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Equation 8 · Part 1 · How Llama's Architecture Actually Works, Generation by Generation

Symbol x

x2i−1′=x2i−1cos⁡(mθi)−x2isin⁡(mθi)x2i′=x2i−1sin⁡(mθi)+x2icos⁡(mθi)\begin{aligned} x'_{2i-1} &= x_{2i-1}\cos(m\theta_i) - x_{2i}\sin(m\theta_i) \\ x'_{2i} &= x_{2i-1}\sin(m\theta_i) + x_{2i}\cos(m\theta_i) \end{aligned}
xx

What this part means

x is part of the quantity the equation computes from the expression on the right.

Its job in the formula

x is part of the quantity the equation computes from the expression on the right.

The passage around this formula

and a vector at sequence position m has its i -th pair rotated by angle mθi\theta_i : x2i−1′=x2i−1cos⁡(mθi)−x2isin⁡(mθi)x2i′=x2i−1sin⁡(mθi)+x2icos⁡(mθi)\begin{aligned} x'_{2i-1} &= x_{2i-1}\cos(m\theta_i) - x_{2i}\sin(m\theta_i) \\ x'_{2i} &= x_{2i-1}\sin(m\theta_i) + x_{2i}\cos(m\theta_i) \end{aligned}. The property that makes this useful rather than merely elegant is that rotating a query at position m and a key at position n by these matrices before taking their dot product leaves a result that depends only on the distance between them:

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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