Overcompleteness
Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952. Formally, a subset of the vectors of a Banach space , sometimes called a "system", is complete if every element in can be approximated arbitrarily well in norm by finite linear combinations of elements in . A complete system is additionally overcomplete if there exists a which can be removed from the system while maintaining completeness (i.e., ). In this sense, the system contains more vectors than necessary to be complete, hence "overcomplete". In research areas such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis. Relation between overcompleteness and frames Overcompleteness is usually discussed as a property of overcomplete frames. The theory of frame originates in a paper by Duffin and Schaeffer on non-harmonic Fourier series. The frame is defined to be a set of non-zero vectors such that for an arbitrary , where denotes the inner product, and are positive constants called bounds of the frame. When and can be chosen such that , the frame is called a tight frame. It can be seen that . An example of frame can be given as follows. Let each of and be an orthonormal basis of , then is a frame of with bounds . Let be the frame operator, A frame that is not a Riesz basis, in which case it consists of a set of functions more than a basis, is said to be overcomplete. In this case, given , it can have different decompositions based on the frame. The frame given in the example above is an overcomplete frame. When frames are used for function estimation, one may want to compare the performance of different frames. The parsimony of the approximating functions by different frames may be considered as one way to compare their performances. Given a tolerance and a frame in , for any function , define the set of all approximating functions that satisfy Then let indicates the parsimony of utilizing frame to approximate . Different may have different based on the hardness to be approximated with elements in the frame. The worst case to estimate a function in is defined as For another frame , if , then frame is better than frame at level . And if there exists a that for each , we have , then is better than broadly. Overcomplete frames are usually constructed in three ways. Combine a set of bases, such as wavelet basis and Fourier basis, to obtain an overcomplete frame. Enlarge the range of parameters in some frame, such as in Gabor frame and wavelet frame, to have an overcomplete frame. Add some other functions to an existing complete basis to achieve an overcomplete frame. An example of an overcomplete frame