Neural Learning of Spiral Structures

Spiral structures are one of the most difficult patterns to classify. In this paper, some important characteristics of the two-spiral problem are discussed. The paper discusses the reasons why linear and non-linear approaches have difficulties with classifying such data. The paper focusses on how structural information about spirals can be useful in providing critical information to a neural network for their recognition. Results are presented on neural network solutions to the classical two-spiral problem by extracting structural and rotational information from the spiral training data. 1. SPIRAL STRUCTURES Spiral data is found in several natural and physical domains. The classic double helix DNA, the motion of particles in cyclotrons, and spiral feed in manufacturing are some of the wellknown examples. Spirals are particularly intriguing because of their high levels of nonlinearity and resistance to shape transformation under rotation, translation or other scalar operations. Spirals structures are also attractive for their temporal properties and are found to be particularly hard to classify. For pattern recognition purposes, spiral recognition problems are specially attractive since we can manipulate their complexity with relative ease and control its size. The spiral problem is a classic example of non-linear data. It is impossible to separate two spirals coiling around each other with a linear method. The spiral data is considered here in two dimension since previous work exists in this area for comparison. However, spirals can be generated in any number of dimensions. The benchmark spiral program, available from the Carnegie Mellon AI repository, generates two sets of points, each set with 96.density + 1 data points (3 revolutions of 32 times the density plus one end point). If a total of N data points are to be generated, then the spiral shape parameters change as follows, 1≤ i ≤ N: angle = ( i.π) / (16.density ) ...(1) radius = maxRadius.((104.density) i) / (104.density)) ...(2) x = radius.cos(angle) ...(3) y = radius.sin(angle) ...(4) Here x and y are the spiral data points generated by the program, and π= 3.14. Since data points are generated in sequence, equations 1-4 are time dependent. The temporal nature of the resultant spiral is shown in Figure 1. Here the angle and radius of the spiral changes as new data is generated in sequence. The spiral is temporal because at a given point in time, the angle of the spiral that determines its position (x, y) is dependent on time (i in equation 1). The two spirals are governed by three parameters: density φ, radius σ, and offset δ. The density variable defines the total number of points generated within an envelope defined by 1 The original spiral was proposed with φ = 1, σ = 6.5 and δ = .1 (ref: Carnegie Mellon AI Repository) the radius. Data belonging to two different classes lie on these two different spirals (represented as a sequence of white and black circles in Figure 1). By manipulating spiral parameters, it is possible to generate different spirals with varying radius and length. Figure 1. 2D Spiral data scatterplot. Two spirals with a maximum radius of 6.5 coil around each other. The two different classes are highlighted in a hypersphere with their training data (white and black points) and a test pattern is illustrated with a black square.