smoothmax

Builtin index · Core maths

result = smoothmax(x, beta=1)

Input: x, beta · Output: result

What it is

Smooth (differentiable) approximation to max(x): (1/beta) * logsumexp(beta * x), which tracks the true maximum more tightly as beta grows (at the cost of a stiffer gradient). x is a scalar, Vec, or Mat/Signal, flattened; beta is an optional positional scalar defaulting to 1 (a non-positive or non-finite beta falls back to the plain hard max).

Overloads

One call shape — this function is not overloaded.

Examples

From the documentation, and run by the build -- the output and the figure below are what the engine actually produced. Simplest first.

crest(x) := max(abs(x)) / rms(x)

xs = [0, 1, 2, 3]
print("hard max = {max(xs)}")
print("logsumexp = {logsumexp(xs):.4f}")       # >= hard max, a proven upper bound

print("smoothmax b=50 = {smoothmax(xs, 50):.4f}")  # -> hard max as beta grows

print("softmax = {softmax(xs)}")                # the gradient of logsumexp

hard max = 3
logsumexp = 3.4402
smoothmax b=50 = 3.0000
softmax = [0.032059, 0.087144, 0.236883, 0.643914]

from numerics-dsp