Title: A Dynamic Low-Rank Fast Gaussian Transform
Primary area: optimization
Keywords: Fast Gaussian Transform, Theory, kernal-density estimation

Abstract: The Fast Gaussian Transform (FGT) enables subquadratic-time multiplication of an $n\times n$ Gaussian kernel matrix $\mathsf{K}_{i,j}= \exp ( -  \lVert x_i - x_j \rVert_2^2 ) $ with an arbitrary vector $h \in \mathbb{R}^n$, where $x_1,\dots, x_n \in \mathbb{R}^d$ 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 ($\gtrsim n$ time for a single source update $\in \mathbb{R}^d$). 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 ($k\leq d$), our main result is an efficient dynamic FGT algorithm, supporting the following operations in $\log^{O(k)}(n/\varepsilon)$ time: (1) Adding or deleting a source point, and (2) Estimating the "kernel-density" of a query point with respect to sources with $\varepsilon$ 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.