TITLE: Are there variations on least-squares approximations?
QUESTION [7 upvotes]: In least-squares approximations the normal equations act to project a vector existing in N-dimensional space onto a lower dimensional space, where our problem actually lies, thus providing the "best" solution we can hope for (the orthogonal projection of the N-vector onto our solution space).  The "best" solution is the one that minimizes the Euclidean distance (two-norm) between the N-dimensional vector and our lower dimensional space. 
There exist other norms and other spaces besides $\mathbb{R}^d$, what are the analogues of least-squares under a different norm, or in a different space?

REPLY [2 votes]: For regression, least sum of squares criterion tried to fit your function through the mean of the data. In other words, with enough degrees of freedom, value of your fitted function for x will be the average of all observed values at x. Least sum of absolute values instead produces a function that'll go through the median of observed values. Some discussion on this