Kernel embedding of distributions: Difference between revisions

Revision as of 18:35, 27 March 2020

Revision as of 18:38, 27 March 2020

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* If <math>k</math> is defined such that <math>f</math> takes values in <math>[0,1]</math> for all <math>f \in \mathcal{H}</math> with <math>\| f\|_\mathcal{H} \le 1 </math> (as is the case for the widely used [[radial basis function]] kernels), then with probability at least <math>1-\delta </math>:<ref name="SongThesis" />

::<math>\|\mu_X - \widehat{\mu}_X \|_\mathcal{H} = \sup_{f \in \mathcal{B}(0,1)} \left| \mathbb{E}_X[f(X)] - \frac{1}{n} \sum_{i=1}^n f(x_i) \right| \le \frac{2}{n} \mathbb{E}_X \left[ \sqrt{\textoperatorname{tr} K} \right] + \sqrt{\frac{\log (2/\delta)}{2n}}</math>

:where <math>\mathcal{B}(0,1)</math> denotes the unit ball in <math>\mathcal{H}</math> and <math>\mathbf{K} =(k_{ij})</math> is the Gram matrix with <math>k_{ij} = k(x_i, x_j).</math>

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DICA measures dissimilarity between domains via '''distributional variance''' which is computed as

:<math>V_\mathcal{H} (\mathcal{P}) = \frac{1}{N} \textoperatorname{tr}(\mathbf{G}) - \frac{1}{N^2} \sum_{i,j=1}^N \mathbf{G}_{ij} </math>

where