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### How come “”mathematically“” that lasso shrinks the coefficient towards zero whereas ridge does not? [duplicate]

I understand them geometrically !
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### Can lasso and ridge regressions theoretically have exact same solution?

Intuitively lasso leads more sparsity, but is that theoretically possible they have exact same solution vector?
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### Implementing Lasso Regression in Numpy

I'm doing a little self study project, and am trying to implement OLS, Ridge, and Lasso regression from scratch using just Numpy, and am having problems getting this to work with Lasso regression. ...
### $L^1$ Regularization
Let $J(w)$ be some cost function. By adding $L^1$ regularization we get $$\tilde{J}(w) = J(w) + \beta\sum_i|w_i|$$ To study the effect of $L^1$ regularization on the optimum weights, we can ...