“Convex Combination”
The term “convex” in mathematics usually refers to a set of points that, if you take any two points within the set and draw a line segment between them, every point on that line segment also belongs to the set. However, in the context of the paragraph, “convex” refers to a specific type of combination of elements, known as a convex combination. In a convex combination, you combine elements such that each element is weighted by a coefficient, and all coefficients are non-negative and sum to 1. Mathematically, given elements x1,x2,…,xn and weights w1,w2,…,wn, a convex combination would look like:
w1⋅x1+w2⋅x2+…+wn⋅xn
where w1+w2+…+wn=1 and wi≥0.
“Softmax”
The softmax function is used in machine learning and statistics, primarily for transforming a vector of real numbers into a probability distribution. Given an input vector x=[x1,x2,…,xn], the softmax function outputs a new vector s=[s1,s2,…,sn] where each element si is calculated as:
Here, e is the base of the natural logarithm. The output values are all between 0 and 1, and they sum to 1, making them suitable as probabilities.
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