Spectral Clustering in Birthday Paradox Time

TL;DR

This work designs a sublinear-time spectral clustering framework for $(k, \varphi, \epsilon)$-clusterable graphs by constructing locally computable embeddings $\widetilde{f}_x$ via a carefully crafted mixture of random-walk samples and a low-coefficient matrix polynomial $p$. The embeddings enable an efficient dot-product-based clustering paradigm, with a Sketch-based nearest-neighbor primitive that achieves nearly linear-time queries and sublinear preprocessing/space under mild parameter constraints. A clustering oracle is obtained that misclassifies only a small fraction of vertices, scales with the key parameters as $O^*(\sqrt{nk}\, n^{O(\epsilon/\varphi^2 \log(1/\epsilon))})$ for preprocessing and $O^*(\sqrt{n/k}\, n^{O(\epsilon/\varphi^2 \log(1/\epsilon))})$ for queries, and can approximate the number of clusters $k$ in sublinear time. The approach bridges birthday-paradox intuition with algebraic techniques (matrix polynomials and Chebyshev approximations) to remove prior $k$-dependent overhead, offering practical sublinear clustering with provable guarantees for broad graph models.

Abstract

Given a vertex in a $(k, \varphi, ε)$-clusterable graph, i.e. a graph whose vertex set can be partitioned into a disjoint union of $\varphi$-expanders of size $\approx n/k$ with outer conductance bounded by $ε$, can one quickly tell which cluster it belongs to? This question goes back to the expansion testing problem of Goldreich and Ron'11. For $k=2$ a sample of $\approx n^{1/2+O(ε/\varphi^2)}$ logarithmic length walks from a given vertex approximately determines its cluster membership by the birthday paradox: two vertices whose random walk samples are `close' are likely in the same cluster. The study of the general case $k>2$ was initiated by Czumaj, Peng and Sohler [STOC'15], and the works of Chiplunkar et al. [FOCS'18], Gluch et al. [SODA'21] showed that $\approx \text{poly}(k)\cdot n^{1/2+O(ε/\varphi^2)}$ random walk samples suffice for general $k$. This matches the $k=2$ result up to polynomial factors in $k$, but creates a conceptual inconsistency: if the birthday paradox is the guiding phenomenon, then the query complexity should decrease with the number of clusters $k$! Since clusters have size $\approx n/k$, we expect to need $\approx (n/k)^{1/2+O(ε/\varphi^2)}$ random walk samples, which decreases with $k$. We design a novel representation of vertices in a $(k, \varphi, ε)$-clusterable graph by a mixture of logarithmic length walks. This representation uses the optimal $\approx (n/k)^{1/2+O(ε/\varphi^2)}$ walks per vertex, and allows for a fast nearest neighbor search: given $k$ vertices representing the clusters, we can find the cluster of a given query vertex $x$ using nearly linear time in the representation size of $x$. This gives a clustering oracle with query time $\approx (n/k)^{1/2+O(ε/\varphi^2)}$ and space complexity $k\cdot (n/k)^{1/2+O(ε/\varphi^2)}$, matching the birthday paradox bound.

Figures (1)

• Figure 1: Illustration of $\left \langle p_{x}^{t} , \left(p_S^{\ell}\right)^2 \right \rangle$ (Left) and $\left \langle \left( p_x^t\right)^2, p_S^{\ell}\right \rangle$ (Right). Intuitively, we can think of $\left \langle p_{x}^{t} , \left(p_S^{\ell}\right)^2 \right \rangle$ and $\left \langle \left( p_x^t\right)^2, p_S^{\ell}\right \rangle$ as the expected three-way collision rates between random walks started from $x$ and random walks started from $S$.

Theorems & Definitions (149)

• Theorem 1.1: Approximating the number of clusters; informal version of \ref{['thm:find_k']}
• Theorem 1.2: Clustering oracle; informal version of \ref{['thm:main']}
• Theorem 1.3: Space/query tradeoffs for clustering oracle; informal
• Remark 1.4
• Remark 1.5
• Definition 2.1: Inner and outer conductance
• Remark 2.2
• Definition 2.3: $(k,\varphi,\epsilon)$-clusterable graph
• Definition 2.4: Spectral Clustering Oracle
• Definition 2.5: Spectral embedding