In statistics and optimization, we often multiply a vector x \mathbf{x} x by the inverse of its covariance matrix Σ \boldsymbol{\Sigma} Σ : Σ − 1 x . (1) \boldsymbol{\Sigma}^{-1} \mathbf{x}. \tag{1} Σ − 1 x . ( 1 ) For example, you see this in the probability density function of the multivariate normal distribution or in the analytic solution to mean–variance optimization. What does this operation…
Imagine that a pollen particle is suspended in a glass of water. If we were to observe and record the vertical position of the particle over time, we would find that its movements were random. And if we were to plot this position, we’d get a jagged path through time (Figure 1 1 1 , left). This path would be just one of many possible paths, and if we were to repeat this observational experiment…
Large language models (LLMs) still feel a bit like magic to me. Of course, I understand the general machinery enough to know that they aren’t, but the gap between my outdated knowledge of the field and the state-of-the-art feels especially large right now. Things are moving fast. So six months ago, I decided to close that gap just a little by digging into what I believed was one of the core…