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This figure represents a perceptron model with a 2 dimensional feature vector input.

enter image description here

The hypothesis space of the perceptron is defined by this set:

$\{y | y = w\cdot x + b\}$

What is the geometrical representation of the y value in this perceptron plot? Is it the perpendicular distance from a point to the separating hyperplane?

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  • $\begingroup$ To my interpretation, if we were to ignore the bias term $b$ for a moment, then $y$ is like a cosine similarity measure of the datapoint $x$ with the weight vector $w$. The smaller the angle between the vector $w$ and $x$, the large is the cosine between the two which is captured in the dot product $w.x$. Now if we add back the interpretation of the bias term $b$, it tells us at which value of cosine similarity would you start labeling a data point as class $1$ vs class $-1$. $\endgroup$
    – Deb Nandy
    Commented Jun 7, 2019 at 3:02

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in the context of binary classification, Y indicate the point at ($x_1,x_2$) belongs to positive(1) or negative(-1)

this piece of code is helpful to understand this

import numpy as np
import random
def generate_data(no_points):
    X = np.zeros(shape=(no_points, 2))
    Y = np.zeros(shape=no_points)
    for ii in range(no_points):
        X[ii][0] = random.randint(1,9)+0.5
        X[ii][1] = random.randint(1,9)+0.5
        Y[ii] = 1 if X[ii][0]+X[ii][1] >= 13 else -1
    return X, Y
X, Y = generate_data(100)
plt.scatter(X[:,0],X[:,1],c=Y)

enter image description here

in this figure, positive(1) class is shown as yellow, and negative(-1) class is shown as purple.

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