Rotary Position Embeddings for Long Context Length

This article is divided into two parts; they are: • Simple RoPE • RoPE for Long Context Length Compared to the sinusoidal position embeddings in the original Transformer paper, RoPE mutates the input tensor using a rotation matrix: $$ begin{aligned} X_{n,i} &= X_{n,i} cos(ntheta_i) – X_{n,frac{d}{2}+i} sin(ntheta_i) \ X_{n,frac{d}{2}+i} &= X_{n,i} sin(ntheta_i) + X_{n,frac{d}{2}+i} cos(ntheta_i) end{aligned} $$ where $X_{n,i}$ is the $i$-th element of the vector at the $n$-th position of the sequence of tensor $X$.