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Abstract:
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In the standard Gaussian linear measurement model $Y=X\mu_0+\xi \in \R^m$ with a fixed noise level $\sigma>0$, we consider the problem of estimating the unknown signal $\mu_0$ under a convex constraint $\mu_0 \in K$, where $K$ is a closed convex set in $\R^n$. We show that the risk of the natural convex constrained least squares estimator (LSE) $\hat{\mu}(\sigma)$ can be characterized exactly in high dimensional limits, by that of the convex constrained LSE $\hat{\mu}_K^{\seq}$ in the corresponding Gaussian sequence model at a different noise level via a fixed point equation. The characterization holds (uniformly) for risks $r_n^2$ in the maximal regime that ranges from constant order all the way down to essentially the parametric rate, as long as certain necessary non-degeneracy condition is satisfied for $\hat{\mu}(\sigma)$.
The precise risk characterization reveals a fundamental difference between noiseless and noisy linear inverse problems in terms of the sample complexity for signal recovery. Several examples, including non-negative least squares, shape-constrained regression and constrained Lasso, are worked out to demonstrate the concrete applicability of the theory.
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