2.4 Geometric interpretation

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Geometric interpretation

The perceptron can be functionally visualized as



In classification, the model tries to separate classes.

Geometrically:

  • In 2D → Decision boundary is a line

  • In 3D → Decision boundary is a plane

  • In nD → Decision boundary is a hyperplane

Example: Perceptron

Equation:

                    w1x1 + w2x2 + b =0

This represents a line in 2D.


  • In this case, the vector X is an input vector of dimensionality n, x⊂R, whose elements are the values x1,x2...xn.
  •  Note that a ‘1’ is concatenated to the front of any input vector.
  • These values are all multiplied by a corresponding weight, w0 to wn. 
  • w1 to wn are called correlating weights, and w0, which is multiplied against the ‘1,’ is called the bias.
  • These products are all summed, generating a weighted sum of the inputs.
  • Note that this weighted sum is equal to the dot product of the x vector and a vector of the weights w, such that 

This weighted sum is then put through an activation function, which is a function with an output space of {0,1} that has the form:

 

Put simply, this function just checks the sign of its input, and returns 0 if the input is negative, 1 otherwise.

Taken all together, the perceptron classification model is simply


where the left side of the equation is the predicted label for a point, s is the activation function described above, and the dot product w⋅x is equal to the weighted sum of the elements of that point.

 

If the weights are used to define some decision boundary, then the above classification function tells us whether data x is above or below the boundary. This is done mathematically by seeing if w⋅x is greater than or less than zero.

 

Let’s consider the decision boundary of the perceptron algorithm that is given by the equation:


This is the place where the model output changes is the place where the linear combination w⋅x changes sides.





















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