LCS: A Learnlet-Based Sparse Framework for Blind Source Separation

TL;DR

The paper addresses blind source separation in multi-frequency astrophysical data, where fixed wavelet dictionaries can fail to capture complex signal statistics. It introduces the Learnlet Component Separator (LCS), a GMCA-like iterative framework that embeds the Learnlet transform—a learned, wavelet-inspired multiscale representation—into the separation loop, with option for shared or per-component bases and a final Learnlet denoiser. Across toy models, textures, and astrophysical datasets (including CMB and SZ), LCS consistently improves source reconstruction quality and mixing-matrix accuracy compared to traditional BSS methods, showing robustness to noise and perturbations in the mixing process. The hybrid approach offers interpretability, efficiency, and scalability for next-generation cosmological surveys (e.g., SKA), and lays groundwork for extensions to nonlinear mixing, uncertainty quantification, and physics-informed priors in broader signal-processing applications.

Abstract

Blind source separation (BSS) plays a pivotal role in modern astrophysics by enabling the extraction of scientifically meaningful signals from multi-frequency observations. Traditional BSS methods, such as those relying on fixed wavelet dictionaries, enforce sparsity during component separation, but may fall short when faced with the inherent complexity of real astrophysical signals. In this work, we introduce the Learnlet Component Separator (LCS), a novel BSS framework that bridges classical sparsity-based techniques with modern deep learning. LCS utilizes the Learnlet transform: a structured convolutional neural network designed to serve as a learned, wavelet-like multiscale representation. This hybrid design preserves the interpretability and sparsity, promoting properties of wavelets while gaining the adaptability and expressiveness of learned models. The LCS algorithm integrates this learned sparse representation into an iterative source separation process, enabling effective decomposition of multi-channel observations. While conceptually inspired by sparse BSS methods, LCS introduces a learned representation layer that significantly departs from classical fixed-basis assumptions. We evaluate LCS on both synthetic and real datasets, demonstrating superior separation performance compared to state-of-the-art methods (average gain of about 5 dB on toy model examples). Our results highlight the potential of hybrid approaches that combine signal processing priors with deep learning to address the challenges of next-generation cosmological experiments.

Figures (10)

• Figure 1: Architecture of the Learnlet network: Example with a number of scales $J=5$. The $n_f$ first filters of the $J-1$ first scales of both the analysis $\mathcal{A}$ and of the synthesis $\mathcal{S}$ layers are learned, as well as the $J-1$ thresholding parameters $k_j$ of the threshold layer $\mathcal{T}$.
• Figure 2: PSNR of the denoised test images for Learnlet (green), UNet-8 (blue), and UNet-64 (orange) plotted against the input noise level $\sigma_\mathrm{N}$. The legend includes the number of trainable parameters for each network.
• Figure 3: Performance comparison of various BSS methods on standard test images: LCS (orange), GMCA (blue), GMCA-biorthogonal (green), and FastICA (red). Left panel: mean PSNR values on the source matrix $\mathbf{S}$ as a function of the input noise level $\mathrm{k}_\mathrm{noise}$. Right panel: corresponding mean Component Angle (CA) values on the mixing matrix $\mathbf{A}$. Each point represents the average and standard error over 10 independent realizations of both the mixing matrices A and the noise.
• Figure 4: Performance comparison of different BSS methods in the astrophysical case, evaluated as a function of input noise level expressed in terms of SNR: LCS is shown in orange, and LPALM in blue. Each data point represents the mean and standard error over 150 realizations from the test set of lpalm. Left panel: average PSNR for the source estimates $\mathbf{S}$. Right panel: average CA for the mixing matrix estimates $\mathbf{A}$.
• Figure 5: Comparison of BSS method performance in the astrophysical case at a fixed SNR of 30, as a function of the percentage of perturbation applied to the mixing matrix $\mathbf{A}$. LCS is shown in orange and LPALM in blue. Each point represents the mean and standard error across 150 realizations from the test set of lpalm. Left panel: average PSNR for the source estimates $\mathbf{S}$. Right panel: average CA for the mixing matrix estimates $\mathbf{A}$.