A Dynamic Low-Rank Fast Gaussian Transform
arxiv(2022)
摘要
The Fast Gaussian Transform (FGT) enables subquadratic-time
multiplication of an n× n Gaussian kernel matrix 𝖪_i,j= exp
( - x_i - x_j _2^2 ) with an arbitrary vector h ∈ℝ^n,
where x_1,…, x_n ∈ℝ^d are a set of fixed source
points. This kernel plays a central role in machine learning and random feature
maps. Nevertheless, in most modern data analysis applications, datasets are
dynamically changing (yet often have low rank), and recomputing the FGT from
scratch in (kernel-based) algorithms incurs a major computational overhead
(≳ n time for a single source update ∈ℝ^d). These
applications motivate a dynamic FGT algorithm, which maintains a dynamic
set of sources under kernel-density estimation (KDE) queries in
sublinear time while retaining Mat-Vec multiplication accuracy and
speed.
Assuming the dynamic data-points x_i lie in a (possibly changing)
k-dimensional subspace (k≤ d), our main result is an efficient dynamic
FGT algorithm, supporting the following operations in
log^O(k)(n/ε) time: (1) Adding or deleting a source point, and
(2) Estimating the “kernel-density” of a query point with respect to sources
with ε additive accuracy. The core of the algorithm is a dynamic
data structure for maintaining the projected “interaction rank”
between source and target boxes, decoupled into finite truncation of Taylor and
Hermite expansions.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要