感知器的几何表示（人工神经网络）

Question

I am taking this course on Neural networks in Coursera by Geoffrey Hinton (not current).

I have a very basic doubt on weight spaces. https://d396qusza40orc.cloudfront.net/neuralnets/lecture_slides%2Flec2.pdf Page 18.

If I have a weight vector (bias is 0) as [w1=1,w2=2] and training case as {1,2,-1} and {2,1,1} where I guess {1,2} and {2,1} are the input vectors. How can it be represented geometrically?

I am unable to visualize it? Why is training case giving a plane which divides the weight space into 2? Could somebody explain this in a coordinate axes of 3 dimensions?

The following is the text from the ppt:

1.Weight-space has one dimension per weight.

2.A point in the space has particular setting for all the weights.

3.Assuming that we have eliminated the threshold each hyperplane could be represented as a hyperplane through the origin.

My doubt is in the third point above. Kindly help me understand.

Answer 1

It's probably easier to explain if you look deeper into the math. Basically what a single layer of a neural net is performing some function on your input vector transforming it into a different vector space.

You don't want to jump right into thinking of this in 3-dimensions. Start smaller, it's easy to make diagrams in 1-2 dimensions, and nearly impossible to draw anything worthwhile in 3 dimensions (unless you're a brilliant artist), and being able to sketch this stuff out is invaluable.

Let's take the simplest case, where you're taking in an input vector of length 2, you have a weight vector of dimension 2x1 , which implies an output vector of length one (effectively a scalar)

In this case it's pretty easy to imagine that you've got something of the form:

input = [x, y]
weight = [a, b]
output = ax + by

If we assume that weight = [1, 3] , we can see, and hopefully intuit that the response of our perceptron will be something like this:

With the behavior being largely unchanged for different values of the weight vector.

It's easy to imagine then, that if you're constraining your output to a binary space, there is a plane, maybe 0.5 units above the one shown above that constitutes your "decision boundary".

As you move into higher dimensions this becomes harder and harder to visualize, but if you imagine that that plane shown isn't merely a 2-d plane, but an nd plane or a hyperplane, you can imagine that this same process happens.

Since actually creating the hyperplane requires either the input or output to be fixed, you can think of giving your perceptron a single training value as creating a "fixed" [x,y] value. This can be used to create a hyperplane. Sadly, this cannot be effectively be visualized as 4-d drawings are not really feasible in browser.

Hope that clears things up, let me know if you have more questions.

Answer 2

I have encountered this question on SO while preparing a large article on linear combinations (it's in Russian, https://habrahabr.ru/post/324736/ ). It has a section on the weight space and I would like to share some thoughts from it.

Let's take a simple case of linearly separable dataset with two classes, red and green:

The illustration above is in the dataspace X, where samples are represented by points and weight coefficients constitutes a line. It could be conveyed by the following formula:

w^T * x + b = 0

But we can rewrite it vice-versa making x component a vector-coefficient and w a vector-variable:

x^T * w + b = 0

because dot product is symmetrical. Now it could be visualized in the weight space the following way:

where red and green lines are the samples and blue point is the weight.

More possible weights are limited to the area below (shown in magenta):

which could be visualized in dataspace X as:

Hope it clarifies dataspace/weightspace correlation a bit. Feel free to ask questions, will be glad to explain in more detail.

Answer 3

The "decision boundary" for a single layer perceptron is a plane (hyper plane)

where n in the image is the weight vector w , in your case w={w1=1,w2=2}=(1,2) and the direction specifies which side is the right side. n is orthogonal (90 degrees) to the plane)

A plane always splits a space into 2 naturally (extend the plane to infinity in each direction)

you can also try to input different value into the perceptron and try to find where the response is zero (only on the decision boundary).

Recommend you read up on linear algebra to understand it better: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces

Answer 4

For a perceptron with 1 input & 1 output layer, there can only be 1 LINEAR hyperplane. And since there is no bias , the hyperplane won't be able to shift in an axis and so it will always share the same origin point. However, if there is a bias, they may not share a same point anymore.

Answer 5

I think the reason why a training case can be represented as a hyperplane because... Let's say [j,k] is the weight vector and [m,n] is the training-input

training-output = jm + kn

Given that a training case in this perspective is fixed and the weights varies, the training-input (m, n) becomes the coefficient and the weights (j, k) become the variables. Just as in any text book where z = ax + by is a plane, training-output = jm + kn is also a plane defined by training-output, m, and n.

Answer 6

Equation of a plane passing through origin is written in the form:

ax+by+cz=0

If a=1,b=2,c=3;Equation of the plane can be written as:

x+2y+3z=0

So,in the XYZ plane,Equation: x+2y+3z=0

Now,in the weight space;every dimension will represent a weight.So,if the perceptron has 10 weights,Weight space will be 10 dimensional.

Equation of the perceptron: ax+by+cz<=0 ==> Class 0

                          ax+by+cz>0  ==> Class 1

In this case;a,b & c are the weights.x,y & z are the input features.

In the weight space;a,b & c are the variables(axis).

So,for every training example;for eg: (x,y,z)=(2,3,4);a hyperplane would be formed in the weight space whose equation would be:

2a+3b+4c=0

passing through the origin.

I hope,now,you understand it.

Answer 7

Consider we have 2 weights. So w = [w1, w2] . Suppose we have input x = [x1, x2] = [1, 2] . If you use the weight to do a prediction, you have z = w1*x1 + w2*x2 and prediction y = z > 0 ? 1 : 0 y = z > 0 ? 1 : 0 .

Suppose the label for the input x is 1. Thus, we hope y = 1, and thus we want z = w1*x1 + w2*x2 > 0 . Consider vector multiplication, z = (w ^ T)x . So we want (w ^ T)x > 0 . The geometric interpretation of this expression is that the angle between w and x is less than 90 degree. For example, the green vector is a candidate for w that would give the correct prediction of 1 in this case. Actually, any vector that lies on the same side, with respect to the line of w1 + 2 * w2 = 0 , as the green vector would give the correct solution. However, if it lies on the other side as the red vector does, then it would give the wrong answer. However, suppose the label is 0. Then the case would just be the reverse.

The above case gives the intuition understand and just illustrates the 3 points in the lecture slide. The testing case x determines the plane, and depending on the label, the weight vector must lie on one particular side of the plane to give the correct answer.