Learning Nonlinear Factor Models with Unknown Monotone Links from Incomplete and Noisy Data
It extends classical linear factor models to a nonlinear regime, addressing identifiability and nonconvexity challenges for practitioners analyzing latent structures in high-dimensional data.
This paper introduces a nonlinear factor model where observed responses depend on low-rank latent factors via an unknown monotone link function, and proposes a projected block coordinate descent algorithm to jointly recover factors, loadings, and the link function from incomplete and noisy data. The method achieves convergence guarantees and sublinear regret bounds, with synthetic experiments showing promising performance.
We study a nonlinear factor model in which observed responses depend on low-rank latent factors through an unknown monotone link function. This setting is challenging and largely underexplored due to severe nonconvexity and identifiability issues. The link function is assumed to lie in a reproducing kernel Hilbert space (RKHS), enabling flexible nonparametric modeling while preserving identifiability. We formulate the problem as the joint recovery of the low-rank factors, loadings, and the nonlinear link function from possibly incomplete and noisy observations and propose a projected block coordinate descent (BCD) algorithm with explicit regularization to address scale and rotational ambiguities. Under mild incoherence of factors and standard sampling conditions, we establish convergence guarantees in both noiseless and noisy regimes, along with sublinear regret bounds for the link-function updates. Our results extend classical linear factor models to a broad nonlinear regime and provide a principled framework for learning nonlinear latent structures. We evaluate the proposed approach using controlled synthetic experiments, indicating promising performance.