This post comprises two parts:
We offer mathematical lessons on Data-Science: Multidimensional statistics and Machine and Deep learning (Neural networks). This field has experienced rapid development over the past decade, particularly under the impetus of computer scientists. Innovative methodologies have been proposed, starting from multidimensional statistics, including neural networks and leading to deep learning.
The objective is to guide the reader to acquire the mathematical bases of machine learning. We pursue a rigorous mathematical formalism by defining the important notions, then by stating and demonstrating the key properties according to the approach: lemma, theorem, proof, corollary. We present structural results, such as the relationship between the number of training examples and the desired precision for the algorithm resulting from the training. Another example is the characterization of the class of functions that can be well approximated by neural networks.
The results we present are of the type of Shannon’s theorem in information theory. They give bounds on attainable performance. A complementary work consists in searching for algorithms adapted to a given problem, in a similar way to the search for coding schemes adapted to a given communication channel.
Some applications illustrate the theoretical results and help the reader to acquire the intuitions and to reinforce his comprehension. We try to ensure a good balance between mathematical rigor and the wealth of application examples.
We hope that this post will help as many people as possible to have pleasant access to this fascinating discipline, and to be able to use machine learning algorithms with clear understanding (and not as a black box).
B. Błaszczyszyn, L. Darlavoix, M.K. Karray (2024). Data science : From multivariate statistics to machine, deep learning. Book in preparation.