New Algorithm Learns Nonlinear Factor Models from Incomplete Noisy Data

A new paper on arXiv by Yutong Chao, Resat Gökhan, Jalal Etesami, and Ali Habibnia tackles the challenging problem of learning nonlinear factor models where observed responses are linked to low-rank latent factors through an unknown monotone function. This setting is notoriously difficult due to severe nonconvexity and identifiability issues. The authors address this by modeling the monotone link function within a reproducing kernel Hilbert space (RKHS), which allows flexible nonparametric modeling while maintaining theoretical identifiability. They formulate the joint recovery of low-rank factors, loadings, and the nonlinear link from possibly incomplete and noisy observations as an optimization problem and solve it using a projected block coordinate descent (BCD) algorithm with explicit regularization to handle scale and rotational ambiguities.

Under mild incoherence conditions on the factors and standard sampling assumptions, the team establishes convergence guarantees for both noiseless and noisy regimes, along with sublinear regret bounds for the link-function updates. This work extends classical linear factor models (e.g., PCA, factor analysis) to a broad nonlinear regime, providing a principled framework for learning nonlinear latent structures from real-world data. Controlled synthetic experiments demonstrate promising performance. The implications are significant for fields like econometrics, finance, and neuroscience where latent variables often exhibit nonlinear relationships with observed data, enabling more accurate modeling of complex systems without requiring the link function to be known a priori.