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Fast Optimization Methods for L1 Regularization: A Comparative Study and Two New Approaches
"... Abstract. L1 regularization is effective for feature selection, but the resulting optimization is challenging due to the non-differentiability of the 1-norm. In this paper we compare state-of-the-art optimization techniques to solve this problem across several loss functions. Furthermore, we propose ..."
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Cited by 84 (2 self)
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Abstract. L1 regularization is effective for feature selection, but the resulting optimization is challenging due to the non-differentiability of the 1-norm. In this paper we compare state-of-the-art optimization techniques to solve this problem across several loss functions. Furthermore, we propose two new techniques. The first is based on a smooth (differentiable) convex approximation for the L1 regularizer that does not depend on any assumptions about the loss function used. The other technique is a new strategy that addresses the non-differentiability of the L1-regularizer by casting the problem as a constrained optimization problem that is then solved using a specialized gradient projection method. Extensive comparisons show that our newly proposed approaches consistently rank among the best in terms of convergence speed and efficiency by measuring the number of function evaluations required. 1
RELATIVE-ERROR CUR MATRIX DECOMPOSITIONS
- SIAM J. MATRIX ANAL. APPL
, 2008
"... Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the ..."
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Cited by 83 (18 self)
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Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and/or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an m × n matrix A and a rank parameter k. In our first algorithm, C is chosen, and we let A ′ = CC + A, where C + is the Moore–Penrose generalized inverse of C. In our second algorithm C, U, R are chosen, and we let A ′ = CUR. (C and R are matrices that consist of actual columns and rows, respectively, of A, and U is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least 1 − δ, ‖A − A ′ ‖F ≤ (1 + ɛ) ‖A − Ak‖F, where Ak is the “best ” rank-k approximation provided by truncating the SVD of A, and where ‖X‖F is the Frobenius norm of the matrix X. The number of columns of C and rows of R is a low-degree polynomial in k, 1/ɛ, and log(1/δ). Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants
A statistical approach to material classification using image patch exemplars
- IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2009
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Collaborative Filtering: A Machine Learning Perspective
, 2004
"... Collaborative filtering was initially proposed as a framework for filtering information based on the preferences of users, and has since been refined in many different ways. This thesis is a comprehensive study of rating-based, pure, non-sequential collaborative filtering. We analyze existing method ..."
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Cited by 78 (3 self)
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Collaborative filtering was initially proposed as a framework for filtering information based on the preferences of users, and has since been refined in many different ways. This thesis is a comprehensive study of rating-based, pure, non-sequential collaborative filtering. We analyze existing methods for the task of rating prediction from a machine learning perspective. We show that many existing methods proposed for this task are simple applications or modi cations of one or more standard machine learning methods for classifi cation, regression, clustering, dimensionality reduction, and density estimation. We introduce new prediction methods in all of these classes. We introduce a new experimental procedure for testing stronger forms of generalization than has been used previously. We implement a total of nine prediction methods, and conduct large scale prediction accuracy experiments. We show interesting new results on the relative performance of these methods.
Spectral feature selection for supervised and unsupervised learning
- In ICML
, 2007
"... Feature selection aims to reduce dimensionality for building comprehensible learning models with good generalization performance. Feature selection algorithms are largely studied separately according to the type of learning: supervised or unsupervised. This work exploits intrinsic properties underly ..."
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Cited by 76 (17 self)
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Feature selection aims to reduce dimensionality for building comprehensible learning models with good generalization performance. Feature selection algorithms are largely studied separately according to the type of learning: supervised or unsupervised. This work exploits intrinsic properties underlying supervised and unsupervised feature selection algorithms, and proposes a unified framework for feature selection based on spectral graph theory. The proposed framework is able to generate families of algorithms for both supervised and unsupervised feature selection. And we show that existing powerful algorithms such as ReliefF (supervised) and Laplacian Score (unsupervised) are special cases of the proposed framework. To the best of our knowledge, this work is the first attempt to unify supervised and unsupervised feature selection, and enable their joint study under a general framework. Experiments demonstrated the efficacy of the novel algorithms derived from the framework. 1.
Result analysis of the NIPS 2003 feature selection challenge
- Advances in Neural Information Processing Systems 17
, 2004
"... The NIPS 2003 workshops included a feature selection competition organized by the authors. We provided participants with five datasets from different application domains and called for classification results using a minimal number of features. The competition took place over a period of 13 weeks and ..."
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Cited by 73 (11 self)
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The NIPS 2003 workshops included a feature selection competition organized by the authors. We provided participants with five datasets from different application domains and called for classification results using a minimal number of features. The competition took place over a period of 13 weeks and attracted 78 research groups. Participants were asked to make on-line submissions on the validation and test sets, with performance on the validation set being presented immediately to the participant and performance on the test set presented to the participants at the workshop. In total 1863 entries were made on the validation sets during the development period and 135 entries on all test sets for the final competition. The winners used a combination of Bayesian neural networks with ARD priors and Dirichlet diffusion trees. Other top entries used a variety of methods for feature selection, which combined filters and/or wrapper or embedded methods using Random Forests, kernel methods, or neural networks as a classification engine. The results of the benchmark (including the predictions made by the participants and the features they selected) and the scoring software are publicly available. The benchmark is available at www.nipsfsc.ecs.soton.ac.uk for post-challenge submissions to stimulate further research. 1
Unsupervised activity recognition using automatically mined common sense
- In AAAI
, 2005
"... A fundamental difficulty in recognizing human activities is obtaining the labeled data needed to learn models of those activities. Given emerging sensor technology, however, it is possible to view activity data as a stream of natural language terms. Activity models are then mappings from such terms ..."
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Cited by 70 (5 self)
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A fundamental difficulty in recognizing human activities is obtaining the labeled data needed to learn models of those activities. Given emerging sensor technology, however, it is possible to view activity data as a stream of natural language terms. Activity models are then mappings from such terms to activity names, and may be extracted from text corpora such as the web. We show that models so extracted are sufficient to automatically produce labeled segmentations of activity data with an accuracy of 42 % over 26 activities, well above the 3.8 % baseline. The segmentation so obtained is sufficient to bootstrap learning, with accuracy of learned models increasing to 52%. To our knowledge, this is the first human activity inferencing system shown to learn from sensed activity data with no human intervention per activity learned, even for labeling.
Unsupervised Learning Techniques for an Intrusion Detection System
, 2004
"... With the continuous evolution of the types of attacks against computer networks, traditional intrusion detection systems, based on pattern matching and static signatures, are increasingly limited by their need of an up-to-date and comprehensive knowledge base. Data mining techniques have been succes ..."
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Cited by 69 (6 self)
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With the continuous evolution of the types of attacks against computer networks, traditional intrusion detection systems, based on pattern matching and static signatures, are increasingly limited by their need of an up-to-date and comprehensive knowledge base. Data mining techniques have been successfully applied in host-based intrusion detection. Applying data mining techniques on raw network data, however, is made di#cult by the sheer size of the input; this is usually avoided by discarding the network packet contents. In this paper, we introduce a two-tier architecture to overcome this problem: the first tier is an unsupervised clustering algorithm which reduces the network packets payload to a tractable size. The second tier is a traditional anomaly detection algorithm, whose e#ciency is improved by the availability of data on the packet payload content.
Efficient and robust feature selection via joint l21-norms minimization. NIPS
, 2010
"... Feature selection is an important component of many machine learning applications. Especially in many bioinformatics tasks, efficient and robust feature selection methods are desired to extract meaningful features and eliminate noisy ones. In this paper, we propose a new robust feature selection met ..."
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Cited by 68 (24 self)
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Feature selection is an important component of many machine learning applications. Especially in many bioinformatics tasks, efficient and robust feature selection methods are desired to extract meaningful features and eliminate noisy ones. In this paper, we propose a new robust feature selection method with emphasizing joint ℓ2,1-norm minimization on both loss function and regularization. The ℓ2,1-norm based loss function is robust to outliers in data points and the ℓ2,1norm regularization selects features across all data points with joint sparsity. An efficient algorithm is introduced with proved convergence. Our regression based objective makes the feature selection process more efficient. Our method has been applied into both genomic and proteomic biomarkers discovery. Extensive empirical studies are performed on six data sets to demonstrate the performance of our feature selection method. 1
Support Vector Channel Selection in BCI
, 2003
"... Designing a Brain Computer Interface (BCI) system one can choose from a variety of features that may be useful for classifying brain activity during a mental task. For the special case of classifying EEG signals we propose the usage of the state of the art feature selection algorithms Recursive Fe ..."
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Cited by 60 (6 self)
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Designing a Brain Computer Interface (BCI) system one can choose from a variety of features that may be useful for classifying brain activity during a mental task. For the special case of classifying EEG signals we propose the usage of the state of the art feature selection algorithms Recursive Feature Elimination [1] and Zero-Norm Optimization [2] which are based on the training of Support Vector Machines (SVM) [3]. These algorithms can provide more accurate solutions than standard filter methods for feature selection [4]. We adapt the methods for the purpose of selecting EEG channels. For a motor imagery paradigm we show that the number of used channels can be reduced significantly without increasing the classification error. The resulting best channels agree well with the expected underlying cortical activity patterns during the mental tasks. Furthermore we show how time dependent task specific information can be visualized.
