 This research was partially supported by the National Science Foundation Grant No. CCR-9902961. * Corresponding author. Tel.: #1-858-534-0688; fax: #1- 858-534-1004. E-mail addresses: scotter@ece.ucsd.edu (S.F. Cotter), kreutz@ece.ucsd.edu (K. Kreutz-Delgado), brao@ece.ucsd.edu (B.D. Rao). Signal Processing 81 (2001) 1849}1864 Backward sequential elimination for sparse vector subset selection  S.F. Cotter, K. Kreutz-Delgado*, B.D. Rao Electrical and Computer Engineering Department, University of California, San Diego, La Jolla, CA 92093-0407, USA Received 27 January 2000; received in revised form 18 March 2001 Abstract Selection of a subset of vectors from a larger dictionary of vectors arises in a wide variety of application areas. This problem is known to be NP-hard and many algorithms have been proposed for the suboptimal solution of this problem. The focus of this paper is the development of a backward sequential elimination algorithm wherein, starting from the full dictionary, elements are deleted until a subset of a desired size is obtained. In contrast to previous formulations, we start with an overcomplete dictionary of vectors which is often the problem faced in a signal representation context. Once enough vectors have been deleted to give a complete system, the algorithm is modi"ed to allow further deletion of vectors. In addition, the derived algorithm gives access to the coe$cients associated with each vector in representing the signal. This allows us to experiment with di!erent criteria, including entropy-based and p-norm criteria, for selection of the vector to be deleted in each iteration. There is also the #exibility to combine criteria or to switch between criteria at a given stage of the algorithm. Following a series of simulations on a test-case system, we are able to conclude that the p-norm close to 1 performs best while the system considered is overcomplete. A minimum representation error criterion gives the best results once the system considered becomes undercomplete. The performance of the algorithm is also compared to that of forward selection algorithms on the test-case dictionary.  2001 Elsevier Science B.V. All rights reserved. Keywords: Subset selection; Sparsity; Backward elimination 1. Introduction The problem of selecting a subset of basis ele- ments from a large set of vectors has a long history and can be traced to the search for optimal regres- sions in the statistical literature [24,28,31,16]. This problem has also been the subject of much research interest in the recent signal processing literature. For example, much work has been done in con- structing signal representation dictionaries which are collections of basic signals suitable for the decomposition of signals of interest. These diction- aries can then be used for compression of audio [22,18] and video signals [3,33,36]. These dictiona- ries are overcomplete in that they form a non- minimal spanning set, and so very many di!erent representations of the same input signal are pos- sible in terms of the dictionary elements. For the 0165-1684/01/$-see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0165-1684(01)00064-0