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**1 - 3**of**3**### Multi-Layer Feature Reduction for Tree Structured Group Lasso via Hierarchical Projection

"... Abstract Tree structured group Lasso (TGL) is a powerful technique in uncovering the tree structured sparsity over the features, where each node encodes a group of features. It has been applied successfully in many real-world applications. However, with extremely large feature dimensions, solving T ..."

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Abstract Tree structured group Lasso (TGL) is a powerful technique in uncovering the tree structured sparsity over the features, where each node encodes a group of features. It has been applied successfully in many real-world applications. However, with extremely large feature dimensions, solving TGL remains a significant challenge due to its highly complicated regularizer. In this paper, we propose a novel MultiLayer Feature reduction method (MLFre) to quickly identify the inactive nodes (the groups of features with zero coefficients in the solution) hierarchically in a top-down fashion, which are guaranteed to be irrelevant to the response. Thus, we can remove the detected nodes from the optimization without sacrificing accuracy. The major challenge in developing such testing rules is due to the overlaps between the parents and their children nodes. By a novel hierarchical projection algorithm, MLFre is able to test the nodes independently from any of their ancestor nodes. Moreover, we can integrate MLFre-that has a low computational cost-with any existing solvers. Experiments on both synthetic and real data sets demonstrate that the speedup gained by MLFre can be orders of magnitude.

### Reduction Techniques for Graph-based Convex Clustering

"... Abstract The Graph-based Convex Clustering (GCC) method has gained increasing attention recently. The GCC method adopts a fused regularizer to learn the cluster centers and obtains a geometric clusterpath by varying the regularization parameter. One major limitation is that solving the GCC model is ..."

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Abstract The Graph-based Convex Clustering (GCC) method has gained increasing attention recently. The GCC method adopts a fused regularizer to learn the cluster centers and obtains a geometric clusterpath by varying the regularization parameter. One major limitation is that solving the GCC model is computationally expensive. In this paper, we develop efficient graph reduction techniques for the GCC model to eliminate edges, each of which corresponds to two data points from the same cluster, without solving the optimization problem in the GCC method, leading to improved computational efficiency. Specifically, two reduction techniques are proposed according to tree-based and cyclic-graph-based convex clustering methods separately. The proposed reduction techniques are appealing since they only need to scan the data once with negligibly additional cost and they are independent of solvers for the GCC method, making them capable of improving the efficiency of any existing solver. Experiments on both synthetic and real-world datasets show that our methods can largely improve the efficiency of the GCC model.

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"... Nuclear norm regularization has been shown very promising for pursing a low rank solution for matrix variable in various machine learning problems. Many efforts have been devoted to develop efficient algorithms for solving the opti-mization problem in nuclear norm regularization. Solving the problem ..."

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Nuclear norm regularization has been shown very promising for pursing a low rank solution for matrix variable in various machine learning problems. Many efforts have been devoted to develop efficient algorithms for solving the opti-mization problem in nuclear norm regularization. Solving the problem for large-scale matrix vari-ables, however, is still a challenging task since the complexity grows fast with the size of ma-trix variable. In this work, we propose a novel method called safe subspace screening (SSS), to improve the efficiency of the solver for nuclear norm regularized least squares problems. Moti-vated by the fact that the low rank solution can be represented by a few subspaces, the proposed method accurately discards a predominant per-centage of inactive subspaces prior to solving the problem to reduce problem size. Consequently, a much smaller problem is required to solve, mak-ing it more efficient than optimizing the original problem. The proposed SSS is safe, in that its so-lution is identical to the solution from the solver. In addition, the proposed SSS can be used togeth-er with any existing nuclear norm solver since it is independent of the solver. We have evaluat-ed the proposed SSS on several synthetic as well as real data sets. Extensive results show that the