The focus of this paper is the linear model with Gaussian design under convex constraints. Specifically, we study the performance of the constrained least squares estimate. We derive two general results characterizing its performance - one requiring a tangent cone structure, and one which holds in a general setting. We use our general results to analyze three functional shape constrained problems where the signal is generated from an underlying Lipschitz, monotone or convex function. In each of the examples we show specific classes of functions which achieve fast adaptive estimation rates, and we also provide non-adaptive estimation rates which hold for any function. Our results demonstrate that the Lipschitz, monotone and convex constraints allow one to analyze regression problems even in high-dimensional settings where the dimension may scale as the square or fourth degree of the sample size respectively.
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