Pattern theory
Pattern Theory, pioneered by Ulf Grenander, is a distinctive approach to the analysis of all forms of real-world signals. At its core is the design of a large variety of probabilistic models whose samples reproduce the look and feel of the real signals, their patterns, and their variability. Bayesian statistical inference then allows you to apply these models in the analysis of new signals. This book treats the mathematical tools, the models themselves, and the computational algorithms for applying statistics to analyze six representative classes of signals of increasing complexity. "Pattern Theory covers six classic attempts at modeling signals from the human and natural world: natural language (written), music, character recognition, texture modeling, face recognition, and natural scenes. These applications, appealing to students and researchers alike, include fourteen "crash courses" giving all the needed basics, exercises, and numerical simulations. Thus, it is a complete pedagogic tool at master or first-year graduate level. I endorse the publication of Pattern Theory, and will actually use it and recommend it to other researchers."--Jean-Michel Morel, CMLA. "This book is fascinating. It develops a statistic approach to finding the patterns in the signals generated by the world. The style is lucid. I'm reminded of Mumford's exposition of Theta functions and Abelian varieties in his Tata lectures. The exposition is thorough.
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- The author, David Mumford, is known for his distinguished work in algebraic geometry, and was awarded the Fields Medal in 1974. [2]
Pattern theory was formulated by Ulf Grenander to describe knowledge of the world as patterns. [3]
Prof. Mumford explains that "[s]everal essential ideas brought me to realize how Grenander's Pattern Theory was the right way to understand almost all cognitive skills and especially vision. One was the emphasis on pattern synthesis as well as pattern analysis." [0]. Second, "was that natural signals given by functions f vary not only by random additive perturbations but often by composition with random rearrangements of their domain. The resulting probability distribution in the vector space of signals is nothing like Gaussian. Its support is usually a twisted snakey submanifold. This puts the lie to all simplistic Gaussian pattern recognition algorithms." [0]. And third, "graphical structures were everywhere in the representations of ideas in cognitive domains" [0].
His most recent work seems to be "Pattern Theory, the Stochastic Analysis of Real World Signals" [1]
[0] https://www.dam.brown.edu/people/mumford/vision/pattern.html
[1] https://www.amazon.com/dp/1568815794/ref=sr_1_1?ie=UTF8&qid=...
- Having once studied just enough algebraic geometry to appreciate the true excellence of Mumford’s work in the field I’ve wondered before if I should study his Pattern Theory book [0] to mine it for insights that the ML community may have missed as a whole. I'd forgotten about that though.
Has anyone here worked through it? It seems like much of it is rooted in methods that have kind of fallen by the wayside due to the incredible success of modern techniques. I’d also be very interested to know David Mumford’s thoughts on modern deep learning and how he thinks it fits into his pattern theory framework. It looks like he goes into this to some extent in a 2019 blog post on whether AI can achieve consciousness [1] and another on the convergence between ML methods and the human brain [2].
[0] https://www.routledge.com/Pattern-Theory-The-Stochastic-Anal...
[1] https://www.dam.brown.edu/people/mumford/blog/2019/conscious...
[2] https://www.dam.brown.edu/people/mumford/blog/2020/Astonishi...
- Insights from famed algebraic geometer and Fields medalist David Mumford, who has also worked (and co-authored a book) on Pattern Theory (statistical computer vision). https://en.wikipedia.org/wiki/David_Mumford
- I'm looking to gain functional familiarity but also at the same time working on real world applications.
Did you mean David Mumford's "Pattern Theory"? It was the only book I could find that's close to the topic. Unless I'm looking at the wrong David Mumford?
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