# Efficient Sum of Outer Products Dictionary Learning (SOUP-DIL) and Its Application to Inverse Problems

arXiv: https://arxiv.org/abs/1511.06333 · HTML: https://mlsanity.com/paper/1511.06333/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Nov 2015.

## Built on by (most cited first, 5 of 5)

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- [Adaptive Local Neighborhood-based Neural Networks for MR Image Reconstruction from Undersampled Data](https://mlsanity.com/paper/2206.00775/) (2022)

## Abstract (arXiv)

The sparsity of signals in a transform domain or dictionary has been exploited in applications such as compression, denoising and inverse problems. More recently, data-driven adaptation of synthesis dictionaries has shown promise compared to analytical dictionary models. However, dictionary learning problems are typically non-convex and NP-hard, and the usual alternating minimization approaches for these problems are often computationally expensive, with the computations dominated by the NP-hard synthesis sparse coding step. This paper exploits the ideas that drive algorithms such as K-SVD, and investigates in detail efficient methods for aggregate sparsity penalized dictionary learning by first approximating the data with a sum of sparse rank-one matrices (outer products) and then using a block coordinate descent approach to estimate the unknowns. The resulting block coordinate descent algorithms involve efficient closed-form solutions. Furthermore, we consider the problem of dictionary-blind image reconstruction, and propose novel and efficient algorithms for adaptive image reconstruction using block coordinate descent and sum of outer products methodologies. We provide a convergence study of the algorithms for dictionary learning and dictionary-blind image reconstruction. Our numerical experiments show the promising performance and speed-ups provided by the proposed methods over previous schemes in sparse data representation and compressed sensing-based image reconstruction.