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Abstract:
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In recent years, researchers have developed the tensor method to approximate closed-form solutions for method of moments estimation procedures involving third-order moments. This approach is an alternative to maximum likelihood estimation, which may be computationally intractable due to a highly multi-modal likelihood surface. However, the tensor method has complications and instabilities that limit its use in practice. One notable issue is the requirement that the true parameters be linearly independent, implying that the number of parameters cannot exceed the data dimension. We present two variants of the tensor method to address this issue. The first approach uses principal components to make the ingredients of the tensor method more familiar and intuitive. The second approach embeds the data in a higher-dimensional space to apply a tensor method to the embedded vectors. We demonstrate these variants in simulations of Gaussian mixture estimation. The principal components approach simplifies the estimation of the mixture components' third-order moments. The basis expansion example demonstrates how to devise additional variables for moment estimation in a higher-dimensional space.
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