Learning Nonlinear Factor Models with Unknown Monotone Links from Incomplete and Noisy Data

2026-05-25

AI Summary

This study addresses the challenging problem of recovering a low-rank latent factor structure linked through an unknown monotonic nonlinear function from incomplete and noisy observations, which is hindered by severe non-convexity and identifiability ambiguities. The authors generalize linear factor models to a nonparametric nonlinear setting by assuming the link function resides in a reproducing kernel Hilbert space (RKHS) and introduce explicit regularization to resolve scale and rotational indeterminacies. They propose a projected block coordinate descent algorithm that jointly estimates the latent factors, loading matrix, and link function, providing theoretical convergence guarantees in both noiseless and noisy regimes. Moreover, the update of the link function enjoys a sublinear regret bound. Synthetic experiments demonstrate the method’s effectiveness and robustness.

Abstract

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.

Problem

• nonlinear factor models
• unknown monotone links
• incomplete data
• noisy data
• identifiability

Innovation

• nonlinear factor model
• unknown monotone link
• reproducing kernel Hilbert space
• projected block coordinate descent
• identifiability