3 ms·

Pattern Theory: The Mathematics of Perception (2002)

79 points by mindcrime 4y ago

sarosh 4y ago
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 https://www.dam.brown.edu/people/mumford/vision/pattern.html [1] https://www.amazon.com/dp/1568815794/ref=sr_1_1?ie=UTF8&qid=1351792472&sr=8-1&keywords=Pattern+Theory+Mumford+Desolneux https://www.amazon.com/dp/1568815794/ref=sr_1_1?ie=UTF8&qid=... [2] https://en.wikipedia.org/wiki/David_Mumford https://en.wikipedia.org/wiki/David_Mumford [3] https://en.wikipedia.org/wiki/Pattern_theory https://en.wikipedia.org/wiki/Pattern_theory
- mindcrime 4y ago
  Something I stumbled across in the last day or two brought Ulf Grenander[1] and his idea of "Pattern Theory" to my attention, and I've been going down a bit of a rabbit-hole the last couple of days digging into this. Discovering David Mumford's work was especially interesting given that he is, as you say, a Field Medalist. Not to mention his also being a MacArthur Fellow and a recipient of a National Medal of Science. Not to go all "appeal to authority", but if he thinks this is valuable territory, then it's probably worth a little bit of my time exploring it. [1]: https://en.wikipedia.org/wiki/Ulf_Grenander https://en.wikipedia.org/wiki/Ulf_Grenander
  - steppi 4y ago
    Would you consider doing a write-up somewhere after getting further into your investigation of pattern theory? I'd certainly be interested in a distillation of any insights you find and I imagine others would too.
    - mindcrime 4y ago
      Sure, but no promises on how long that might take. :-)
missfroggy 4y ago
Hasn't deep learning answered the question of perception? Like the breakthrough application was pattern recognition, and for low numbers of nodes (synapses?) you can see how it becomes more sophisticated as the net grows. Like it doesn't just explain human pattern recognition but weak nets model like chicken pattern recognition.
steppi 4y ago
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-Analysis-of-Real-World-Signals/Mumford-Desolneux/p/book/9781568815794 https://www.routledge.com/Pattern-Theory-The-Stochastic-Anal... [1] https://www.dam.brown.edu/people/mumford/blog/2019/conscious.html https://www.dam.brown.edu/people/mumford/blog/2019/conscious... [2] https://www.dam.brown.edu/people/mumford/blog/2020/Astonishing.html https://www.dam.brown.edu/people/mumford/blog/2020/Astonishi...
- mindcrime 4y ago
  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. TBH, that's one reason I find this interesting to begin with. It's a pet interest (obsession?) of mine to look for ideas that have been "left behind" and forgotten about prematurely for whatever reason. I suspect that from time to time there's a technique / idea / whatever lying out there in the sands of history just waiting for somebody to pick it up, dust it off, and turn it into something amazing.
  - steppi 4y ago
    Along those lines, have you ever checked out the book Neurocomputing, Foundations of Research [0]? It's a compendium of old foundational papers that I think would be up your alley. Researchers will find Neurocomputing an essential guide to the concepts employed in this field that have been taken from disciplines as varied as neuroscience, psychology, cognitive science, engineering, and physics. A number of these important historical papers contain ideas that have not yet been fully exploited, while the more recent articles define the current direction of neurocomputing and point to future research. Each article has an introduction that places it in historical and intellectual perspective. [0] https://mitpress.mit.edu/9780262510486/neurocomputing/ https://mitpress.mit.edu/9780262510486/neurocomputing/
e-dant 4y ago
So, we have three (umbrella) paths to modeling intelligence: - Pattern theory - Abductive logic - Machine learning Do they converge?
- mindcrime 4y ago
  I don't know, but if they do, perhaps they converge to Marcus Hutter's "Universal Artificial Intelligence"[1]. [1]: http://hutter1.net/ai/uaibook.htm http://hutter1.net/ai/uaibook.htm
fuzzfactor 4y ago
Last line in the paper: >Perhaps the ultimate dream is a fully unsupervised learning machine which is given only signals from the world and which finds their statistically significant patterns with no assistance: something like a baby in its first 6 months. Keep in mind how a baby does with some supervision and assistance rather than none. Until dreams come true I would think different levels of supervision and assistance would be key to somewhat functional operation. Once perfection is achieved then the ideal judgement could be made about overall system maturity, and people should be able to plainly see if it is wise to go with less supervision or assistance, and to what degree.

79 points by mindcrime 4y ago

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 https://www.dam.brown.edu/people/mumford/vision/pattern.html [1] https://www.amazon.com/dp/1568815794/ref=sr_1_1?ie=UTF8&qid=1351792472&sr=8-1&keywords=Pattern+Theory+Mumford+Desolneux https://www.amazon.com/dp/1568815794/ref=sr_1_1?ie=UTF8&qid=... [2] https://en.wikipedia.org/wiki/David_Mumford https://en.wikipedia.org/wiki/David_Mumford [3] https://en.wikipedia.org/wiki/Pattern_theory https://en.wikipedia.org/wiki/Pattern_theory

mindcrime 4y ago
Something I stumbled across in the last day or two brought Ulf Grenander[1] and his idea of "Pattern Theory" to my attention, and I've been going down a bit of a rabbit-hole the last couple of days digging into this. Discovering David Mumford's work was especially interesting given that he is, as you say, a Field Medalist. Not to mention his also being a MacArthur Fellow and a recipient of a National Medal of Science. Not to go all "appeal to authority", but if he thinks this is valuable territory, then it's probably worth a little bit of my time exploring it. [1]: https://en.wikipedia.org/wiki/Ulf_Grenander https://en.wikipedia.org/wiki/Ulf_Grenander
- steppi 4y ago
  Would you consider doing a write-up somewhere after getting further into your investigation of pattern theory? I'd certainly be interested in a distillation of any insights you find and I imagine others would too.
  - mindcrime 4y ago
    Sure, but no promises on how long that might take. :-)

missfroggy 4y ago

Hasn't deep learning answered the question of perception? Like the breakthrough application was pattern recognition, and for low numbers of nodes (synapses?) you can see how it becomes more sophisticated as the net grows. Like it doesn't just explain human pattern recognition but weak nets model like chicken pattern recognition.

steppi 4y ago

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-Analysis-of-Real-World-Signals/Mumford-Desolneux/p/book/9781568815794 https://www.routledge.com/Pattern-Theory-The-Stochastic-Anal... [1] https://www.dam.brown.edu/people/mumford/blog/2019/conscious.html https://www.dam.brown.edu/people/mumford/blog/2019/conscious... [2] https://www.dam.brown.edu/people/mumford/blog/2020/Astonishing.html https://www.dam.brown.edu/people/mumford/blog/2020/Astonishi...

mindcrime 4y ago
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. TBH, that's one reason I find this interesting to begin with. It's a pet interest (obsession?) of mine to look for ideas that have been "left behind" and forgotten about prematurely for whatever reason. I suspect that from time to time there's a technique / idea / whatever lying out there in the sands of history just waiting for somebody to pick it up, dust it off, and turn it into something amazing.
- steppi 4y ago
  Along those lines, have you ever checked out the book Neurocomputing, Foundations of Research [0]? It's a compendium of old foundational papers that I think would be up your alley. Researchers will find Neurocomputing an essential guide to the concepts employed in this field that have been taken from disciplines as varied as neuroscience, psychology, cognitive science, engineering, and physics. A number of these important historical papers contain ideas that have not yet been fully exploited, while the more recent articles define the current direction of neurocomputing and point to future research. Each article has an introduction that places it in historical and intellectual perspective. [0] https://mitpress.mit.edu/9780262510486/neurocomputing/ https://mitpress.mit.edu/9780262510486/neurocomputing/

e-dant 4y ago

So, we have three (umbrella) paths to modeling intelligence: - Pattern theory - Abductive logic - Machine learning Do they converge?

mindcrime 4y ago
I don't know, but if they do, perhaps they converge to Marcus Hutter's "Universal Artificial Intelligence"[1]. [1]: http://hutter1.net/ai/uaibook.htm http://hutter1.net/ai/uaibook.htm

fuzzfactor 4y ago

Last line in the paper: >Perhaps the ultimate dream is a fully unsupervised learning machine which is given only signals from the world and which finds their statistically significant patterns with no assistance: something like a baby in its first 6 months. Keep in mind how a baby does with some supervision and assistance rather than none. Until dreams come true I would think different levels of supervision and assistance would be key to somewhat functional operation. Once perfection is achieved then the ideal judgement could be made about overall system maturity, and people should be able to plainly see if it is wise to go with less supervision or assistance, and to what degree.