This paper investigates channel estimation in stacked intelligent metasurface (SIM)-assisted multi-user holographic multiple-input multiple-output (HMIMO) communication systems. The SIM, with its architecture resembling an artificial neural network, enables wave-based processing and precoding/combining in the wave domain, reducing hardware costs and energy consumption. However, the challenge lies in estimating the channel state information (CSI) due to the large number of meta-atoms per layer compared to the number of antennas at the base station (BS). To address this, the authors propose a channel estimation protocol that collects multiple copies of uplink pilot signals through the SIM, leveraging array geometry to identify the subspace spanning spatial correlation matrices. Four channel estimation techniques—least squares (LS), minimum mean square error (MMSE), reduced-subspace LS (RSLS), and RSLS based on isotropic scattering statistics (RSLS-iso)—are presented. The mean square error (MSE) of these estimators is derived, and the phase shifts of the meta-atoms in the SIM are optimized to minimize the MSE. Numerical results demonstrate the effectiveness of the proposed channel estimation schemes, showing significant performance gains over the LS estimator and improved robustness with the MMSE and RSLS estimators.This paper investigates channel estimation in stacked intelligent metasurface (SIM)-assisted multi-user holographic multiple-input multiple-output (HMIMO) communication systems. The SIM, with its architecture resembling an artificial neural network, enables wave-based processing and precoding/combining in the wave domain, reducing hardware costs and energy consumption. However, the challenge lies in estimating the channel state information (CSI) due to the large number of meta-atoms per layer compared to the number of antennas at the base station (BS). To address this, the authors propose a channel estimation protocol that collects multiple copies of uplink pilot signals through the SIM, leveraging array geometry to identify the subspace spanning spatial correlation matrices. Four channel estimation techniques—least squares (LS), minimum mean square error (MMSE), reduced-subspace LS (RSLS), and RSLS based on isotropic scattering statistics (RSLS-iso)—are presented. The mean square error (MSE) of these estimators is derived, and the phase shifts of the meta-atoms in the SIM are optimized to minimize the MSE. Numerical results demonstrate the effectiveness of the proposed channel estimation schemes, showing significant performance gains over the LS estimator and improved robustness with the MMSE and RSLS estimators.