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块稀疏信号 ℓ₁ 恢复的精度保证AI 解读

We introduce a general framework to handle structured models (sparse and block-sparse with possibly overlapping blocks). We discuss new methods for their recovery from incomplete observation, corrupted with deterministic and stochastic noise, using block-$\ell_{1}$ regularization. While the current theory provides promising bounds for the recovery errors under a number of different, yet mostly hard to verify conditions, our emphasis is on verifiable conditions on the problem…

正性、单调性与凸性的非参数检验AI 解读

We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the $L_r$-distance, $1 \leq r < \infty$, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity,…