Are there applications of Grenander's pattern theory in pattern recognition or for implementing algorithms?

Question

I came across Grenander's work "Probabilities on Algebraic Structures" recently, and I 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.

Answer 1

Grenader's Pattern Theory sees AI from a different perspective from other techniques- one of pattern recognition (as opposed to algorithms, analogies to the human brain, etc...). Like anything that sees an existing problem from a radically different perspective, it's difficult to unpack all the implications.

It has considerable overlap with existing AI algorithms. For example, formulations with Bayes' theorem and information theory. The idea of basing AI on patterns is not unique to this theory. Terms like "signals" are thrown around in AI development (especially computer vision and time series analysis), and the most famous machine learning textbook is called Pattern Recognitin.

That being said, I can think that the way that this formula would impact AI is by giving us a series of tools for machine learning that would allow for handling of problems from a pattern recognition perspective. For example:

1. ways to decompose noisy images into separate image + noise signals with mathematical formulations for each
2. better ways of creating embeddings based on discrete signals in input data, and more mathematical operations that could be performed on each
3. finding a way to compress data by turning an image (essentially a very dense and large graph where each pixel is a node with an edge to adjacent ones) to compressed graphs where the nodes are signals.

There are a lot of different possibilities. One of the joys and hardships of AI and machine learning is that there are often many mathematical formulations for the same idea, and knowing which one to select and why is one of the great challenges of the field.