ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Map
▼ Operation
▼ N-operation
▼ Binary operation
▼ Enclosed binary operation
▼ Groupoid
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▼ Standard N-operation
▼ Indexed sum
▼ Series
▼ Power series
▼ Convergent power series
▼ Convergent basic real power series
▼ Standard natural real exponential function
Definition D5752
Softmax function
Formulation 0
Let $\exp$ be the D1932: Standard natural real exponential function.
The softmax function with respect to $N \in 2, 3, 4, \ldots$ is the D4363: Euclidean real function \begin{equation} \mathbb{R}^N \to (0, 1)^N, \quad x \mapsto \frac{1}{\sum_{n = 1}^N \exp(x_n)} \left( \exp (x_1), \ldots, \exp(x_N) \right) \end{equation}
Formulation 1
Let $t \mapsto e^t$ be the D1932: Standard natural real exponential function.
The softmax function with respect to $N \in 2, 3, 4, \ldots$ is the D4363: Euclidean real function \begin{equation} \mathbb{R}^N \to (0, 1)^N, \quad x \mapsto \frac{1}{\sum_{n = 1}^N e^{x_n}} \left( e^{x_1}, \ldots, e^{x_N} \right) \end{equation}