  When estimating finite mixture models, it is common to make assumptions on
the mixture components, such as parametric assumptions. In this work, we make
no distributional assumptions on the mixture components and instead assume that
observations from the mixture model are grouped, such that observations in the
same group are known to be drawn from the same mixture component. We precisely
characterize the number of observations $n$ per group needed for the mixture
model to be identifiable, as a function of the number $m$ of mixture
components. In addition to our assumption-free analysis, we also study the
settings where the mixture components are either linearly independent or
jointly irreducible. Furthermore, our analysis considers two kinds of
identifiability -- where the mixture model is the simplest one explaining the
data, and where it is the only one. As an application of these results, we
precisely characterize identifiability of multinomial mixture models. Our
analysis relies on an operator-theoretic framework that associates mixture
models in the grouped-sample setting with certain infinite-dimensional tensors.
Based on this framework, we introduce general spectral algorithms for
recovering the mixture components and illustrate their use on a synthetic data
set.
