Dynamic Kernel Graph Sparsifiers

TL;DR

The paper tackles dynamic kernel graphs formed by a point set $P\subset \mathbb{R}^d$ with kernel $\mathsf{K}$, aiming to maintain a $$(1\pm \epsilon)$$-spectral sparsifier of the graph Laplacian $L_G$ under point updates. It introduces DynamicGeoSpar, a fully-dynamic sparsification data structure built on a JL projection, a well-separated pair decomposition (WSPD), and resampling from bicliques to achieve $n^{o(1)}$ update times and $n^{1+o(1)}$ initialization, with robustness to adaptive adversaries under certain conditions. The framework further develops low-dimensional sketches for Laplacian-vector products and Laplacian-system solves, enabling subpolynomial-time query/update handling for both $L_G v$ and $L_G^{\dagger} b$ while controlling error via spectral guarantees. The methods combine spectral sparsification with dimension-reduction techniques to support real-time dynamic updates in geometric graphs, with potential impact on dynamic kernel methods, spectral clustering, N-body simulations, and semi-supervised learning. Overall, the work delivers near-optimal dynamic primitives for geometry-based linear algebra, including robust sparsification, adaptive-adversary resilience, and efficient Laplacian sketches.

Abstract

A geometric graph associated with a set of points $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ and a fixed kernel function $\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^d\to\mathbb{R}_{\geq 0}$ is a complete graph on $P$ such that the weight of edge $(x_i, x_j)$ is $\mathsf{K}(x_i, x_j)$. We present a fully-dynamic data structure that maintains a spectral sparsifier of a geometric graph under updates that change the locations of points in $P$ one at a time. The update time of our data structure is $n^{o(1)}$ with high probability, and the initialization time is $n^{1+o(1)}$. Under certain assumption, our data structure can be made robust against adaptive adversaries, which makes our sparsifier applicable in iterative optimization algorithms. We further show that the Laplacian matrices corresponding to geometric graphs admit a randomized sketch for maintaining matrix-vector multiplication and projection in $n^{o(1)}$ time, under sparse updates to the query vectors, or under modification of points in $P$.

Figures (3)

• Figure 1: Division of the new biclique ($A' \times B'$): divided it into two parts (Blue part: $(A' \times B')\backslash(A \times B)$ and red part $(A \times B)\cap(A' \times B')$). And we sample from them respectively.
• Figure 2: Resampling the biclique: $E$ (The red edges) is uniformly sampled from $\mathrm{Biclique}(A, B)$. After $A \times B$ becomes $A' \times B'$, we resample from $E \cap (A' \times B')$ with specific probabilities.
• Figure 3: The net argument in our problem. Let $d = 2$. Here we restrict all the points to be in the $\ell_2$ unit ball. By the definition of aspect ratio $\alpha$, we know the minimum distance between two points is $1/\alpha$ ($A$ and $B$ in the figure). Thus, by setting $\epsilon = C \cdot \alpha^{-1}$ for some constant $C$ small enough, every pair of points is distinguishable.

Theorems & Definitions (39)

• Definition 1.1: Dynamic spectral sparsifier of geometric graph
• Definition 1.2: Sketch of approximation to matrix multiplication
• Definition 1.3: Sketch of approximation to Laplacian solving
• Theorem 2.1: Informal version of Theorem \ref{['thm:formal_dynamic_kernel_sparsifier']}
• Theorem 2.2: Informal version of Theorem \ref{['thm:formal_adversarial_sparsifier']}
• Theorem 2.3: Informal version of Theorem \ref{['thm:formal_solve_implies_multiply']}
• Theorem 2.4: Informal version of Theorem \ref{['thm:formal_multiply_implies_solve']}
• Definition B.1
• Definition B.2: Laplacian of graph
• proof