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In continuation to this question, Non linear to linearly seperable I am trying to understand if we do not want to transform to a higher dimension is it possible to still find a feature map that maps non linearly separable data into linearly separable data. If yes whats the intuition. For example in the image below I am trying to check if there is any 1D transformation I can use to find linearly separable data. Also can we find transformation in lower dimension. Any reference material to read would be helpful as well.

Example if it can be transformed to linearly separable data

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  • $\begingroup$ Good question; note that it's not enough just to have some map, we need one that's continuous(ly differentiable). $\endgroup$
    – John Madden
    Nov 14, 2022 at 14:41
  • $\begingroup$ by assumption there is a mapping from a given data point to its class, and using the mapping, the problem is linearly separable. ie if my feature already solves the class mapping. $\endgroup$
    – seanv507
    Dec 5, 2022 at 1:03

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Every case is different, and it's not generally possible for every data set, but there's a very simple transformation that makes your data linearly separable:

x = 1:20
y = c(0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1)
f = function(x) (x - 1) %% 4

plot(x, y, main = 'Not linearly separable')

enter image description here

plot(f(x), y, main = 'Linearly seperable')

enter image description here

x %% 4 here is the modulo or remainder operator: the remainder when x is divided by 4.

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Yes, it is sometimes possible. In the example you give, a point $t$ is class X precisely when $\sin(\pi t)>0$.

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