NeuroScale: Novel Topographic Feature Extraction using RBF Networks
Authors: David Lowe, Michael E. Tipping
Published in: Advances in Neural Information Processing Systems 9  (NIPS 1996) (Main Conference Track), 1996

Abstract: Dimension-reducing feature  extraction  neural  network  techniques  which also preserve neighbourhood  relationships in data have tra(cid:173) ditionally  been  the  exclusive  domain  of  Kohonen  self organising  maps.  Recently, we  introduced a  novel dimension-reducing feature  extraction process, which is  also topographic, based upon a  Radial  Basis Function  architecture.  It has been observed that the gener(cid:173) alisation performance of the system is  broadly insensitive to model  order  complexity  and  other  smoothing factors  such  as  the  kernel  widths,  contrary to  intuition  derived  from  supervised  neural  net(cid:173) work models.  In this paper we  provide an effective demonstration  of this property and give a theoretical justification for the apparent  'self-regularising' behaviour of the 'NEUROSCALE'  architecture. 
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'NeuroScale':  A  Feed-forward Neural Network  Topographic Transformation 
Recently an important class of topographic neural network based feature extraction  approaches, which  can be related to the traditional statistical methods of Sammon  Mappings  (Sammon,  1969)  and  Multidimensional  Scaling  (Kruskal,  1964),  have  been introduced (Mao and Jain, 1995; Lowe,  1993; Webb,  1995; Lowe and Tipping,  1996).  These novel alternatives to Kohonen-like approaches for topographic feature  extraction  possess  several  interesting properties.  For  instance,  the  NEuROSCALE  architecture  has  the empirically observed  property  that the generalisation perfor-
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mance does  not  seem  to  depend  critically on  model order  complexity,  contrary to  intuition based upon knowledge of its supervised counterparts.  This paper presents  evidence for their 'self-regularising' behaviour and provides an explanation in terms  of the curvature of the trained models. 
We  now  provide a  brief introduction to the  NEUROSCALE  philosophy of nonlinear  topographic feature extraction.  Further details may be found  in  (Lowe,  1993;  Lowe  and  Tipping,  1996).  We  seek  a  dimension-reducing,  topographic  transformation of  data for the purposes of visualisation and analysis.  By 'topographic', we imply that  the geometric structure of the data be optimally preserved  in  the transformation,  and the embodiment of this constraint is that the inter-point distances in the feature  space should correspond as closely as  possible to those distances in  the data space.  The implementation of this principle by  a  neural network is  very simple.  A  Radial  Basis  Function  (RBF)  neural  network  is  utilised  to predict the coordinates of the  data point in the transformed feature space.  The locations of the feature points are  indirectly determined by adjusting the weights of the network.  The transformation  is  determined by optimising the network parameters in order to minimise a suitable  error measure that embodies the topographic principle. 
The  specific  details  of  this  alternative  approach  are  as  follows.  Given  an  m(cid:173) dimensional  input  space  of N  data  points  x q ,  an  n-dimensional  feature  space  of  points  Yq  is  generated  such  that the  relative  positions of the feature  space points  minimise the error, or  'STRESS',  term: 
N 
E  = 2: 2:(d~p - dqp)2,