在壓縮感知這個領域中，我們嘗試一個模型架構來解決稀疏還原問題，透過這樣的概念可以減少所需的測量數量來還原稀疏訊號。我們參考 [Hegde, Indyk, and Schmidt, SODA 2014, pp. 1544--1561 and ICML 2015, pp. 928--937] 中提出關於稀疏結構的圖形模型，並使用隨機漫步的方法來尋找非零位置以改善最著名的CoSaMP演算法。再利用譜聚類的方法讓我們在某些稀疏模型下可以得到更好的還原效果。另外，我們的實驗結果也顯示我們所提出的演算法確實有得到更好的結果。我們也提供理論分析證明出我們的演算法可以達到收斂的效果。無論是從理論上或者是實務上的觀點，都可以看出我們的演算法可以在各種應用上明顯的減少儲存空間。

We consider the model-based compressive sensing framework for sparse recovery, which can reduce the number of required measurements to recover sparse signals. We refer to the graphical model proposed in [Hegde, Indyk, and Schmidt, SODA 2014, pp. 1544--1561 and ICML 2015, pp. 928--937] for sparsity structures, and improve the best known CoSaMP algorithm by using random walks to identify supports. Based on such spectral clustering technique, our approach can perform better in some sparsity models. Moreover, the experimental results demonstrate the usefulness of the proposed recovery algorithm. We also provide theoretical analysis to show its fast convergence. From both theoretical and practical perspectives, our algorithm can significantly reduce the storage space required for a variety of applications.

摘要 I
Abstract II
誌謝 III
Contents IV
List of Figures and Tables V
1 Introduction 1
1.1 Preliminary 1
1.2 Sparse recovery revisited 3
1.2.1 Model projection 3
1.2.2 Approximating model projection 4
1.3 Contribution 5
2 Proposed algorithm 7
2.1 Node-weighted model (NWM) 7
2.2 New recovery algorithm 9
2.2.1 Head approximation algorithm 11
2.2.2 Tail approximation algorithm 13
3 Analysis 15
4 Experiments 18
5 Concluding Remarks 24
References 24

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