High-dimensional Clustering and Signal Recovery under Block Signals (2504.08332v1)

Abstract: This paper studies computationally efficient methods and their minimax optimality for high-dimensional clustering and signal recovery under block signal structures. We propose two sets of methods, cross-block feature aggregation PCA (CFA-PCA) and moving average PCA (MA-PCA), designed for sparse and dense block signals, respectively. Both methods adaptively utilize block signal structures, applicable to non-Gaussian data with heterogeneous variances and non-diagonal covariance matrices. Specifically, the CFA method utilizes a block-wise U-statistic to aggregate and select block signals non-parametrically from data with unknown cluster labels. We show that the proposed methods are consistent for both clustering and signal recovery under mild conditions and weaker signal strengths than the existing methods without considering block structures of signals. Furthermore, we derive both statistical and computational minimax lower bounds (SMLB and CMLB) for high-dimensional clustering and signal recovery under block signals, where the CMLBs are restricted to algorithms with polynomial computation complexity. The minimax boundaries partition signals into regions of impossibility and possibility. No algorithm (or no polynomial time algorithm) can achieve consistent clustering or signal recovery if the signals fall into the statistical (or computational) region of impossibility. We show that the proposed CFA-PCA and MA-PCA methods can achieve the CMLBs for the sparse and dense block signal regimes, respectively, indicating the proposed methods are computationally minimax optimal. A tuning parameter selection method is proposed based on post-clustering signal recovery results. Simulation studies are conducted to evaluate the proposed methods. A case study on global temperature change demonstrates their utility in practice.

High-dimensional Clustering and Signal Recovery under Block Signals

The paper presents a comprehensive examination of computational methods optimized for performing high-dimensional clustering and signal recovery in the context of block signal structures. Targeted especially at data manifesting non-Gaussian traits, heterogeneous variances, and complex correlation structures, two algorithms are proposed: the Cross-block Feature Aggregation PCA (CFA-PCA) and the Moving Average PCA (MA-PCA). These algorithms cater to sparse and dense block signals, respectively, leveraging adaptive gap statistics for signal identification.

Methods and Theoretical Contributions

The CFA-PCA algorithm bases its feature selection on block-wise U-statistics, aiding in the pre-clustering phase by identifying meaningful signal blocks from data with unknown cluster labels. This approach departs from traditional methods, particularly influential feature principal component analysis (IF-PCA), by avoiding assumptions of Gaussian distributions and requiring the covariance matrix to be diagonal.

Contrastingly, the MA-PCA algorithm addresses computational challenges in dense signal regimes by using moving averages to smooth data, followed by spectral clustering techniques. This algorithm integrates post-clustering signal recovery motivated by spatial scan statistics, enabling effective identification of signals amidst noise.

The paper provides proofs demonstrating that its proposed methods are consistent for clustering and signal recovery under conditions less stringent than prior methods that disregard signal block structures. Specifically, statistical and computational minimax lower bounds are derived, focusing on algorithms constrained by polynomial computation complexity. This establishes clear regions of signal strengths that are infeasible for clustering or recovery, illustrating a partition into regions of impossibility and possibility.

Numerical Results and Practical Implications

Simulation analyses show these algorithms excel beyond existing clustering methods, highlighting their adaptability to heterogenous environments and capability in detecting weaker signals within block structures. In particular, compared to conventional clustering and recovery methods which typically fail under sparsity assumptions, both CFA-PCA and MA-PCA show pronounced efficiency in their respective signal regimes.

A case paper assessing global temperature fluctuations illustrates these methods' practical competencies. The algorithms successfully identified spatially contiguous regions with significant climate patterns, elucidating their real-world applicability in environmental data contexts.

Future Directions

While the paper establishes foundational work in block signal recovery, several avenues remain open for exploration. Extensions to tensor data models have been proposed, inviting further investigation into applications spanning more complex multi-dimensional environments, such as medical imaging and genomic studies.

Moreover, additional research could explore further reducing computational overhead while preserving algorithmic fidelity, making these methods more accessible for large-scale data operations prevalent in fields like remote sensing and neuroscience.

In conclusion, this paper advances the paper of high-dimensional clustering by introducing computational methods adept at navigating unique data structures presented by block signals. By marrying theoretical innovation with empirical validation, it charts a path forward for the development of clustering algorithms sensitive to high-dimensional data complexities encountered across diverse scientific domains.