softmax not defined

"softmax" is a mathematical function commonly used in machine learning and neural networks. It takes a vector of real numbers and normalizes them to a probability distribution, with each element in the output vector representing the probability of the corresponding input element. The formula for the softmax function is:

softmax(x_i) = exp(x_i) / sum(exp(x_j)) for all j

where x_i is the i-th element of the input vector, and j ranges over all elements in the input vector.

In order to use the softmax function in your code, you need to define it explicitly. Here's an example implementation in Python:

import numpy as np

def softmax(x):
    """Compute softmax values for each row of x."""
    e_x = np.exp(x - np.max(x))
    return e_x / e_x.sum(axis=0)

This implementation uses NumPy to perform the necessary mathematical operations. Given an input vector `x`, it first subtracts the maximum element from all elements to avoid overflow, then exponentiates each element and divides by the sum of all exponentiated elements to obtain the output probabilities.