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By the triangle inequality, this gives the desired contradiction d2S (x, x ) < d2S (t(x), x ) ≤ d2S (t(x), x) + d2S (x, x ) = 0 + d2S (x, x ). Based on the total invariance, d2S (·, x ) is constant on each E ∈ X /∼ : For all x ∼ x transformations t¯, t¯ exist such that t¯(x) = t¯(x ). e. this induces a well defined function on X /∼ by d¯2S (E, E ) := d2S (x(E), x(E )). Here x(E) denotes one representative from the equivalence class E ∈ X /∼ . Obviously, d¯2S ¯ is a Hilbertian metric. via )(E) := )(x(E)).

Iii) If T = T¯ and T¯−1 = T¯ , then unregularized IDS-kernels are totally invariant with respect to T¯. iv) If d is the ordinary Euclidean distance, then limO→f kIDS = k. Proof. Statement i) is obvious from the definition, as d2S = d in this case. Similarly, iv) follows as limO→f d2S = d. For statement ii), we note that if x ∈ (T −1 ◦ T )x , then there exist transformations t,t ∈ T such that t(x) = t (x ) and consequently d2S (x, x ) = 0. So any distance-based kernel kIDS is constant on this set (T −1 ◦ T )x .

This is more likely if the parameter GCC is large. Therefore the classification performance depends heavily on the choice of GCC . In order to calculate the class posterior probabilities the parameters T j and the priors S jk and Pk := P(ck ) are estimated based on maximum likelihood and the EM algorithm. Typically, fT j (x | j) is a normal density with parameters T j = {z j , 6 j }. A derivation of the EM steps for the gaussian case is given in Titsias and Likas (2001), p. 989. 2 Hierarchical Mixture Classifier – HM Classifier The HM Classifier (Titsias and Likas (2002)) can be considered as extension of the CC Classifier.