A class of error tolerant pattern discrimination functions

-A general pattern recognition problem is posed in Hilbert space. Two new solutions are then given and it is shown that the sensitivity of the pattern recognition functions to pattern perturbation can be a priori controlled. A series of examples demonstrate the principal results in a variety of settings. Pattern recognition Multilinearity Hilbert space Functional sensitivity I N T R O D U C T I O N The term pattern recognition is used in a diversity of settings ~'~) and hence it is only proper that we delineate the problem to be considered here. For this we choose a hypothetical automated manufacturing application as a vehicle for the discussion. In short, consider a machining unit which is presented with a parts flow drawn from a large but finite number of distinct parts. Each part must be recognized, oriented, secured and machined in an appropriate way. For simplicity, suppose that each part can be physically oriented and secured in the field of view of a set of optical scanners. The scanner outputs constitute a time-varying set of patterns from which the part must be identified, and oriented for machining. Our interest here is in the part identification problem which has an obvious pattern recognition flavor. To model our problem in mathematical form consider first a discrete set of optical sensors with n outputs {xi}. At the instant, t, we may view {xi(t)} as an n-tuple or as a geometric array. For convenience we adopt the n-tuple viewpoint and let x(t) = (x l ( t ) , X z ( t ) . . . . . xn( t ) ) be the instantaneous pattern. The case where the optical scan is continuous in a spatial variable is also of interest and we shall use the notation x(c~, t) to denote the instantaneous scan function as it depends on the spatial variable e. Even more generally ~ could be interpreted as a triplet of 3 spatial variables in which case the scanner would have three dimensional perception. In any of the above cases a static pattern recognition problem can be easily formulated. For each of the possible parts we assume an a priori pattern which may be viewed as a signature of the part. We have then a set of signatures, F = {7~ . . . . ,?,.}, where each yie F is the same type of entity as the scanner * Sponsored in part by the Air Force Office of Scientific Research under Grant AFOSR 73-2427. output. That is if x(t) is an n-tuple then each 7j is also an n-tuple. More precisely, let X be the universe of admissible patterns. Let Y be a set containing at least m distinct points {Yl . . . . . Y~, }. Problem I. Find a function (or funct ions)f : X ~-, Y such that f ( T i ) = Y i i = 1,2 . . . . . m. Problem l, which has been studied in a variety of settings, provides a function with a distinct output for each of the distinct 7ie F. It does not, however, take into account the effects of noise or variations in the patterns due, for example, to manufacturing tolerance. One way of modeling this would be to equip X and Y with a topology and then try to solve Problem 2, namely: Problem 2. Find a function f : X--~ Y such that f(Yl) = Yi, i = 1,2 . . . . . m and if w is close to 7i, then f ( w ) is close to tO'i). It is noted that existing minimum distance, maximum probability, maximum dot product and committee machine techniques 14~ generate solutions to Problem 2. In the present study we develop a linear and a polynomic solution neither of which appears to have been considered by other authors. Both solutions present a function which has the f (~,~)= Yi, i = 1 , . . . ,m property plus its derivative ( F r e c h e t ) f ' satisfies f'(Yi) = 0. The net effect is that f has zero first order sensitivity to additive disturbances. Of course this results in very effective suppression of small errors. In some applications it might actually be desirable to have a large derivative at ?i. Indeed, in the manufacturing setting, the errors may be due to inaccurate part alignment and a high sensitivity to this could be useful in the alignment procedure. The techniques described in the following can set the derivative value arbitrarily. We consider only f '(7i) = 0 for simplicity of exposition. 59