Neural networks used in machine learning (vii)

A geometrical view of the Perceptron sensor geometry

Weight-space Weight Space

In this space, each of the weights in the perceptron represents one dimension, whereas a point in space represents a specific set of ownership values, assuming the elimination threshold, each training sample can be viewed as a super-plane through the starting point. So, points in the space correspond to weight vectors and training cases correspond to planes. In other words, the points in space correspond to the weights, and the hyper-plane corresponds to the training samples. Therefore, the weights must be on one side of the super plane to get the correct result (classification) of that training sample.

Take the following figure as an example to illustrate:

Consider only one training sample, this training sample defines a plane, in this 2d figure is the black line, the plane through the origin, and perpendicular to the input vector (blue vector arrows). Weight vectors need to be on the right side of the super plane to get the correct results. He needs to be on the same side as the input vector. The green weight vector and the input vector are not more than 90 degrees, so their point set is positive, so the correct result can be obtained. Conversely, if we have a weighted value such as red, on the wrong side, with an input angle of more than 90 degrees,

The weighted value and the input point set are negative, less than 0, so the perceptron will say no, or 0, in this case the wrong answer.

Another example, the correct result is 0.

In this example, any weight vector with input less than 90 degrees gets more than 0, so it is not the desired result.

Now put these two test samples in a picture

You can see that there is a cone region in which any weight vector can get the correct results for both test samples. If there is a weight vector that can get the correct result for all samples, these weights must be in the hyper-vertebral region and its vertices at the origin. So what the learning algorithm needs to do is consider each of these training samples one at a time, and move the vectors so that they finally reach the cone area. Any two vectors are averaged, and the result is within the conical region of the two vectors. This is a convex problem (convex). In machine learning, if you get a convex learning problem, it's a good solution to make life easy.

Neural networks used in machine learning (vii)