  Many regularization schemes for high-dimensional regression have been put
forward. Most require the choice of a tuning parameter, using model selection
criteria or cross-validation schemes. We show that a simple non-negative or
sign-constrained least squares is a very simple and effective regularization
technique for a certain class of high-dimensional regression problems. The sign
constraint has to be derived via prior knowledge or an initial estimator but no
further tuning or cross-validation is necessary. The success depends on
conditions that are easy to check in practice. A sufficient condition for our
results is that most variables with the same sign constraint are positively
correlated. For a sparse optimal predictor, a non-asymptotic bound on the
L1-error of the regression coefficients is then proven. Without using any
further regularization, the regression vector can be estimated consistently as
long as \log(p) s/n -> 0 for n -> \infty, where s is the sparsity of the
optimal regression vector, p the number of variables and n sample size. Network
tomography is shown to be an application where the necessary conditions for
success of non-negative least squares are naturally fulfilled and empirical
results confirm the effectiveness of the sign constraint for sparse recovery.
