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Inner product spaces for Bayesian networks AbstractWe consider the idea of combining the key advantages of Bayesian networks and of kernel-based learning systems. In connection with two-label classification tasks over the Boolean domain, we study the question whether the class of decision functions induced by a given Bayesian network can be represented within a low-dimensional inner product space. For Bayesian networks with an explicitly given (full or reduced) parameter collection, we establish tight bounds on the dimension of the ``natural'' inner product space. Further, we consider a variant of the logistic autoregressive Bayesian network and show that every sufficiently expressive inner product space must have dimension at least $2^{\Omega(n)}$, where $n$ is the number of network nodes. As the main technical contribution, this work reveals combinatorial and algebraic structures within Bayesian networks such that known techniques for proving lower bounds on the dimension of inner product spaces can be brought into play.
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