The push for real-time autonomous AI systems has been sought for decades. The DoD has spent considerable R&D budgets looking for systems that can operate with no or little supervision. These systems must process incredible amounts of heterogeneous information looking for information. In order to achieve these goals, we must affect real learning, or "learning with experience," in autonomous AI systems [10]. The goal of having machines that learn with experience is one of the most intriguing problems in computer science and computer engineering. As the types of problems we would like AI systems to solve get more complex and more diverse, it is becoming a necessary task as well. Unfortunately, by its nature, learning is somewhat fuzzy, and random in nature, for information comes at us in stochastic fashion [22]. In fact, the overall goal is to learn things we do not yet know, and in doing so find patterns that we can learn. This constitutes not patter matching, or pattern recognition, but is, in fact, pattern discovery. Nonetheless, we would like a mathematical framework for machine learning to aid in our understanding and improve our ability to make progress toward autonomous AI systems.
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