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Win the Hutter Prize using modern deep learning
Viscerally feel the connection between prediction and compression
Predictive coding:
A Machine Learning Perspective on Predictive Coding with PAQ https://arxiv.org/pdf/1108.3298.pdf
P-adic arithmetic coding https://arxiv.org/pdf/0704.0834v1.pdf
Arithmetic Code Discussion and Implementation http://michael.dipperstein.com/arithmetic/index.html
Language modeling:
Also requires using techniques for compressing networks, and running under computational constraints.