Consistent spectral clustering in sparse tensor block models

TL;DR

This work develops a sparse tensor block model for count-valued data and introduces a simple yet effective hollow Gram spectral clustering algorithm with a trimming step to mitigate noise in sparse regimes. A new sub-Poisson concentration framework yields a near-optimal polynomial-time consistency threshold ρ ≫ n^{-d/2}, closing the gap with computational lower bounds. The results extend from binary to integer-valued TBMs and establish an aggregation-based framework that preserves information while reducing density requirements. Numerical experiments demonstrate the method's robustness across sparsity regimes and illustrate the trade-offs between aggregation, initialization, and clustering accuracy. The work provides a unified, nearly optimal approach to multiway clustering in sparse count data with practical implications for high-dimensional tensor analysis.

Abstract

High-order clustering aims to classify objects in multiway datasets that are prevalent in various fields such as bioinformatics, recommendation systems, and social network analysis. Such data are often sparse and high-dimensional, posing significant statistical and computational challenges. This paper introduces a tensor block model specifically designed for sparse integer-valued data tensors. We propose a simple spectral clustering algorithm augmented with a trimming step to mitigate noise fluctuations, and identify a density threshold that ensures the algorithm's consistency. Our approach models sparsity using a sub-Poisson noise concentration framework, accommodating heavier than sub-Gaussian tails. Remarkably, this natural class of tensor block models is closed under aggregation across arbitrary modes. Consequently, we obtain a comprehensive framework for evaluating the tradeoff between signal loss and noise reduction incurred by aggregating data. The analysis is based on a novel concentration bound for sparse random Gram matrices. The theoretical findings are illustrated through numerical experiments.

Figures (5)

• Figure 1: Binary TBM of order $d=3$ with dimensions $n_1=n_2=n_3=40$, cluster counts $r_1=r_2=r_3=2$, and identical clustering across modes ($z_1 = z_2 = z_3$).
• Figure 2: Core tensors $\mathcal{S}_{\rm uninformative}$ (left) and $\mathcal{S}_{\rm informative}$ (right) used in simulations, with black = 1, white = 0.
• Figure 3: Comparison of clustering algorithms when the aggregate matrix is uninformative. The number of nodes varies logarithmically between $30$ and $180$, and the density parameter varies logarithmically between $0.002$ and $0.027$. The dashed black line shows the theoretical phase-transition boundary corresponding to $\gamma = 1.33$, $1.33$, and $1.5$ for Vanilla SVD, HSC, and Hollow SVD, respectively. The corresponding estimated values are $1.29$, $1.26$, and $1.43$.
• Figure 4: Comparison of clustering algorithms when the aggregate matrix is uninformative. The number of nodes varies logarithmically between $30$ and $180$, and the density parameter varies logarithmically between $0.001$ and $0.019$. The dashed black line shows the theoretical phase-transition boundary corresponding to $\gamma = 2$, $1.33$, $1.33$ and $1.5$ for Aggregate SVD, Vanilla SVD, HSC and Hollow SVD, respectively. The corresponding estimated values are $2.00$, $1.31$, $1.24$, and $1.43$.
• Figure 5: Node embeddings computed with (a)Hollow SVD and (b)HSC initialized with Hollow SVD. The number of nodes is fixed to $n=200$ and the density parameter $\rho$ is varied around the phase transition.

Theorems & Definitions (40)

• Example 2.1: Bernoulli TBM
• Example 2.2: Poisson TBM
• Theorem 3.1: Weak consistency
• Theorem 3.2: Aggregation
• proof
• Corollary 3.3: Weak consistency in Aggregated TBM
• proof
• Example 3.1
• Theorem 6.1
• proof