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  1. State-of-the-art subspace clustering methods are based on the self-expressive model, which represents each data point as a linear combination of other data points. However, such methods are designed for a finite sample dataset and lack the ability to generalize to out-of-sample data. Moreover, since the number of self-expressive coefficients grows quadratically with the number of data points, their ability to handle large-scale datasets is often limited. In this paper, we propose a novel framework for subspace clustering, termed Self-Expressive Network (SENet), which employs a properly designed neural network to learn a self-expressive representation of the data. We show that our SENet can not only learn the self-expressive coefficients with desired properties on the training data, but also handle out-of-sample data. Besides, we show that SENet can also be leveraged to perform subspace clustering on large-scale datasets. Extensive experiments conducted on synthetic data and real world benchmark data validate the effectiveness of the proposed method. In particular, SENet yields highly competitive performance on MNIST, Fashion MNIST and Extended MNIST and state-of-the-art performance on CIFAR-10. 
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  2. Classical results in sparse recovery guarantee the exact reconstruction of s-sparse signals under assumptions on the dictionary that are either too strong or NP-hard to check. Moreover, such results may be pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We derive necessary and sufficient conditions for sparse recovery, which depend on properties of the cycles of the graph that can be checked in polynomial time. We also derive support-dependent conditions for sparse recovery that depend only on the intersection of the cycles of the graph with the support of the signal. Finally, we exploit sparsity properties on the measurements and the structure of incidence matrices to propose a specialized sub-graph-based recovery algorithm that outperforms the standard l1 -minimization approach. 
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