{"type":"rich","version":"1.0","title":"Formula: Artificial Neuron","html":"<span class=\"wb-embed wb-embed-math\" data-latex=\"\\begin{aligned}\text{Let } \\mathbf{x} = (x_1,\\dots,x_n)^\top \\in \\mathbb{R}^n \text{ be the input vector} \\\text{Let } \\mathbf{w} = (w_1,\\dots,w_n)^\top \\in \\mathbb{R}^n \text{ be the weight vector} \\\text{Let } b \\in \\mathbb{R} \text{ be the scalar bias} \\\text{The affine pre‑activation is } z = \\mathbf{w}^\top \\mathbf{x} + b = \\sum_{i=1}^n w_i x_i + b \\\text{Choose a non‑linear activation } \\sigma: \\mathbb{R} \to \\mathbb{R} \\\text{The artificial neuron output is } y = \\sigma(z) \\in \\mathbb{R} \\\text{Typical activations: } \\sigma(z) = \\mathbf{1}_{z \\ge 0} \text{ (step)}, \\quad \\sigma(z) = (1+e^{-z})^{-1} \text{ (sigmoid)}, \\quad \\sigma(z) = \\max(0,z) \text{ (ReLU)}\\end{aligned}\">\\begin{aligned}\text{Let } \\mathbf{x} = (x_1,\\dots,x_n)^\top \\in \\mathbb{R}^n \text{ be the input vector} \\\text{Let } \\mathbf{w} = (w_1,\\dots,w_n)^\top \\in \\mathbb{R}^n \text{ be the weight vector} \\\text{Let } b \\in \\mathbb{R} \text{ be the scalar bias} \\\text{The affine pre‑activation is } z = \\mathbf{w}^\top \\mathbf{x} + b = \\sum_{i=1}^n w_i x_i + b \\\text{Choose a non‑linear activation } \\sigma: \\mathbb{R} \to \\mathbb{R} \\\text{The artificial neuron output is } y = \\sigma(z) \\in \\mathbb{R} \\\text{Typical activations: } \\sigma(z) = \\mathbf{1}_{z \\ge 0} \text{ (step)}, \\quad \\sigma(z) = (1+e^{-z})^{-1} \text{ (sigmoid)}, \\quad \\sigma(z) = \\max(0,z) \text{ (ReLU)}\\end{aligned}</span><footer><span class=\"wb-embed-source\">Rong Martin-Siebler ZHOU</span></footer>","width":640,"height":320}