Understanding Linear Perceptrons and their Decision Boundary in Neural Networks

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In summary, the conversation discusses the relationship between an 8x3 matrix X and its rows spanning R3. It also mentions how the solution to Xw=t can be represented as each row forming a unique plane in R3. This is relevant in understanding Perceptrons, which use linear parameters and decision boundaries to classify data into two classes. The conversation also touches on how Xw=0 forms a hyperplane in x-space depending on the matrix's rank.
  • #1
3.141592654
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^{}This may be a silly question, but if I have an 8x3 matrix X, for example, then the rows of this matrix will span R3 (and will be linearly dependent). When we find the solution to:

Xw=t

where t is an 8x1 matrix of t's. Then each row can be represented as

[itex]w_{1}[/itex][itex]x_{i1}[/itex]+[itex]w_{2}[/itex][itex]x_{i2}[/itex]+[itex]w_{3}[/itex][itex]x_{i3}[/itex] = [itex]t[/itex].

Each row then forms a unique plane in R3, correct? Does the matrix Xw form a plane? I'm learning about Perceptrons, a form of Artificial Neural Network, in which each row of data is classified as either

[itex]y^{'}[/itex] = +1 or -1 depending on if [itex]w_{1}[/itex][itex]x_{i1}[/itex]+[itex]w_{2}[/itex][itex]x_{i2}[/itex]+[itex]w_{3}[/itex][itex]x_{i3}[/itex] > [itex]t[/itex] or [itex]w_{1}[/itex][itex]x_{i1}[/itex]+[itex]w_{2}[/itex][itex]x_{i2}[/itex]+[itex]w_{3}[/itex][itex]x_{i3}[/itex] < [itex]t[/itex].

The book states that in the above situation, "The perceptron model [in the example above] is linear in its parameters w and x. Because of this, the decision boundary of a perceptron, which is obtained by setting [itex]y^{'}[/itex]=0, is a linear hyperplane that separates the data into two classes, -1 and +1."

I'm having a really hard time understanding what this quote is trying to say, because I don't see how Xw=0 "forms a hyperplane" in x-space.
 
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  • #2
Hey 3.141592654.

The answer will depend on the rank of the matrix.

If you row reduce the matrix and get a consistent system with n non-zero rows then the system will form an n-dimensional place from those vectors.

It may be a point, line, or n-dimensional plane depending on the above.
 

Related to Understanding Linear Perceptrons and their Decision Boundary in Neural Networks

1. What is the basic concept behind "Understanding Ax=b"?

The basic concept behind "Understanding Ax=b" is solving a system of linear equations. This involves finding the values of the variables that satisfy all of the equations in the system. The solution can be represented in the form of the matrix equation Ax=b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix.

2. What are the key components needed to solve a system of linear equations using the "Understanding Ax=b" method?

The key components needed to solve a system of linear equations using the "Understanding Ax=b" method are the coefficient matrix, the variable matrix, and the constant matrix. The coefficient matrix contains the coefficients of the variables in each equation, the variable matrix contains the values of the variables, and the constant matrix contains the constants in each equation.

3. How do you determine if a system of linear equations has a unique solution using the "Understanding Ax=b" method?

A system of linear equations has a unique solution if the coefficient matrix has full rank, meaning that the number of linearly independent rows is equal to the number of variables. In other words, there are no redundant equations in the system and it can be solved without any contradictions or inconsistencies.

4. Can the "Understanding Ax=b" method be used to solve non-linear systems of equations?

No, the "Understanding Ax=b" method can only be used to solve systems of linear equations. Non-linear systems of equations require different methods, such as substitution or elimination, to find the solutions.

5. Are there any limitations to the "Understanding Ax=b" method?

Yes, the "Understanding Ax=b" method can only be used to solve systems of linear equations. It also assumes that the variables are directly proportional to each other, meaning that a change in one variable will result in a proportional change in another variable. This may not always be the case in real-world problems, so other methods may need to be used to solve more complex systems of equations.

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