The perceptron convergence theorem played an important role in the early development of machine learning. Mathematically, the perceptron learning algorithm is an iterative procedure for finding a separating hyperplane for a finite set of linearly separable vectors, or equivalently, for finding a separating hyperplane for a finite set of linearly contained vectors. In this paper, the author shows that the perceptron algorithm can be extended to a more general algorithm, called the cone algorithm, for finding a covering cone for a finite set of linearly contained vectors. A proof of the convergence of the cone algorithm is given. The relationship between the cone algorithm and other related algorithms is discussed. The equivalence of the problem of finding a covering cone for a set of linearly contained vectors and the problem of finding a solution cone for a system of homogeneous linear inequalities is established.>

An important aspect of human learning is the ability to select effective samples to learn and utilize the experience to infer the outcomes of new events. This type of learning is characterized as partially supervised learning. A learning algorithm of this type is suggested for linearly separable systems. The algorithm selects a subset S from a finite set X of linearly separable vectors to construct a linear classifier that can correctly classify all the vectors in X. The sample set S is chosen without any prior knowledge of how the vectors in X-S are classified. The computational complexity of the algorithm is analyzed, and the lower bound on the size of the sample set is established.>

A new divisive algorithm for multidimensional data clustering is suggested. Based on the minimization of the sum-of-squared-errors, the proposed method produces much smaller quantization errors than the median-cut and mean-split algorithms. It is also observed that the solutions obtained from our algorithm are close to the local optimal ones derived by the k-means iterative procedure.