Provable Subspace Identification of Nonlinear Multi-view CCA
Authors: Zhiwei Han, Stefan Matthes, Hao Shen
Organizations: Chair of Data Processing, Technical University of Munich, Germany · fortiss GmbH, Munich, Germany
Abstract
We investigate the identifiability of nonlinear canonical correlation analysis (CCA) in a multi-view setup, in which each view is generated by applying an unknown nonlinear map to a linear mixture of shared latent variables plus view-private noise. Rather than pursuing exact unmixing, which is known to be ill-posed under general nonlinear mixing, we instead reframe multi-view CCA as a basis-invariant subspace identification problem. Under suitable latent priors and spectral separation conditions, we prove that the pairwise population CCA objective recovers correlated signal subspaces up to view-wise orthogonal ambiguity. For N≥3 views, their multi-view aggregation provably isolates the jointly correlated subspaces shared across all views while eliminating view-private variation. We further establish finite-sample statistical consistency guarantees by translating the concentration of empirical cross-covariances into explicit subspace error bounds via spectral perturbation theory. Experiments on synthetic and rendered image datasets support our theoretical findings and illustrate the necessity of the assumed conditions.
In this work, we establish the sufficient conditions under which nonlinear Canonical Correlation Analysis (CCA) recovers ground-truth latent factors up to an affine transformation. By transporting the analysis from the observation space to the source space, we extend classical statistical results on orthogonal polynomial expansions of bivariate distributions to representation learning, proving affine identifiability under specific distributional priors. We formally demonstrate that whitening is strictly necessary to ensure the boundedness and well-conditioning of the learned mappings. Furthermore, we bridge the gap between theory and practice by proving that ridge-regularized empirical CCA converges to its population counterpart in the finite-sample regime. Finally, our findings provide a rigorous theoretical foundation explaining the empirical success of recent correlation-based non-contrastive learning methods. Experiments on synthetic and rendered image datasets, alongside systematic ablations, validate the predicted recovery behavior and illustrate the failure modes that arise when the assumptions are violated.
We derive the linear union-of-subspaces (UoS) model for subspace clustering (SC) from the nonlinear mixture model (NMM) used in blind source separation (BSS) to represent a D-dimensional observation vector as an unknown multivariate nonlinear mapping of C latent variables. Assuming the mapping is differentiable up to an unknown order K, we approximate NMM by a K-th order Taylor expansion, yielding a model equivalent to the linear UoS framework underlying SC. This establishes that: (i) the smoothness order K corresponds to the unknown subspace dimension d; (ii) KC equals the number of anchors; and (iii) the sparsity of the representation vector equals K (i.e., d). These relationships enable estimation of bounds on subspace dimension, and that is validated on six benchmark datasets using five established SC algorithms. Established theoretical results are important for post-processing of self-representation matrices estimated by SC algorithms.
Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.