L1 Norm
The sum of absolute values of a vector's components, also called Manhattan or taxicab length.
Think of It Like This
The distance a taxi drives on a grid, not the distance a bird flies.
The L1 norm is . For it is 7, against an L2 norm of 5.
Its defining behaviour as a penalty is that it produces exact zeros, not just small values. The reason is geometric: the set of vectors with L1 norm 1 is a diamond whose corners sit precisely on the axes, and a corner is where one coordinate is exactly zero. A constrained optimum is far more likely to land on a spike than on a flat face, so L1 regularisation performs feature selection as a side effect. That is what lasso regression is.
The cost is that those same corners are points where the derivative does not exist, so L1 penalties need optimisers tolerant of a subgradient and are less smooth to train than L2. It is also more robust to outliers than L2, because it grows linearly with error rather than quadratically.