Euclidean Representation of RoPE Rotation

Element-wise Formula for RoPE Rotation

A positional encoding method can be described in two ways. In one view, a d-dimensional vector is treated as d/2 complex numbers, and each complex number is rotated by an angle. In another view, the same d-dimensional vector is treated as d/2 pairs of real numbers, and each pair is rotated as a 2D vector by the same corresponding angle. What is the core principle that makes these two views equivalent?

Equivalence of 2D Vector Rotation Representations

Verifying Equivalence of 2D Rotational Methods

Element-wise Formula for RoPE Rotation

In a system that encodes positional information, a 'cosine vector' is defined for each position $i$ based on a set of frequency parameters $\theta = [\theta_1, \dots, \theta_{d/2}]$. The formula for this vector is: $[\cos i\theta_1, \dots, \cos i\theta_{d/2}]$.

Given a total dimension $d=4$, a position $i=2$, and frequency parameters $\theta_1 = 0.01$ and $\theta_2 = 0.0001$, what is the correct cosine vector for this position?

A developer is implementing a positional encoding scheme where a 'cosine vector' is needed for each position $i$. Given a set of frequency parameters $\theta = [\theta_1, \dots, \theta_{d/2}]$ and a total embedding dimension $d$, which of the following formulas correctly defines this vector?

Properties of Positional Cosine Vectors

Element-wise Formula for RoPE Rotation

Consider a system where positional information is encoded using periodic functions. A key component is a vector constructed for each position. Given a position index i = 2 and a set of frequency parameters θ = [π/2, π/4], what is the resulting 2-dimensional sine vector, defined as [sin(iθ₁), sin(iθ₂)]?

Consider a method for encoding positional information where a d-dimensional input vector is modified using periodic functions. In this method, a 'sine vector' is constructed for each position i. This sine vector has a dimensionality equal to d.

Determining Position Index from a Sine Vector