Neural networks promise to approximate extraordinarily complex multivariate functions, yet the representation must still fit into physical compute. Hilbert’s thirteenth problem offers a powerful lens on that tension between expressive composition and executable scale.
Brian Greenforest connects that mathematical lineage to Q-learning, convolutional and recurrent networks, Monte Carlo methods, Markov chains, and the next possibilities in quantum machine learning.
Representation Determines Scale
High-dimensional functions can demand explosive resources when a system represents them directly. Compositions of lower-dimensional functions, sums of products, and learned hierarchies offer different ways to organize the same complexity.
Deep reinforcement learning adds sampled experience and statistical updates, allowing useful policies to emerge without enumerating an entire phase space. Its practical success makes the underlying mechanism worth understanding in exact computational terms.
Carry the Question Into Quantum Learning
Quantum systems offer a radically different state space and measurement process. Combining them with learning raises new questions about representation, sampling, correlations, and which computations produce useful advantage.
Mathematicians, ML researchers, and hardware builders can study those questions together. The prize reaches beyond a faster model: it could reveal a new way to organize multivariate computation.
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I think scalability of Hilbert's 13th problem in the form of a neural network or sum of products function of many, many, really many variables! We will be finally able to curve-fit ANY arbitrary multivariate function. I honestly am very impressed that Q-learning combined with deep convolutional and recursive networks DOES WORK at all, because it's a pure magic, given that classical neural networks do not scale. The methods for Monte Carlo, used in deep reinforcement learning, allowing deep Markov chains to represent and model real systems without introducing huge piece-wise discontinuities and phase space transition chaotic attractor bifurcations, are FASCINATING. I swear, I do not understand why these methods should even work in theory.
Now simply imagine, what can we do with QUANTUM machine learning!!!! <3 <3 <3
#machinelearning #neuralnetworks #scalability
