source: arxiv statistics ml: learnability and competition in high-dimensional multi-component ica
level: research
independent component analysis is a key method for unsupervised learning, but its theory in high dimensions mostly covers recovering just one component. this work presents a mean-field theory for online ica that handles multiple components at once. it accounts for the coupling that comes from learning and orthogonalizing several directions simultaneously. in the limit of large dimensions, the joint distribution of learned estimates and true components follows a deterministic process. this leads to a closed system of ordinary differential equations for the overlap matrix between learned and true directions.
the analysis uncovers two distinct regimes depending on initialization. in the decoupled regime, estimates align with separate true components and evolve almost independently. in the competition regime, overlapping initializations cause conflicts due to orthogonality constraints. this slows down reorientation and delays convergence. the phase structure is genuinely multi-component, meaning the behavior cannot be reduced to single-component dynamics. the findings show how initial conditions critically shape the learning trajectory.
the results provide a rigorous understanding of how online ica behaves when many components are learned together. the mean-field equations offer a tool to predict convergence rates and final alignment. this can guide practical choices like initialization strategies and step sizes. the theory also highlights potential pitfalls when components are not well separated at the start. it extends the theoretical foundation of ica beyond the single-component case.
why it matters: understanding the phase structure helps practitioners avoid slow convergence in high-dimensional ica by choosing better initializations.
source: arxiv statistics ml: learnability and competition in high-dimensional multi-component ica