A robust guarantee we will use for Jennrich's algorithm [ Finally, recall Fact A.3, the derivatives of a Gaussian p.d.f. First, 8 k 2 N, by Turán's inequality [ Tur50 ] we have He This yields the required contradiction for the first claim. Since there is no error in the tensor, Jennrich's From Lemma 4.3, we know that Hence, Theorem 4.2 can be applied to recover for all All the Hermite coefficients are hence equal (and the functions are squared-integrable w.r.t. the Gaussian measure for bounded From Lemma 3.5, it is easy to verify that all the even Hermite coefficients are equal, and the odd In this section, we prove Algorithm 4 and its algorithmic guarantee in Theorem D.1 (and Theorems 3.1 We break down the proof into multiple parts. Equipped with the essential claim, we are now ready to prove Lemma D.3 . On the other hand, from Hölder's inequality, we have Hence, again combined with Claim D.2, we have E [ Z D.2 Recovering the parameters under errors Suppose "> 0 is the desired recovery error. Before we proceed to the proof of this lemma, we first state and prove a couple of simple claims. Hence, we get that " We now proceed to the proof of Lemma D.4 .

Hierarchical Bayesian methods enable information sharing across regression problems on multiple groups of data.

This constant can sometimes be prohibitively large. We show that this approach has advantages in auto-tuning to the ill-posedness of the problem. Darolles et al. [ 2011 ] consider the closed form solution to the minimization problem, which takes the Darolles et al. [ 2011 ] take the latter approach to estimation by first estimating the conditional operators The crucial assumption in this line of work (see e.g. Hall et al. [ 2005 ], Darolles et al. [ 2011 ]) is what Our estimator adapts to these two quantities and automatically and optimally balances them, by imposing an RKHS norm penalty. Our work on sparse linear hypotheses provides a minimax formulation alternative to the Dantzig selector.

PO is often higher than the cost of an FO call.