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Equation 8 · Part 10 · How Llama's Architecture Actually Works, Generation by Generation

subscript

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}
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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 subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

Sources cited in the article section

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