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 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 in particular argued that the hierarchical/compositional structure of the human brain exploits the Manifold Hypothesis as an inductive bias [7], so any efficiently computable function that is of interest to humans is most likely PAC-learnable. A strong form of this hypothesis, which is generally assumed by OpenAI and DeepMind, suggests that deep learning may be the ultimate path to AGI.

However, a plausible exception to Bengio's heuristic is the mysterious distribution of prime numbers that has captured the imagination of the best mathematicians of all ages:

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-Euler

Though perhaps if Euler had access to GPUs he would have generalised this statement to the minds of machines. In fact, after careful deliberations with Steve Brunton, Marcus Hutter and Hector Zenil regarding the empirical observation that deep learning models fail to approximate the Prime Counting Function...it appears that not all efficiently computable functions that are of human interest are PAC-learnable.

In order to clarify the nature of these observations, Sasha Kolpakov and myself rigorously formulated the Monte Carlo Hypothesis which implies that machine learning alone is not a complete path to human-level intelligence. If Bernhard Riemann knew of the Prime Counting Function, it would have had to be by other means than data compression for reasons that are clarified below.

The Monte Carlo Hypothesis:

The Prime Coding Theorem:

F...

Dhruv Ghulati

Dhruv Ghulati

Dhruv Ghulati

Dhruv Ghulati