Apply Softmax Function to a 3D Array

This function applies the softmax transformation along the third dimension of a 3D array. The softmax function converts raw scores into probabilities such that they sum to 1 for each slice along the third dimension.

Usage

softmax_3d(x)

Arguments

x

A 3D array. The input array on which the softmax function will be applied.

Value

A 3D array of the same dimensions as x, where the values along the third dimension are transformed using the softmax function.

Details

The softmax transformation is computed as: $$\text{softmax}(x_{ijk}) = \frac{\exp(x_{ijk})}{\sum_{l} \exp(x_{ijl})}$$ This is applied for each pair of indices (i, j) across the third dimension (k).

The function processes the input array slice-by-slice for the first two dimensions (i, j), normalizing the values along the third dimension (k) for each slice.

Examples

# Example: Apply softmax to a 3D array
x <- array(runif(24), dim = c(2, 3, 4)) # Random 3D array (2x3x4)
softmax_result <- softmax_3d(x)