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

arXiv: https://arxiv.org/abs/2605.26271 · HTML: https://mlsanity.com/paper/2605.26271/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 53 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 3 of 3)

- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent](https://mlsanity.com/paper/1605.07051/) (2016)
- [Nonlinear Factor Models for Network and Panel Data](https://mlsanity.com/paper/1412.5647/) (2014)

## Abstract (arXiv)

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.