Justification of RoPE Dot Product Simplification
In the derivation simplifying the dot product of two 2D vectors encoded with rotational positions, the expression x * R_tθ * (R_sθ)^T * y^T is reduced to x * R_(t-s)θ * y^T. Explain the two key properties of rotation matrices that justify the simplification of the term R_tθ * (R_sθ)^T into R_(t-s)θ.
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Ch.2 Generative Models - Foundations of Large Language Models
Foundations of Large Language Models
Foundations of Large Language Models Course
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A student is simplifying the dot product between two 2D vectors, x and y, which have been encoded with rotational positional information at positions t and s, respectively. The encoded vectors are given by x' = xRtθ and y' = yRsθ, where Rα is a rotation matrix for angle α. The student's derivation for the dot product x' ⋅ y' is shown below. Identify the step that contains a mathematical error.
Step 1: x' ⋅ y' = (xRtθ)(yRsθ)T Step 2: = xRtθ(Rsθ)TyT Step 3: = xRtθRsθyT Step 4: = xR(t+s)θyT
A key property of certain positional embeddings is that the dot product between two encoded vectors depends on their relative position. The derivation for this property involves several steps. Arrange the following mathematical expressions to show the correct logical sequence for simplifying the dot product of two 2D vectors, x and y, which have been rotated by angles tθ and sθ respectively. The rotated vectors are given by xRtθ and yRsθ.
Justification of RoPE Dot Product Simplification