# Determining the Dimension and Structure of the Subspace Correlated Across Multiple Data Sets

## TL;DR

This paper presents a method to fully identify the correlation structure of components across multiple data sets by analyzing eigenvalues and eigenvectors, advancing beyond just estimating the number of correlated components.

## Contribution

It introduces a novel approach using eigenanalysis and bootstrap hypothesis testing to determine the complete correlation structure across data sets.

## Key findings

- Eigenvalues and eigenvectors characterize correlation structures
- Bootstrap hypothesis testing effectively identifies correlated components
- Method improves understanding of relationships among data set components

## Abstract

Detecting the components common or correlated across multiple data sets is challenging due to a large number of possible correlation structures among the components. Even more challenging is to determine the precise structure of these correlations. Traditional work has focused on determining only the model order, i.e., the dimension of the correlated subspace, a number that depends on how the model-order problem is defined. Moreover, identifying the model order is often not enough to understand the relationship among the components in different data sets. We aim at solving the complete modelselection problem, i.e., determining which components are correlated across which data sets. We prove that the eigenvalues and eigenvectors of the normalized covariance matrix of the composite data vector, under certain conditions, completely characterize the underlying correlation structure. We use these results to solve the model-selection problem by employing bootstrap-based hypothesis testing.