Zen Sep 2 '12 at 3:49
@Zen I want to see if there are any connections over the minimization problem with RKHS. –  PraneethVepakomma Sep 2 '12 at 16:17
OK. But just to make it clear: my description without the introduction of a RKHS is correct? –  Zen Sep 2 '12 at 17:33
Just read section 1.1. –  Zen Sep 2 '12 at 17:40
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## 1 Answer

Hilbert-Schmidt norm in the RKHS as you describe has been used in statistics for a while.

Hilbert-Schmidt Independence Criterion (HSIC) has been successful in capturing statistical dependences. See Gretton and coworkers' papers:

• A Gretton, O Bousquet, A Smola, B Schölkopf. Measuring statistical dependence with Hilbert-Schmidt norms. Algorithmic learning theory, 2005

• K Fukumizu, A Gretton, X Sun, B Schölkopf. Kernel measures of conditional dependence. 2008

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