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Representation: Learning What to Represent
Before a problem can be solved it must be described, and the description decides the difficulty.
Stories by Mahendra
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The Ergodic Hypothesis: Time Averages and Ensemble Averages
To predict a system's behavior one can watch it for a long time or imagine many copies at once.
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Learning From Finite Data: The Problem of Induction
No finite set of examples logically compels a general conclusion.
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The Constructive Role of Noise
Noise is usually the enemy of signal, yet in learning and in nature it can help.
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Optimization Landscapes: Hills, Valleys, and Saddles
To train a model is to descend a landscape of error, seeking its lowest point.
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High-Dimensional Geometry: Where Intuition Breaks Down
Our intuitions are built for three dimensions and betray us in many.
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Regression to the Mean
An extreme result tends to be followed by a less extreme one, not because of any force but because extremes owe much to luck.
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Probability as Degree of Belief
What does it mean to say an event is probable? One view holds that probability is not a property of the world but a measure of our confidence.
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Infinity: Sizes Beyond Counting
Infinity is not a single vast number but a hierarchy of sizes, some larger than others.
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Gödel: Truths a System Cannot Prove
Within any sufficiently rich system of mathematics there are true statements it cannot prove.