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The general form of divergence invariant to transformations

Speech signals inevitably include non-linguistic information, such as, gender, age, noise etc. Finding measures (or features) invariant to the inevitable variations caused by the non-linguistical factors (transformations) is a fundamental yet important problem in speech recognition. Recently, Minematsu [16], [17] proved that Bhattacharyya distance (BD) between two distributions is invariant to invertible transforms on feature space, and developed an invariant structural representation of speech based on it. There is a question: which kind of measures can be invariant? In this paper, we prove that f -divergence yields a generalized family of invariant measures, and show that all the invariant measures have to be written in the form of f -divergence. Many famous measures and divergences in information and statics theory, such as Bhattacharyya distance, KL-divergence, Hellinger distance, belong to f -divergence. We carried out experiments on recognizing utterances of connected Japanese vowels. The experimental results indicate that BD and KL have the best performance.