Positional Encoding

Generate the sinusoidal positional encodings of the original Transformer for n positions and model dimension dim. Even columns hold $\sin (\text{pos}\cdot r_k)\backslash sin(\backslash text\{pos\}\backslash cdot\; r\_k)$ and odd columns hold $\cos (\text{pos}\cdot r_k)\backslash cos(\backslash text\{pos\}\backslash cdot\; r\_k)$, where $r_k={10000}^{-2\lfloor k\mathrm{/}2\rfloor \mathrm{/}\text{dim}}r\_k=10000^\{-2\backslash lfloor\; k/2\backslash rfloor/\backslash text\{dim\}\}$ for column $kk$.

Implement solve(n,dim).

Returns. Return a float array of shape (n, dim).

Examples

Example 1

Input:

solve(3, 4)

Output:

[[0.0, 1.0, 0.0, 1.0], [0.841471, 0.540302, 0.01, 0.99995], [0.909297, -0.416147, 0.019999, 0.9998]]

Example 2

Input:

solve(1, 5)

Output:

[[0.0, 1.0, 0.0, 1.0, 0.0]]