  This work considers the problem of estimating the parameters of negative
mixture models, i.e. mixture models that possibly involve negative weights. The
contributions of this paper are as follows. (i) We show that every rational
probability distributions on strings, a representation which occurs naturally
in spectral learning, can be computed by a negative mixture of at most two
probabilistic automata (or HMMs). (ii) We propose a method to estimate the
parameters of negative mixture models having a specific tensor structure in
their low order observable moments. Building upon a recent paper on tensor
decompositions for learning latent variable models, we extend this work to the
broader setting of tensors having a symmetric decomposition with positive and
negative weights. We introduce a generalization of the tensor power method for
complex valued tensors, and establish theoretical convergence guarantees. (iii)
We show how our approach applies to negative Gaussian mixture models, for which
we provide some experiments.
