PIECEWISE-POLYNOMIAL APPROXIMATIONS OF FUNCTIONS OF THE CLASSES $ W_{p}^{\alpha}$

By M. Birman, M. Solomjak. 1967

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Variance-based Regularization with Convex Objectives (2016, background): “Kernel operators underlying Sobolev spaces with different smoothness orders [9, 23] typically have eigenvalues scaling as λj .”
Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization (2008, neutral): “Defining the “distance” d(g0, g) := ∫ (g − g0)dQ − log gg0dP, the following statements hold: (i) For any g ∈ G, we have d(g0, g) ≥ 2h2Q(g, g0).”
On the mathematical foundations of learning (2001, neutral): “A main result —of a kind going back to the work of Birman and Solomyak [5]— concerning entropy numbers of Sobolev spaces states that, if X ⊂ R is a compact domain with smooth (C∞) boundary and s > n/2, then, for all k ≥ 1,”
Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates (2013, background): “Then the mean-squared error is bounded as The upper bound (19) is unimprovable up to constant factors, as shown by known minimax bounds on estimation error in Sobolev spaces (Stone, 1982; Tsybakov, 2009); see also Theorem 2(b) of Raskutti et al. (2012).”
Chapter 76 Large Sample Sieve Estimation of Semi-Nonparametric Models (2007, unknown): “…the approximation error kπ n θ o − θ o k = 0, we have k b θ n − θ o k = O P (δ n ), where δ n solves: 1 √ nδ 2 n Z δ n bδ 2 n v u u t q X j=1 log N (w, H j , || · || ∞ )dw ≤ 1 √ nδ 2 n Z δ n bδ 2 n v u u t q X j=1 ³ c j w ´ 1/p j dw by Birman and Solomjak (1967) ≤ 1 √ nδ 2 n max j=1,...,q const.”
Lower bounds on minimax rates for nonparametric regression with additive sparsity and smoothness (2009, unknown): “For the sparse additive model(2) with univariate function spaceH such that such that log M(ǫ;H) = Θ(ǫ−1/m), we have min bf max f∗∈F0(s) E‖f̂ − f∗‖2L2(P) ≥ max [ σ2s log(p/s) 32n ,C s (σ2 n ) 2m 2m+1 ] , (8) for someC > 0.”
Divide and Conquer Kernel Ridge Regression (2013, unknown): “Kernels with polynomially-decaying eigenvalues include those that underlie for the Sobolev spaces with different smoothness orders (e.g. Birman and Solomjak, 1967; Gu, 2002).”
Testing Conditional Independence via Rosenblatt Transforms (2009, background): “Hence by the result of Birman and Solomjak (1967), for any q ≥ 1 logN[·](ε,T ,Lq(P )) ≤ C2 ε (13) for a constant C2 > 0 that does not depend on P .”
Estimating conditional quantiles with the help of the pinball loss (2011, background): “Analogously, ifm > d/2 is some integer, then the Sobolev spaceH := W(X) is an RKHS that satisfies (6) for % := d 2m , and this estimate is also asymptotically sharp, see [5, 11].”
Fast learning rates for plug-in classifiers (2005, background): “It follows from [4] that Assumption (CAR) for this class is satisﬁed with ρ = 1 for any 1 ≤ p < ∞ .”