Boolean factor analysis: A review of a novel method of matrix decomposition and neural network Boolean factor analysis.

摘要

Data reductionism is a key area of focus across many a discipline owing to the massive amount of data available nowadays. Raw Data which is mostly redundant is expressed in terms of fewer entities called factors. Factor Analysis is a method of reducing data dimensionality by grouping and combining variables, expressing them in a highly structured manner without losing the information value of data. Boolean Factor Analysis is a method of representing binary data in terms of its principal factors. This thesis addresses two entirely different approaches, their methodologies and implementation in performing Boolean factor analysis.;The first of these approaches is a novel method of matrix decomposition that uniquely decomposes a given data matrix into successive smaller matrices entailing a fixed number of factors in the process. An n x m binary data matrix I is expressed as a boolean product A ∘ B of n x k binary matrix A and k x m binary matrix B, keeping factors k minimal. I is known as an object-attribute matrix, while A and B are the object-factor and factor-attribute matrices respectively. The approach is built on a theorem and accompanying approximation algorithm which finds optimal decompositions based on the framework provided by the theorem.;The second approach is based on a modified hopfield neural network that uses correlational hebbian learning and the native nature of recurrent networks in identifying factors of a given data set. The original data whose factors are to be found are mapped into the space of factors allowing the hopfield-like network to find them. Due to hebbian learning, neurons belonging to one common factor tend to be more correlated than the other neurons and fire collectively when the factor is found, thus constituting attractors of network dynamics. The hopfield network follows a step by step procedure in the factor search, building attraction basins proceeding towards convergence and ultimately factor revelation.;Both approaches albeit oriented towards factor analysis follow entirely different methodologies with some parallels along the way, making for a comparative research study which this thesis comprises of. Data sets, both real and artificial are tested for either approach and compared in order to validate Boolean Factor Analysis