Learning Nonlinear Overcomplete Representations for Efficient Coding
Authors: Michael S. Lewicki, Terrence J. Sejnowski
Published in: Advances in Neural Information Processing Systems 10  (NIPS 1997) (Main Conference Track), 1997

Abstract: We  derive a  learning algorithm for  inferring an overcomplete basis  by  viewing  it  as  probabilistic  model  of the  observed data.  Over(cid:173) complete  bases  allow  for  better  approximation  of the  underlying  statistical density.  Using a Laplacian prior on the basis coefficients  removes  redundancy  and  leads  to  representations that  are  sparse  and  are  a  nonlinear  function  of the data.  This  can  be  viewed  as  a  generalization of the technique  of independent  component  anal(cid:173) ysis  and  provides  a  method  for  blind  source  separation  of  fewer  mixtures  than  sources.  We  demonstrate  the  utility  of  overcom(cid:173) plete  representations  on  natural  speech  and  show  that  compared  to the traditional Fourier basis the inferred representations poten(cid:173) tially have much greater coding efficiency. 
A  traditional  way  to represent  real-values  signals is  with  Fourier or wavelet  bases.  A  disadvantage  of these  bases,  however,  is  that  they  are  not  specialized  for  any  particular  dataset.  Principal  component  analysis  (PCA)  provides  one  means  for  finding  an  basis  that is  adapted for  a  dataset,  but the basis  vectors are  restricted  to  be  orthogonal.  An  extension  of  PCA  called  independent  component  analysis  (Jutten  and  Herault,  1991;  Comon  et  al.,  1991;  Bell  and  Sejnowski,  1995)  allows  the learning of non-orthogonal bases.  All  of these bases  are complete in  the sense  that they span the input space,  but they are limited in terms of how  well  they can  approximate the dataset's statistical density. 
Representations that are overcomplete,  i. e. more basis vectors than input variables,  can provide a better representation, because the basis vectors can be specialized for 
Learning Nonlinear Overcomplete Representations for Efficient Coding 
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a  larger variety of features  present  in  the entire  ensemble  of data.  A  criticism  of  overcomplete representations is that they are redundant, i.e.  a given data point may  have  many  possible  representations,  but  this  redundancy  is  removed  by  the  prior  probability of the basis coefficients which specifies the probability  of the alternative  representations. 
Most  of the  overcomplete  bases  used  in  the  literature are  fixed  in  the  sense  that  they  are not adapted to the structure in  the data.  Recently  Olshausen  and  Field  (1996)  presented an algorithm that allows an overcomplete basis to be learned. This  algorithm relied on an approximation to the desired probabilistic objective that had  several  drawbacks, including tendency to breakdown in the case of low  noise levels  and when learning bases with higher degrees of overcompleteness.  In this paper, we  present an improved approximation to the desired probabilistic objective and show  that this leads to a simple  and robust algorithm for  learning optimal overcomplete  bases. 
1 
Inferring the representation 
The data,  X 1 :L '  are modeled with an overcomplete linear basis plus additive noise: