Clarification on bias of a perceptron - neural-network

Question

Isn't it true that if a bias is not present, a line passing through origin should be able to linearly separate the two data sets??
But the most popular answer in this -->> question says
y
^
| - + \\ +
| - +\\ + +
| - - \\ +
| - - + \\ +
---------------------> x
stuck like this
I am confused about it. Do you mean the origins in figure above are somewhere in middle of x-axis and y-axis? Can somebody please help me and clarify this?

Answer 1

Alright, so perhaps the original ASCII graph was not 100% accurate! Let me try to depict this again:
y y
^ ^
- + \\ | + -\\+ | +
- +\\ | + + - \\ + | + +
- - \\ | + - - \\ | +
- - + \\| + - - \\+ | +
------------------------> x ---------------------------> x
- - |\\ + - - \\ | +
- - + | \\ + - - \\ + | +
- - - | \\ + + - - -\\ | + +
-- - - | +\\ ++ -- - - \\ | + ++
stuck like this needs to get like this
y = ax y = ax + b
(w0*x + w1*y = 0) (w0*x + w1*y + w2*1 = 0)

Answer 2

I think your intuition is correct on this issue:
Do you mean the origins in figure above are somewhere in middle of x-axis and y-axis?
In my reading of the graph, yes.
I think the ASCII graph, as cool as it is, is a bit confusing here, because it shows a line that is not traveling through what would normally be considered as the origin. Normally one would think of the intersection of the x- and y-axis lines as the origin, but in this diagram the separating line is clearly not passing through said intersection. As you've noted, a perceptron without a bias term can only define a separating line that passes through the origin, so the ASCII graph must have some sort of odd origin that is floating out in space somewhere.
Also, note that a standard perceptron always defines a linear separator, but a linear separator is not guaranteed to be able to partition a given dataset correctly -- that depends completely on the dataset. There are also variants of the perceptron that use the "kernel trick" to define nonlinear separators, but that's a whole different story. :)
Hope that helps.