An introduction to computational learning theory
An introduction to computational learning theory
Towards a mathematical theory of machine discovery from facts
Theoretical Computer Science - Special issue on algorithmic learning theory
Computable analysis: an introduction
Computable analysis: an introduction
A comparison of identification criteria for inductive inference of recursive real-valued functions
Theoretical Computer Science - Algorithmic learning theory
Foundations of Logic Programming
Foundations of Logic Programming
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Inferability of Recursive Real-Valued Functions
ALT '97 Proceedings of the 8th International Conference on Algorithmic Learning Theory
Inference of omega-Languages from Prefixes
ALT '01 Proceedings of the 12th International Conference on Algorithmic Learning Theory
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
Pattern Recognition and Machine Learning (Information Science and Statistics)
Pattern Recognition and Machine Learning (Information Science and Statistics)
Uncountable automatic classes and learning
ALT'09 Proceedings of the 20th international conference on Algorithmic learning theory
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Discretization is a fundamental process for machine learning from analog data such as continuous signals. For example, the discrete Fourier analysis is one of the most essential signal processing methods for learning or recognition from continuous signals. However, only the direction of the time axis is discretized in the method, meaning that each datum is not purely discretized. To give a completely computational theoretical basis for machine learning from analog data, we construct a learning framework based on the Gold-style learning model. Using a modern mathematical computability theory in the field of Computable Analysis, we show that scalable sampling of analog data can be formulated as effective Gold-style learning. On the other hand, recursive algorithms are a key expression for models or rules explaining analog data. For example, FFT (Fast Fourier Transformation) is a fundamental recursive algorithm for discrete Fourier analysis. In this paper we adopt fractals, since they are general geometric concepts of recursive algorithms, and set learning objects as nonempty compact sets in the Euclidean space, called figures, in order to introduce fractals into Gold-style learning model, where the Hausdorff metric can be used to measure generalization errors. We analyze learnable classes of figures from informants (positive and negative examples) and from texts (positive examples), and reveal the hierarchy of learnabilities under various learning criteria. Furthermore, we measure the number of positive examples, one of complexities of learning, by using the Hausdorff dimension, which is the central concept of Fractal Geometry, and the VC dimension, which is used to measure the complexity of classes of hypotheses in the Valiant-style learning model. This work provides theoretical support for machine learning from analog data.