Compressed BC-LISTA via Low-Rank Convolutional Decomposition

We study Sparse Signal Recovery (SSR) methods for multichannel imaging with compressed {forward and backward} operators that preserve reconstruction accuracy. We propose a Compressed Block-Convolutional (C-BC) measurement model based on a low-rank Convolutional Neural Network (CNN) decomposition that is analytically initialized from a low-rank factorization of physics-derived forward/backward operators in time delay-based measurements. We use Orthogonal Matching Pursuit (OMP) to select a compact set of basis filters from the analytic model and compute linear mixing coefficients to approximate the full model. We consider the Learned Iterative Shrinkage-Thresholding Algorithm (LISTA) network as a representative example for which the C-BC-LISTA extension is presented. In simulated multichannel ultrasound imaging across multiple Signal-to-Noise Ratios (SNRs), C-BC-LISTA requires substantially fewer parameters and smaller model size than other state-of-the-art (SOTA) methods while improving reconstruction accuracy. In ablations over OMP, Singular Value Decomposition (SVD)-based, and random initializations, OMP-initialized structured compression performs best, yielding the most efficient training and the best performance.

💡 Research Summary

The paper addresses the challenge of sparse signal recovery (SSR) in multichannel imaging systems where the forward and backward operators are large, dense matrices that dominate computational and memory costs. Focusing on time‑delay based measurements such as full‑matrix capture (FMC) in multichannel ultrasound, the authors observe that the physics‑derived forward matrix A possesses a block‑Toeplitz structure that can be exactly expressed as a set of one‑dimensional convolutional layers. Moreover, empirical analysis shows that the collection of convolution kernels (the weight matrix W) is highly redundant: many filters are scaled or shifted versions of each other, and the overall rank is bounded by the number of transceiver elements or scatterers.

To exploit this redundancy, the authors propose a low‑rank factorization W ≈ C B, where B contains a small set of basis filters (BFs) selected directly from the rows of W, and C linearly mixes these BFs to reconstruct the original filters. Two factorization strategies are compared: singular value decomposition (SVD) and orthogonal matching pursuit (OMP). While SVD yields the smallest Frobenius error, it destroys the sparsity and physical interpretability of the filters. OMP, by contrast, iteratively selects rows of W that have the highest correlation with the current residual, guaranteeing that the chosen BFs are genuine physical kernels and preserving sparsity. The mixing matrix C is obtained in closed form via least‑squares, and the process stops when a predefined number M of BFs is reached or the residual falls below a threshold. This reduces the parameter count from C_out × K to M × (K + C_out), achieving substantial compression when M ≪ C_out.

The low‑rank factorization is then mapped onto a trainable neural module using the well‑known low‑rank CNN decomposition technique. The first convolutional layer implements the 1‑D convolution with the basis filters B, producing M intermediate feature maps z_j. The second layer is a 1 × 1 convolution (i.e., a matrix multiplication) with weights C, which linearly combines the intermediate maps to generate the original C_out output channels. This two‑layer construction is mathematically equivalent to the original single convolutional layer but requires far fewer parameters and memory. Importantly, the adjoint (transpose) operation needed for back‑propagation is obtained by swapping the order of the two layers and using transposed convolutions with the same weights, ensuring that gradients are exact up to numerical precision.

Building on this compressed forward model, the authors develop Compressed BC‑LISTA (C‑BC‑LISTA), an unrolled LISTA architecture where each iteration block consists of (1) the compressed forward convolution, (2) its transpose, and (3) a learnable shrinkage/threshold parameter λ/L. Parameter sharing is employed: the basis filters B and mixing matrix C are fixed across all blocks (or optionally fine‑tuned), while each block learns its own threshold. This design dramatically reduces the number of trainable parameters compared with standard LISTA variants (MLP‑LISTA, ALIST‑A, BC‑LISTA).

Experimental evaluation uses synthetic FMC data with 32 transceiver elements, varying the number of scatterers (5–10) and SNR levels (5, 10, 15, 20 dB). C‑BC‑LISTA is trained with 10 unrolled blocks under both noiseless and noisy conditions, testing three values of M (16, 32, 64). Results show that increasing M accelerates early loss reduction, but all three settings converge to comparable final validation loss. Quantitatively, C‑BC‑LISTA achieves a parameter reduction of up to two orders of magnitude (e.g., from 1.7 billion parameters in MLP‑LISTA to ~0.1 million in C‑BC‑LISTA) and a storage saving from several gigabytes to a few megabytes. In terms of reconstruction quality, measured by Percentual Amplitude Error (PAE) and Support Error (SE), C‑BC‑LISTA consistently outperforms the baselines across all SNRs, delivering lower amplitude errors and more accurate scatterer localization.

Ablation studies compare three initialization schemes for the compressed model: OMP‑derived matrices, SVD‑derived matrices, and random initialization. OMP initialization yields the fastest convergence and the best final performance, confirming that preserving the physical structure of the filters is beneficial for training efficiency. Additional experiments demonstrate that sharing the convolutional weights across blocks while learning individual thresholds provides the best trade‑off between model size and flexibility.

In summary, the paper makes four key contributions: (1) a principled, physics‑aware compression of large forward operators using low‑rank, block‑Toeplitz convolutional structure; (2) an OMP‑based selection of basis filters that retains sparsity and interpretability; (3) a two‑layer low‑rank CNN implementation that serves as a drop‑in replacement for the original convolutional layer, enabling exact forward and adjoint operations; and (4) the C‑BC‑LISTA architecture, which demonstrates superior reconstruction accuracy, faster convergence, and dramatically reduced memory footprint compared with state‑of‑the‑art LISTA variants. The methodology is readily extensible to other time‑delay measurement modalities such as radar, sonar, and non‑destructive testing, wherever the forward model exhibits low‑rank convolutional structure.