Formula for RoPE with Linear Positional Interpolation

A researcher defines a new rotary position embedding function, Ro_new, for a token x_i at position i. The new function is defined as Ro_new(x_i, iθ) = Ro(x_i, (i+c)θ), where Ro is the original function and c is a constant offset. According to the general equivalence principle, this can be written as Ro_new(x_i, iθ) = Ro(x_i, iθ'). What is the correct expression for the transformed position parameter iθ'?

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Equivalence of RoPE Modification Strategies

Analysis of a Flawed RoPE Modification