In the following analysis, a machine learning challenge is proposed by Sasha Kolpakov, Managing Editor of the Journal of Experimental Mathematics, and myself to verify a strong form of the Manifold Hypothesis due to Yoshua Bengio. Motivation: In the Deep Learning book, Ian Goodfellow, Yoshua Bengio and Aaron Courville credit...
Motivation: Before the Pandemic of 2020, I had a number of constructive discussions with several neuroscientists including Konrad Körding, a computational neuroscientist and pioneer exploring connections between deep learning and human learning, and Alex Gomez-Marin, an expert on behavioural neuroscience, on the hypothetical boundary between Human and Machine Learning. One...
Motivation: In 1970, Andrey Kolmogorov presented a remarkable vision for the combinatorial foundations of probability theory at the International Congress of Mathematicians in Nice which anticipates the modern development of Artificial Intelligence: > Using his brain, as given by the Lord, a mathematician may not be interested in the combinatorial...
In the following analysis, a machine learning challenge is proposed by Sasha Kolpakov, Managing Editor of the Journal of Experimental Mathematics, and myself to verify a strong form of the Manifold Hypothesis due to Yoshua Bengio.
In the Deep Learning book, Ian Goodfellow, Yoshua Bengio and Aaron Courville credit the unreasonable effectiveness of Deep Learning to the Manifold Hypothesis as it implies that the curse of dimensionality may be avoided for most natural datasets [5]. If the intrinsic dimension of our data is much smaller than its ambient dimension, then sample-efficient PAC learning is possible. In practice, this happens via the latent space of a deep neural network which auto-encodes the input.
Yoshua Bengio... (read 1035 more words →)
Motivation:
Before the Pandemic of 2020, I had a number of constructive discussions with several neuroscientists including Konrad Körding, a computational neuroscientist and pioneer exploring connections between deep learning and human learning, and Alex Gomez-Marin, an expert on behavioural neuroscience, on the hypothetical boundary between Human and Machine Learning. One promising avenue for making progress along these lines appeared to be Kolmogorov’s theory of Algorithmic Probability [4] which Alexander Kolpakov(Sasha) and myself recently used to probe this boundary, motivated in large part by open problems in Machine Learning for Cryptography [1].
However, more progress appears to be possible based on the analyses of Marcus Hutter and Shane Legg who established that Algorithmic Probability defines... (read 381 more words →)
Motivation:
In 1970, Andrey Kolmogorov presented a remarkable vision for the combinatorial foundations of probability theory at the International Congress of Mathematicians in Nice which anticipates the modern development of Artificial Intelligence:
Using his brain, as given by the Lord, a mathematician may not be interested in the combinatorial basis of his work. But the artificial intellect of machines must be created by man, and man has to plunge into the indispensable combinatorial mathematics. - Kolmogorov
The full text was eventually published in 1983 [1] and Kolmogorov's papers on the subject, where he formulated the fundamental notion of Kolmogorov Complexity, represent a partial fulfilment of his original vision. However, it would be his PhD student... (read 541 more words →)
I agree. I don't think he had to attempt to address this problem, if it may be addressed at all.
He has since taken into account the work of experimentalists doing related work, that validates his thesis of Quantum Theory essentially predicting what an observer will see next:
Thank you for sharing this. 👌
Thank you for bringing up these points:
Either way, I believe that additional experiments may be enlightening as the applied mathematics that mathematicians do is only true to the extent that it has verifiable consequences.
Feel free to reach me via email. However, I must note that Sasha and myself are currently oriented towards existing projects of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation.
If your research proposal may be formulated from the vantage point of that research program, that would improve the odds of a collaboration in the medium term.
Regarding your remark on finding low-dimensional representations, I have added a section on physical intuitions for the challenge. Here I explain how the prime recognition problem corresponds to reliably finding a low-dimensional representation of high-dimensional data.
The best physicists on Earth, including Edward Witten and Alain Connes, believe that the distribution of primes and Arithmetic Geometry encode mathematical secrets that are of fundamental importance to mathematical physics. This is why the Langlands program and the Riemann Hypothesis are of great interest to mathematical physicists.
If number theory, besides being of fundamental importance to modern cryptography, allows us to develop a deep understanding of the source code of the Universe then I believe that such advances are a critical part of human intelligence, and would be highly unlikely if the human brain had a different architecture.
