Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Finding the Mode of a Kernel Density Estimate   Given points $p_1, \dots, p_n$ in $\mathbb{R}^d$, how do we find a point $x$
which maximizes $\frac{1}{n} \sum_{i=1}^n e^{-\|p_i - x\|^2}$? In other words,
how do we find the maximizing point, or mode of a Gaussian kernel density
estimation (KDE) centered at $p_1, \dots, p_n$? Given the power of KDEs in
representing probability distributions and other continuous functions, the
basic mode finding problem is widely applicable. However, it is poorly
understood algorithmically. Few provable algorithms are known, so practitioners
rely on heuristics like the "mean-shift" algorithm, which are not guaranteed to
find a global optimum. We address this challenge by providing fast and provably
accurate approximation algorithms for mode finding in both the low and high
dimensional settings. For low dimension $d$, our main contribution is to reduce
the mode finding problem to a solving a small number of systems of polynomial
inequalities. For high dimension $d$, we prove the first dimensionality
reduction result for KDE mode finding, which allows for reduction to the low
dimensional case. Our result leverages Johnson-Lindenstrauss random projection,
Kirszbraun's classic extension theorem, and perhaps surprisingly, the
mean-shift heuristic for mode finding.

13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
2
The categories of this paper are: math
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph