K. Eswaran

K. Eswaran

The problem of optimizing a linear objective function,given a number of linear constraints has been a long standing problem ever since the times of Kantorovich, Dantzig and von Neuman. These developments have been followed by a different approach pioneered by Khachiyan and Karmarkar. In this paper we present an entirely new method for solving an old optimization problem in a novel manner, a technique that reduces the dimension of the problem step by step and interestingly is recursive. A theorem which proves the correctness of the approach is given. The method can be extended to other types of optimization problems in convex space, e.g. for solving a linear optimization problem subject to nonlinear constraints in a convex region.

K. Eswaran, K. Damodhar Rao

Recently an algorithm, was discovered, which separates points in n-dimension by planes in such a manner that no two points are left un-separated by at least one plane{[}1-3{]}. By using this new algorithm we show that there are two ways of classification by a neural network, for a large dimension feature space, both of which are non-iterative and deterministic. To demonstrate the power of both these methods we apply them exhaustively to the classical pattern recognition problem: The Fisher-Anderson's, IRIS flower data set and present the results. It is expected these methods will now be widely used for the training of neural networks for Deep Learning not only because of their non-iterative and deterministic nature but also because of their efficiency and speed and will supersede other classification methods which are iterative in nature and rely on error minimization.

K. Eswaran, Vishwajeet Singh

In this paper we introduce a new method which employs the concept of "Orientation Vectors" to train a feed forward neural network and suitable for problems where large dimensions are involved and the clusters are characteristically sparse. The new method is not NP hard as the problem size increases. We `derive' the method by starting from Kolmogrov's method and then relax some of the stringent conditions. We show for most classification problems three layers are sufficient and the network size depends on the number of clusters. We prove as the number of clusters increase from N to N+dN the number of processing elements in the first layer only increases by d(logN), and are proportional to the number of classes, and the method is not NP hard. Many examples are solved to demonstrate that the method of Orientation Vectors requires much less computational effort than Radial Basis Function methods and other techniques wherein distance computations are required, in fact the present method increases logarithmically with problem size compared to the Radial Basis Function method and the other methods which depend on distance computations e.g statistical methods where probabilistic distances are calculated. A practical method of applying the concept of Occum's razor to choose between two architectures which solve the same classification problem has been described. The ramifications of the above findings on the field of Deep Learning have also been briefly investigated and we have found that it directly leads to the existence of certain types of NN architectures which can be used as a "mapping engine", which has the property of "invertibility", thus improving the prospect of their deployment for solving problems involving Deep Learning and hierarchical classification. The latter possibility has a lot of future scope in the areas of machine learning and cloud computing.