Minimax Risk Over Hyperrectangles, and Implications

By David L. Donoho, Richard C. Liu, B. MacGibbon. 1990

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Additive models with trend filtering (2017, methodology): “In fact, we conjecture that the techniques from Donoho et al. (1990), Donoho & Johnstone (1998) could be used to establish for random inputs the same lower bound rate as in (36), but do not pursue this.”
Prior information and uncertainty in inverse problems (2001, background): “Although its exact value is dificult to compute, R(fl)• minffl 2 ; 1g. In addition, Donoho et al. (1990) show that”
Gaussian model selection (2001, background): “(6.7) On the other hand, since p ≥ 2, E(p, a) is orthosymmetric, compact, convex and quadratically convex, according to the terminology of Donoho et al. (1990).”
Kernel Projection Machine: a New Tool for Pattern Recognition (2004, background): “At stage k k g this , it is interesting to realize that the balls in the RKHS space can be viewed as ellipsoids in the original Hilbert space L 2 ([0 ; 1]) .”
Nonquadratic estimators of a quadratic functional (2005, background): “ONQUADRATIC ESTIMATORS OF A QUADRATIC FUNCTIONAL 7 convex hulls of general Besov balls and L p balls. In fact, when p<2 it is easy to check that Q.Hull(L p(α,M))=L 2 α+ 1 2 − 1 p ,M (12) . See [10]. Similarly it is easy to check that Q.Hull(Bα p,q(M))=B α+1/2−1/p (13) 2,q (M). Write r∗(Θ) for t”
Wavelet Shrinkage: Asymptopia? (1995, background): “Actually, it is nearly minimax among all estimates [36, 25, 24, 14].”
Thresholding estimators for linear inverse problems and deconvolutions (2003, background): “Wavelet thresholding estimators have been applied by Donoho [8] to deconvolution problems Y [n] = θ u[n] + ν[n].”
On automatic boundary corrections (1997, background): “Furthermore, with p+1; de ned by (8), Tp+1; (K0) = p+1; p+1; !(2p+3)=2 ; (10) where p+1; = 2p + 3 2 + 1 (p + + 2)! (p + 1)! !!2 r 2(p+ + 2)!r (p + 1)! (2p + 3)!!2s : (11) Let the resulting equivalent kernel (see (2)) of K0 be Kopt p+1; (t) !eT S 1 (1; t; :::; tp)TK0 (t) =p+1 Xj=0 jtjI[0;1] (t) : (12”
Minimax estimation via wavelet shrinkage (1998, methodology): “We observe data according to the sequence model (6), only now ( I) is a random variable, which may be arbitrary except for the single constraint that jj jjbsp;q C; (10) where is a moment sequence de ned by I = (Ej I jp^q)1=p^q I 2 I: (if p ^ q = 1 we put I = ess sup j I j.”