I have given a glance to:
- Artificial Intelligence: A Modern Approach (Russel & Norvig)
- Machine Learning: The Art and Science of Algorithms that Make Sense of Data (Flach)
- Learning From Data (Abu-Mostafa et al.)
- Introduction to Statistical Learning (James et al.)
- Elements of Statistical Learning (Hastie et al.)
- Pattern Recognition and Machine Learning (Bishop)
Now it is difficult to evaluate if they would fit my needs because only a few pages are generally available online. However my first impression is that they do not. In the appendices of Artificial Intelligence: A Modern Approach I can read:
Mathematicians define a vector as a member of a vector space, but we will use a more concrete definition: a vector is an ordered sequence of values.
This is exactly the kind of approach I am not looking for.
I'm looking for a book which assumes the reader has a good understanding in set theory, abstract algebra, measure and probability theory, statistics, topology, graph theory, complexity theory, etc and a preference for formal and axiomatic explanations rather than lenghty and so-called "intuitive" approaches based on basic mathematical objects and examples. Furthermore I don't want something that looks like a recipe book from the very beginning. I want a book that formalizes the abstract and common shape of all data science methods as well as their common aim first. Only after that it can start to explain the different categories by explicitely stating which further hypotheses each category is assuming and which cases/problems/domains they are known to handle efficiently or not.
At last, to be clear, I have no problem with being shown concrete examples and their treatment via a specific programming language for example. I just want this to come second as an illustration for the conceptual explanation, not as a substitute.