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Given a vector of errors $e(x)$ obtained by variable $x$

In the following problem :

$min_x || e(x) ||$

Besides the robustness, consider only convergence speed, is it l1 norm works better than l2 norm using projected gradient method?

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l1 norms typically work better in Hamming space (boolean, binary lattices-space) whereas l2 norms typically work better for real numbers in a real valued space.

The reason for this is that in an integer lattice, a square root value may not make sense as the shortest path...

A norm typically defines a shortest path between two points. To figure out what is the best norm, you would need to figure out the ambient space where your data lies.

For a min type of function, an l_\infty norm can be practical whenever the problem can be reexpressed from min to max by flipping the values according to the highest possible value in the function space.

In CS, that bound is nothing more than the max integer value sometimes...

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