- Behavior Of A Perceptron A Perceptron Is A Very Simple 1 Layer Neural Network It Takes As Its Inputs A Vector Of Real V 1 (111.65 KiB) Viewed 177 times
Behavior of a Perceptron A perceptron is a very simple 1-layer neural network. It takes as its inputs a vector of real-valued scalars x = (1, 2,...,n), multiplies each input by a weight w = = (wi, w2,..., wn), and then processes it through an activation function f to produce an output between 0 and 1. This process can be thought of taking a set of input data, taking a weighted sum of data, and making an approximately-binary decision based on said data. Alternatively, the perceptron can be thought of as a linear classifier. The following figure depicts the layout of this perceptron. This perceptron can thus be expressed by the following expression: z = f(w.x) 1 X₂ ⠀ (2n) weichts W₁ V/₂ Activation function fry) Z Sum cutput Inputs Figure 1: Basic Perceptron 1. Consider a network with three inputs 2₁, 22, 23. (a) Find the general formula for the 3 first partial derivatives of z w.r.t. the inputs z,. (b) Give the general formula for the total differential of z. (Note: I don't want the definition of a total differential.) (c) Consider the differentiable activation function given by f(y) = y/√1+ y² and the weight vector w= (2,-2,1). What are he first partial derivatives of z w.r.t. each ₁. (d) Say that there is some error in the inputs to the perceptron in part (c) given by Ax = (Axı, Ax2, Ax3) = (±0.01, +0.005, +0.01). Find the approximate error in the output z when x = (3,-3, -1/2), x = =(3,3, -1/2), and x = (-3, 3, -1/2). For which point (s) is z the most sensitive to error in its inputs? 2. Consider next an extended network like the one depicted below, where the variables #₁, #2, 3 are now governed by space and time by z = gi(s, t). (a) Find the general formula for the 2 first partial derivatives of z w.r.t. the inputs s and t. (b) Give the general formula for the total differential of z. (c) Using the same activation function and weight vector as in
question 1, let the g₁ (s, t) = st, 92(s, t) = s-t, and g3 (s, t) = s²+t². What are the first partial derivatives of z w.r.t. s and t. (d) Say that there is some error in the inputs to the perceptron in part (c) given by As = ±0.01 and At = ±0.05. Find the approximate error in the output z when (s, t) = (1,1) and (s, t) (2,-2). 3
g.15,6) Time/Space weights W₁ (an) V/₂ V Activation function fry) Sum Imputs Figure 2: Extended Perceptron Z output