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

Uncovering latent variables and their causal relations from observed data is a fundamental yet challenging problem. Existing methods often rely on restrictive assumptions, such as linear relations or invertible mixing functions. To better address this problem under general nonlinear mixing procedures, we propose a condition called the cross-Hessian Rank Constraint (HRC), which serves as a primitive rank-based tool for nonlinear latent causal discovery. In particular, we show that a rank-based property arises from the cross-Hessian of the observed-data log-density in the nonlinear case, revealing information about the latent variables, and reduces to the Tetrad constraints in the linear Gaussian case. More specifically, when two groups of observed variables are d-separated by a set of lower-dimensional latent variables, the rank of this cross-Hessian is equal to the dimension of the latent variables, under a mild affine derivative assumption on the conditional log-density derivatives. This assumption can be naturally satisfied when the noise level is low or the relevant nonlinearity is moderate. As a downstream application, we instantiate HRC in the pure one-factor measurement setting for locating latent variables and recovering their causal structure up to Markov equivalence. Experimental results on synthetic and real-world datasets support the theoretical claims.

We consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition: $y_i=g(\inner{\ba_i}{\bPhi\bw^\ast+\bPsi\bz^\ast})+e_i$, $i=1,\dots,m$, with $m\ll n$, incoherent orthonormal bases $\bPhi,\bPsi$, a scalar link $g$, and noise $e_i$ that may be heavy-tailed or contaminated. We propose a regularization-based framework combining a Huberized data fidelity with generalized folded-concave penalties (SCAD, MCP), and a two-block proximal alternating algorithm with backtracking (NLD-PALM) whose whole iterate sequence provably converges to critical points under the Kurdyka--Łojasiewicz property, with local linear rates. On the statistical side we establish restricted strong convexity of the Huberized nonlinear loss through an exact sign-definite decomposition, and derive estimation error bounds of order $σ\sqrt{s\log(n)/m}$ that hold at \emph{every} localized stationary point, an oracle rate $σ\sqrt{s/m}$ free of $\log n$ and shrinkage bias under a beta-min condition, and a co-equal recovery theorem for \emph{unknown} monotone links via a linear surrogate and a clipped Plan--Vershynin decoupling. The estimator requires no knowledge of the sparsity levels, and its guarantees hold under symmetric noise with only finite variance. Experiments at $n=512$ under a frozen data-driven regularization rule show an earlier phase transition than convex $\ell_1$ demixing and greedy hard-thresholding baselines, a $35\times$ accuracy advantage over squared-loss estimation under $5\%$ gross outliers, and successful demixing of spike-plus-background signals observed through a saturating amplifier.

In this paper, we propose a robust subspace-constrained quadratic model (SCQM) for learning low-dimensional structure from high-dimensional data. Building upon the subspace-constrained quadratic matrix factorization (SQMF) framework, the proposed model accommodates a broad class of noise distributions, including generalized Gaussian and radial Laplace models. This generalization enables reliable performance under both heavy-tailed and light-tailed noise, thereby substantially enhancing robustness across diverse data regimes. To efficiently address the resulting nonconvex optimization problem, we develop a gradient-based algorithm equipped with a backtracking line-search strategy that ensures stable and efficient convergence. In addition, we present a sensitivity analysis of the $\ell_p^p$ and $\ell_2$ loss functions, elucidating their distinct behaviors under varying noise characteristics. Extensive numerical experiments corroborate the theoretical analysis and demonstrate that the proposed approach consistently outperforms existing methods in terms of robustness and reconstruction accuracy.