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{{Main|Sparse dictionary learning}}
Sparse dictionary learning is a feature learning method where a training example is represented as a linear combination of [[basis function]]s and assumed to be a [[sparse matrix]]. The method is [[strongly NP-hard]] and difficult to solve approximately.<ref>{{cite journal |first=A. M. |last=Tillmann |title=On the Computational Intractability of Exact and Approximate Dictionary Learning |journal=IEEE Signal Processing Letters |volume=22 |issue=1 |year=2015 |pages=45–49 |doi=10.1109/LSP.2014.2345761|bibcode=2015ISPL...22...45T |arxiv=1405.6664 |s2cid=13342762 }}</ref> A popular [[heuristic]] method for sparse dictionary learning is the [[k-SVD|''k''-SVD]] algorithm. Sparse dictionary learning has been applied in several contexts. In classification, the problem is to determine the class to which a previously unseen training example belongs. For a dictionary where each class has already been built, a new training example is associated with the class that is best sparsely represented by the corresponding dictionary. Sparse dictionary learning has also been applied in [[image de-noising]]. The key idea is that a clean image patch can be sparsely represented by an image dictionary, but the noise cannot.<ref>[[Michal Aharon|Aharon, M]], M Elad, and A Bruckstein. 2006. "[http://sites.fas.harvard.edu/~cs278/papers/ksvd.pdf K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation] {{Webarchive|url=https://web.archive.org/web/20181123142158/http://sites.fas.harvard.edu/~cs278/papers/ksvd.pdf |date=2018-11-23 }}." Signal Processing, IEEE Transactions on 54 (11): 4311–4322</ref>
Dictionary learning has been applied to detect weak signals in sparse time-spreading distortion (TSD) channels (multipath channels) by learning a dictionary from received signal samples <ref>Rami Rashid, Ali Abdi, and Zoi-Heleni Michalopoulou. 2025. "[https://pubs.aip.org/asa/jel/article/5/6/064803/3350769/Blind-weak-signal-detection-via-dictionary Blind weak signal detection via dictionary learning in time-spreading distortion channels using vector sensors]." *JASA Express Letters*, 5, 064803.</ref> <ref>Rami Rashid, Ali Abdi, and Zoi-Heleni Michalopoulou. 2024. "[https://pubs.aip.org/asa/jasa/article/155/3_Supplement/A84/3301959/Blind-passive-signal-detection-via-dictionary Blind passive signal detection via dictionary learning in unknown multipath time-spreading distortion underwater channels]." *J. Acoust. Soc. Am.*, 155, A84.</ref>. This method enables blind detection and separation of the unknown signal from the unknown channel impulse response, using the estimated signal as a matched filter for passive detection. Due to the sparse nature of TSD channels, where many paths carry minimal energy, dictionary learning effectively estimates the transmitted signal, enabling detection without prior knowledge. High detection probabilities of the blind method, comparable to the optimal detector, demonstrate the approach's effectiveness for passive detection of unknown signals in unknown channels.
==== Anomaly detection ====
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