  Finding the mode of a high dimensional probability distribution $D$ is a
fundamental algorithmic problem in statistics and data analysis. There has been
particular interest in efficient methods for solving the problem when $D$ is
represented as a mixture model or kernel density estimate, although few
algorithmic results with worst-case approximation and runtime guarantees are
known. In this work, we significantly generalize a result of (LeeLiMusco:2021)
on mode approximation for Gaussian mixture models. We develop randomized
dimensionality reduction methods for mixtures involving a broader class of
kernels, including the popular logistic, sigmoid, and generalized Gaussian
kernels. As in Lee et al.'s work, our dimensionality reduction results yield
quasi-polynomial algorithms for mode finding with multiplicative accuracy
$(1-\epsilon)$ for any $\epsilon > 0$. Moreover, when combined with gradient
descent, they yield efficient practical heuristics for the problem. In addition
to our positive results, we prove a hardness result for box kernels, showing
that there is no polynomial time algorithm for finding the mode of a kernel
density estimate, unless $\mathit{P} = \mathit{NP}$. Obtaining similar hardness
results for kernels used in practice (like Gaussian or logistic kernels) is an
interesting future direction.
