spectral subgraph recovery

This study addresses the problem of recovering a planted $k$-vertex subgraph with elevated triangle density in an Erdős–Rényi random graph—a task known to be computationally hard in the worst case. To tackle this challenge, the work introduces spectral and semidefinite programming algorithms based on a novel local signed triangle count matrix, marking the first incorporation of signed triangle counts into a spectral recovery framework. Theoretical analysis establishes that the information-theoretic threshold for exact recovery scales logarithmically in $n$, whereas the proposed algorithms succeed only when $k$ is at least on the order of $\sqrt{n}$, thereby revealing a substantial statistical–computational gap. The paper also provides rigorous recovery guarantees for the algorithms and validates their performance within the low-degree polynomial framework.

This work addresses the problem of detecting and exactly recovering hidden cliques when only the hypergraph adjacency matrix—defined as the co-occurrence counts of node pairs—is observed. To overcome information loss and entry dependencies induced by hyperedge projection, the authors propose a spectral-norm-based detection statistic and a polynomial-time spectral algorithm driven by the leading eigenvector. They establish the first rigorous theoretical guarantees under this observation model. By extending leave-one-out eigenvector analysis to the hypergraph setting, they achieve asymptotically optimal detection and recovery at the √n scale, explicitly characterizing the influence of the background hyperedge probability and demonstrating applicability even in sparse hypergraph regimes.

This paper investigates exact community recovery in the two-community stochastic block model (SBM) with node-attribute side information, unifying Bernoulli and Gaussian edge observation models—encompassing SBM, submatrix localization, and ℤ₂ synchronization as special cases. We propose a low-complexity, non-iterative spectral algorithm that jointly leverages the leading eigenvector of the observed adjacency matrix and a channel model for side information. For general side information, we establish, for the first time, that this spectral method achieves the information-theoretic threshold for exact recovery. Using entrywise eigenvector analysis (Abbe et al., 2020), we show the algorithm is equivalent to an oracle-assisted estimator. Our results hold across the full spectrum of graph sparsity—from sparse to dense regimes—thereby substantially extending both the theoretical limits and practical applicability of spectral methods for community detection.

This paper investigates pseudo-clique detection in random dot product graphs (RDPGs), focusing on the ability of adjacency spectral embedding (ASE) and graph encoder embedding (GEE) to identify model-injected pseudo-cliques in the absence of clean auxiliary network data. Theoretically and empirically, both ASE and GEE underperform the optimal spectral method for detecting medium-sized pseudo-cliques, revealing fundamental performance limits; yet surprisingly, they exhibit strong robustness against model contamination induced by pseudo-clique generation—challenging the conventional assumption of universal superiority for spectral methods. Moreover, when independent clean network data become available, ASE and GEE achieve asymptotically consistent pseudo-clique localization. This work provides the first systematic characterization of the coexistence of failure and robustness in embedding-based methods under structural contamination, offering new theoretical foundations and practical insights for using graph embeddings in subgraph discovery.

Multi-partite network spectral embeddings reside in high-dimensional spaces, yet their intrinsic node representations lie within group-specific low-dimensional subspaces—a geometric structure previously uncharacterized. Method: This paper introduces the first post-processing dimensionality reduction method with theoretical consistency guarantees to recover the intrinsic dimensionality of such embeddings. Grounded in a low-rank inhomogeneous random graph model, the method jointly leverages subspace estimation and matrix perturbation theory to provably achieve consistent subspace recovery; it further unifies and generalizes the bipartite spectral embedding framework. Results: Extensive experiments demonstrate that the proposed method significantly outperforms standard spectral embedding and conventional bipartite embedding approaches on clustering and visualization tasks, achieving both theoretical rigor and practical effectiveness.

This work investigates the sample complexity lower bound for estimating the spectral density of the normalized adjacency matrix of unweighted graphs via random walks. Addressing a long-standing open problem, it extends— for the first time—the exponential lower bound previously established for weighted graphs to the unweighted setting. By integrating tools from information theory, probabilistic methods, approximation theory under the Wasserstein-1 distance, and careful analysis of random walk trajectories, the study proves that any algorithm achieving an ε-approximation of the spectral density with constant success probability requires at least $2^{\Omega(1/\varepsilon^{1/6})}$ random walk trajectories of comparable length. This result reveals the inherent difficulty of high-precision spectral density estimation on unweighted graphs.

This work addresses the problem of locally approximating the leading eigenvector of a symmetric bounded matrix while querying only a small number of its entries. It proposes the first local computation algorithm for this task, operating in a preprocessing-and-query model and achieving a preprocessing complexity of Õ(1/ε⁴) and a per-coordinate query complexity of Õ(1/ε²), under the condition that |λ_min(A)| = O(λ_max(A)). The study establishes the first tight, error-dependent upper and lower bounds on query complexity for this problem. Furthermore, it demonstrates the practical impact of the proposed method by applying it to sparsest cut and max-cut problems in dense graph models, significantly enhancing the efficiency of local spectral methods.

This work addresses the challenge in graph signal processing where spectral-based filtering methods are often inapplicable due to incomplete knowledge of the full graph topology. To overcome this limitation, we propose the first data-driven algebraic framework for subgraph filtering, constructing a distance-aware Laplacian-based subgraph filtering algebra that defines a structured and controllable class of filters capable of approximating full-graph filters. Leveraging statistical learning theory, we establish risk bounds on the approximation performance under least-squares loss, providing rigorous theoretical guarantees. Empirical evaluations demonstrate that our approach significantly outperforms polynomial filters, distribution-agnostic operators, and end-to-end numerical learning baselines on real-world datasets.

This study investigates the low-degree polynomial approximation of the leading eigenpair of random symmetric matrices. Focusing on the Spiked Gaussian Orthogonal Ensemble (GOE) and standard GOE models, it integrates exact spectral methods with the low-degree algorithmic framework. By leveraging the extremal properties of Chebyshev polynomials and random matrix theory, this work rectifies prevailing misconceptions regarding the required number of iterations in classical power methods. It establishes a critical degree threshold for approximating the leading eigenpair and derives an exact expression for the asymptotic overlap. The resulting theoretical predictions significantly improve upon existing bounds, offering new insights into the fundamental limits of polynomial-based algorithms for random matrix computations.

本文解决了无权图归一化邻接矩阵的谱密度估计问题，通过局部访问模型证明了经典算法的最优性，并提出了量子算法将误差依赖从指数级改进到多项式级。