We introduce a novel framework to reconstruct highly undersampled signals from their measurements using a correlated signal as an aid. The correlated signal, called side information, need not be close or similar to the signal to reconstruct. Thus, our framework applies to the case in which the signals are multimodal. We use twomain ingredients: the theory of ℓ1-ℓ1 minimization, which establishes precise reconstruction guarantees of sparse signals using a similar signal as an aid, and a set of training data consisting of several examples of pairs of the signal to reconstruct and the side information. We adopt a statistical framework where the training and the test data are drawn from the same joint distribution, which is assumed unknown. Our main insight is that a quantity arising in the ℓ1-ℓ1 minimization theory to measure the quality of the side information can be written as the 0-1 loss of a classification problem. Therefore, our problem can be solved with classification methods, such assupport vector machines. Furthermore, using statistical learning theory, we provide guarantees for our method. Specifically, the expected value of the side information quality decreases with O(1/√T), where T is the number of training samples. Simulations with synthetic data validate our approach.
Mota, J, Tsiligianni, E & Deligiannis, N 2017, A framework for learning affine transformations for multimodal sparse reconstruction. in YM Lu, D Van De Ville, D Van De Ville & M Papadakis (eds), SPIE Wavelets and Sparsity XVII: Optimization and Sparse Inverse Problems II. vol. 10394, 103941T, Proceedings of SPIE, SPIE, pp. 1-12, SPIE Wavelets and Sparsity XVII, 24/08/17. https://doi.org/10.1117/12.2272728
Mota, J., Tsiligianni, E., & Deligiannis, N. (2017). A framework for learning affine transformations for multimodal sparse reconstruction. In Y. M. Lu, D. Van De Ville, D. Van De Ville, & M. Papadakis (Eds.), SPIE Wavelets and Sparsity XVII: Optimization and Sparse Inverse Problems II (Vol. 10394, pp. 1-12). Article 103941T (Proceedings of SPIE). SPIE. https://doi.org/10.1117/12.2272728
@inproceedings{35bbaca542874eb791005f6a5c7f2f65,
title = "A framework for learning affine transformations for multimodal sparse reconstruction",
abstract = "We introduce a novel framework to reconstruct highly undersampled signals from their measurements using a correlated signal as an aid. The correlated signal, called side information, need not be close or similar to the signal to reconstruct. Thus, our framework applies to the case in which the signals are multimodal. We use twomain ingredients: the theory of ℓ1-ℓ1 minimization, which establishes precise reconstruction guarantees of sparse signals using a similar signal as an aid, and a set of training data consisting of several examples of pairs of the signal to reconstruct and the side information. We adopt a statistical framework where the training and the test data are drawn from the same joint distribution, which is assumed unknown. Our main insight is that a quantity arising in the ℓ1-ℓ1 minimization theory to measure the quality of the side information can be written as the 0-1 loss of a classification problem. Therefore, our problem can be solved with classification methods, such assupport vector machines. Furthermore, using statistical learning theory, we provide guarantees for our method. Specifically, the expected value of the side information quality decreases with O(1/√T), where T is the number of training samples. Simulations with synthetic data validate our approach.",
author = "Jo{\~a}o Mota and Evangelia Tsiligianni and Nikolaos Deligiannis",
year = "2017",
month = aug,
doi = "10.1117/12.2272728",
language = "English",
isbn = "978-1-5106-1245-7",
volume = "10394",
series = "Proceedings of SPIE",
publisher = "SPIE",
pages = "1--12",
editor = "Lu, {Yue M.} and {Van De Ville}, Dimitri and {Van De Ville}, Dimitri and Manos Papadakis",
booktitle = "SPIE Wavelets and Sparsity XVII",
address = "United States",
note = "SPIE Wavelets and Sparsity XVII : Optimization and Sparse Inverse Problems II ; Conference date: 24-08-2017 Through 25-08-2017",
}