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Sunday, January 3, 2016

Random thoughts on the Lasso (to be continued...)

For any n-by-p matrix X (with non-zero rows), consider the following objects
\mathcal P_X := \{z \in \mathbb R^n | \|X^Tz\|_\infty \le 1\},\; D_X := \text{diag}(\|X_1\|_\infty,\ldots, \|X_n\|_\infty),\; Z_{D_X} := D_X^{-1}\mathbb B_1,
where \mathbb B_1 is the unit-ball for the \ell_1-norm.  Note that Z_{D_X} \subseteq \mathcal P_X.



Given a closed convex set K \subseteq \mathbb R^n, we've defined the euclidean projection
\text{proj}_K(a) := \text{the unique point of }K\text{ minimizing distance from }a \text{ to } K.
It's not (too) hard prove that 0 \le QP \le QA.

What more (of geometric taste) can be said about the picture ?