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  • Mathematics
    Statistics & Probability: Softmax & LogSumExp
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                      Softmax and LogSumExp functions

                      Given the data set (vector) x = (x1, x2, ... , xn) , calculate the Softmax function (distribution) σ : ℝn → (0,1)n, such that
                          σ(xi) =
                      exi
                      ∑
                      nj=1
                      exj

                      and the LogSumExp function LSE : ℝn → ℝ, such that
                          LSE(x) = ln (
                      ∑
                      nj=1
                      exj
                      )
                      Please Note:
                      • All values are calculated with the precision of 10-15, but are displayed with the precision of 10-9.
                      • If any of the values xi, from the data set, is greater then any other value xj by 35, or more, its σ(xi) value will become
                        dominant over all other σ values by the precision greater then the maximal precision of the floating point arihmetic.
                      Data Set x:
                      max σ(xi)   =
                      xi   =
                      LSE(x)   =
                      xσ(x)
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