Studies on Computational Learning via Discretization

This thesis presents cutting-edge studies on computational learning. The key issue throughout the thesis is amalgamation of two processes; discretization of continuous objects and learning from such objects provided by data. Machine learning, or data mining and knowledge discovery, has been rapidly developed in recent years and is now becoming a huge topic in not only research communities but also businesses and industries. Discretization is essential for learning from continuous objects such as real-valued data, since every datum obtained by observation in the real world must be discretized and converted from analog (continuous) to digital (discrete) form to store in databases and manipulate on computers. However, most machine learning methods do not pay attention to the process: they use digital data in actual applications whereas assume analog data (usually real vectors) in theories. To bridge the gap, we cut into computational aspects of learning from theory to practice through three parts in this thesis. Part I addresses theoretical analysis, which forms a disciplined foundation of the thesis. In particular, we analyze learning of igures, nonempty compact sets in Euclidean space, based on the Gold-style learningmodel aiming at a computational basis for binary classi ication of continuous data. We use fractals as a representation system, and reveal a learnability hierarchy under various learning criteria in the track of traditional analysis of learnability in the Gold-style learning model. We show a mathematical connection between machine learning and fractal geometry by measuring the complexity of learning using the Hausdorff dimension and the VC dimension. Moreover, we analyze computability aspects of learning of igures using the framework of Type-2 Theory of Effectivity (TTE). Part II is a way from theory to practice. We start from designing a new measure in a computational manner, called coding divergence, which measures the difference between two sets of data, and go further by solving the typical machine learning tasks: classi ication and clustering. Speci ically, we give two novel clustering algorithms, COOL (COding Oriented cLustering) and BOOL (Binary cOding Oriented cLustering). Experiments show that BOOL is faster than the K-means algorithm, and about two to three orders of magnitude faster than two state-ofthe-art algorithms that can detect non-convex clusters of arbitrary shapes. Part III treats more complex problems: semi-supervised and preference learning, by bene iting from Formal Concept Analysis (FCA). First we construct a SELF (SEmi-supervised Learning via FCA) algorithm, which performs classi ication and label ranking of mixed-type data containing both discrete and continuous variables. Finally, we investigate a biological application; we challenge to ind ligand candidates of receptors from databases by formalizing the problem as multi-label classi ication, and develop an algorithm LIFT (Ligand FInding via Formal ConcepT Analysis) for the task. We experimentally show their competitive performance.