I came across Grenander's work "Probabilities on Algebraic Structures" recently and found that much of Grenander's work focused on what he called "Pattern Theory." He's written many texts on the matter and from what I've seen, they seem like an attempt to unify some mathematical underpinnings of pattern representation. However, I'm not sure what this really means in practice, nor how it relates to results we already have in learning theory. The mathematical aspect of the work is really quite intriguing, but I am skeptical as to its practicality.

Are there any applications of Grenander's pattern theory? Either for getting a better theoretical understanding of certain methods of pattern recognition or for directly implementing algorithms?

Some links to what I'm referring to: Wikipedia entry, book, first book of 3 set volume, and research group, including fields medalist David Mumford.


Pattern-matching is a classical sub-problem of AI systems. Read about the Rete algorithm, or pattern matching in Prolog. Or about Unification. Or about the Warren Abstract Machine. Or about topological data analysis like in GUDHI.

A typical example is the CLIPSrules expert system shell. Or the match expression construct of Ocaml or in Haskell.

In some ways, pattern matching is related to compilers: the GCC compiler (or the Stalin one, or Bigloo, or SBCL) is somehow doing pattern matching on ASTs. So parser generators (e.g. GNU bison, or Iburg) are related to pattern matching.

Patterns also have a different meaning in computer vision.

A classical paper is of course Derivation of a Pattern-Matching Compiler.

A more interesting approach is described in Jacques Pitrat's last book: Artificial Beings: the conscience of a conscious machine ISBN-13: 978-1848211018

You could also look into Pitrat's blog here.

PS. Jacques Pitrat passed away in oct. 2019.

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    $\begingroup$ Is this the problem that pattern theory is addressing? For reference, I am specifically talking about pattern theory in this sense, not general pattern matching. $\endgroup$ – Jason Dec 18 '20 at 16:42

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