<p>We study a nonlinear structured low-rank approximation problem arising in the data-driven identification and denoising of nonlinear dynamical trajectories. Given a time series and a collection of nonlinear feature maps, we construct a lifted block-Hankel matrix whose blocks collect Hankel embeddings of the signal and of its nonlinear transformations. In the noise-free setting, such lifted Hankel matrices are rank-deficient and encode nonlinear dynamical relations in a behavioral form, while measurement noise typically destroys this rank structure. We propose a projection-based algorithm that operates directly in signal space and enforces low-rank structure in lifted coordinates through a linearized consensus update. Each iteration consists of: (i) projection of the lifted block-Hankel matrix onto a fixed-rank set, (ii) blockwise projection onto the Hankel structure, and (iii) a single coupled least-squares solve that enforces first-order consistency across all lifted features. The resulting correction is applied through a relaxed Krasnosel’skiĭ–Mann fixed-point step. Unlike classical Cadzow-type methods, signal reconstruction is not obtained from any individual lifted block, but from a joint consensus update that couples all nonlinear features. We interpret the method as a fixed-point iteration induced by a composite nonlinear operator and establish local linear convergence under smoothness and spectral-gap assumptions on the lifted low-rank projector. Numerical experiments illustrate how the reconstruction accuracy and the convergence behavior depend on the alignment between the lifted representation and the underlying nonlinear dynamics.</p>