Sparsification for Sums of Exponentials and its Algorithmic Applications

Many works in signal processing and learning theory operate under the assumption that the underlying model is simple, e.g. that a signal is approximately $k$-Fourier-sparse or that a distribution can be approximated by a mixture model that has at most $k$ components. However the problem of fitting the parameters of such a model becomes more challenging when the frequencies/components are too close together. In this work we introduce new methods for sparsifying sums of exponentials and give various algorithmic applications. First we study Fourier-sparse interpolation without a frequency gap, where Chen et al. gave an algorithm for finding an $\epsilon$-approximate solution which uses $k' = \mbox{poly}(k, \log 1/\epsilon)$ frequencies. Second, we study learning Gaussian mixture models in one dimension without a separation condition. Kernel density estimators give an $\epsilon$-approximation that uses $k' = O(k/\epsilon^2)$ components. These methods both output models that are much more complex than what we started out with. We show how to post-process to reduce the number of frequencies/components down to $k' = \widetilde{O}(k)$, which is optimal up to logarithmic factors. Moreover we give applications to model selection. In particular, we give the first algorithms for approximately (and robustly) determining the number of components in a Gaussian mixture model that work without a separation condition.