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Suppose that our data matrix X has a duplicated column, i.e, there is a duplicated feature and the matrix is not full column rank. What happpens?

I guess that we can not find a unique solution because that's the case for the close form in linear regression, but I do not see how to show that intuitively, or even if it is true or not.

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  • $\begingroup$ In what context? What kinds of model? $\endgroup$ – Ben Reiniger Jan 26 at 18:23
  • $\begingroup$ Linear regression $\endgroup$ – vicase98 Jan 26 at 18:39
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In 'Efficient Backprop' by Lecun and others (http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf), they explain why correlated variables are bad (§ 4.3 normalizing the inputs).

Duplicated data is a special case of linear dependence, which is a special case of correlation. Say you have duplicated variables $X1 = X2$, so the network output is constant over the weight lines $w_2 = c - w_1$, where c is a constant. It means that the gradient of the error is 0 along those lines : moving along those lines has no effect on the learning. In that sense the solution won't be unique.

It's bad because you could be solving a problem of lesser dimension. Removing one of the input will leave you with a network of lesser dimension.

Other than the dimension problem, for simple learner I don't think it will affect the learning process that much. For more complex learning processes (learning rate depending on time / on variables), it might get more complex.

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