Note on the Rademacher-Walsh Polynomial Basis Functions


  •  Hamse Mussa    
  •  Jonathan Tyzack    
  •  Robert Glen    

Abstract

Over the years, one of the methods of choice to estimate probability density functions for a given random variable (defined on binary input space) has been the expansion of the estimation function in Rademacher-Walsh Polynomial basis functions. For a set of $L$ features (often considered as an ``$L$-dimensional binary vector''), the Rademacher-Walsh Polynomial approach requires $2^{L}$ basis functions. This can quickly become computationally complicated and notationally clumsy to handle whenever the value of $L$ is large. In current pattern recognition applications it is often the case that the value of $L$ can be 100 or more.

In this paper we show that the expansion of the probability density function estimation in Rademacher-Walsh Polynomial basis functions is equivalent to the expansion of the estimation function in a set of Dirac kernel functions. The latter approach is not only able to eloquently allay the computational bottle--neck and notational awkwardness mentioned above, but may also be naturally neater and more ``elegant'' than the Rademacher-Walsh Polynomial basis function approach even when this latter approach is computationally feasible.


This work is licensed under a Creative Commons Attribution 4.0 License.
  • Issn(Print): 1916-9795
  • Issn(Onlne): 1916-9809
  • Started: 2009
  • Frequency: bimonthly

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