2,881 research outputs found
Generalized Dantzig Selector: Application to the k-support norm
We propose a Generalized Dantzig Selector (GDS) for linear models, in which
any norm encoding the parameter structure can be leveraged for estimation. We
investigate both computational and statistical aspects of the GDS. Based on
conjugate proximal operator, a flexible inexact ADMM framework is designed for
solving GDS, and non-asymptotic high-probability bounds are established on the
estimation error, which rely on Gaussian width of unit norm ball and suitable
set encompassing estimation error. Further, we consider a non-trivial example
of the GDS using -support norm. We derive an efficient method to compute the
proximal operator for -support norm since existing methods are inapplicable
in this setting. For statistical analysis, we provide upper bounds for the
Gaussian widths needed in the GDS analysis, yielding the first statistical
recovery guarantee for estimation with the -support norm. The experimental
results confirm our theoretical analysis.Comment: Updates to boun
Positive Semidefinite Metric Learning with Boosting
The learning of appropriate distance metrics is a critical problem in image
classification and retrieval. In this work, we propose a boosting-based
technique, termed \BoostMetric, for learning a Mahalanobis distance metric. One
of the primary difficulties in learning such a metric is to ensure that the
Mahalanobis matrix remains positive semidefinite. Semidefinite programming is
sometimes used to enforce this constraint, but does not scale well.
\BoostMetric is instead based on a key observation that any positive
semidefinite matrix can be decomposed into a linear positive combination of
trace-one rank-one matrices. \BoostMetric thus uses rank-one positive
semidefinite matrices as weak learners within an efficient and scalable
boosting-based learning process. The resulting method is easy to implement,
does not require tuning, and can accommodate various types of constraints.
Experiments on various datasets show that the proposed algorithm compares
favorably to those state-of-the-art methods in terms of classification accuracy
and running time.Comment: 11 pages, Twenty-Third Annual Conference on Neural Information
Processing Systems (NIPS 2009), Vancouver, Canad
Support matrix machine: A review
Support vector machine (SVM) is one of the most studied paradigms in the
realm of machine learning for classification and regression problems. It relies
on vectorized input data. However, a significant portion of the real-world data
exists in matrix format, which is given as input to SVM by reshaping the
matrices into vectors. The process of reshaping disrupts the spatial
correlations inherent in the matrix data. Also, converting matrices into
vectors results in input data with a high dimensionality, which introduces
significant computational complexity. To overcome these issues in classifying
matrix input data, support matrix machine (SMM) is proposed. It represents one
of the emerging methodologies tailored for handling matrix input data. The SMM
method preserves the structural information of the matrix data by using the
spectral elastic net property which is a combination of the nuclear norm and
Frobenius norm. This article provides the first in-depth analysis of the
development of the SMM model, which can be used as a thorough summary by both
novices and experts. We discuss numerous SMM variants, such as robust, sparse,
class imbalance, and multi-class classification models. We also analyze the
applications of the SMM model and conclude the article by outlining potential
future research avenues and possibilities that may motivate academics to
advance the SMM algorithm
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