please i wonder whether this question has been answered but i wanted to know if there is any comprehensive mathematical framework on HTM
This is some of the deep underpinnings of the HTM theory:
This puts these SDRs to work so you can see some context:
I’ve also found two non-Numenta papers of note that examine the mathematics:
What do you mean by “Mathematical Framework”? Can you please cite an analogy?
If you’re looking for formalization of the HTM theory, the closest I have read so far is the one posted above by @jacobeverist, specifically the first resource about the spatial pooler.
i meant the mathematical base where we can stand for further modification in order to match it to other algorithms
I think one of the things to note when dealing with the spatial pooler and temporal memory in HTM is that they exhibit behavior of a complex system wherein it is not straightforward to reduce the emergent behavior (e.g. learning) to its tinier rules and formulate a mathematical framework out of it. Firstly IMO the HTM was mainly biological-constraints-driven and computation is only second. Of course you can approximate the HTM’s operations using math and in my experience there are related posts about this in this forum if you’ll have a search about (objective functions, learning, cost function, linear combinations, etc).
That said, I do think that set theory may have the right tools to analyze the behavior of sparse systems in general, and HTM in particular.
I have actually put together a nice mathematical framework using set theory focused on conceptualizing the computational and representational properties of HTM-like systems. It actually lends itself well to understanding what is actually going on with the distributed representation, what is being achieved with a spatial pooler/temporal learner, how you would construct systems using modular components, and how you would construct algorithms that do other types of operations.
I am currently working on a paper about this that I hope to release in a month or two. I think you will find it very interesting.
I find using set theory and graph theory very convenient in describing and analyzing the SP.
I look forward to it. Awesome. I’m also trying to come up with a narrative about what’s really going on in an SP. But no guarantee on its progress yet. I’m also currently reading about Judea Pearl’s causal framework and see if I can relate this to HTM. This is of course done in my spare time.
Some of the papers cited above rely on this paper from 2015: Encoding Reality: Prediction-Assisted Cortical Learning Algorithm in Hierarchical Temporal Memory which gives a pretty rigorous treatment of the CLA machinery. There have been a number of theses which also base their treatment on the paper Citations
It sounds like there are a number of us trying to come up with a more formal mathematical description and/or narrative of the HTM algorithms. Would any of you be interested in collaborating on a paper where we all bring our specific interests and expertise to bear on the problem? I think one benefit of such a collaboration might be to come up with a consistent set of notations and nomenclature that would allow us to talk to each other in common terms rather than referring to our own respective silos.
Post what you currently have?
Okay. I’ve got a rough start on a paper here. Let me know if you want to edit it, and I will add you as a collaborator. (DM me the email address you use to register on the Overleaf site.)
@fergalbyrne It’s been a while since I read your papers and I appreciate being reminded about where they situate in the larger context. I will have to return to them with fresh eyes again to see what insights I can get again.