Smooth Discrimination Analysis

By Enno Mammen, Alexandre B. Tsybakov. 1999

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Citations in the corpus

Variance-based Regularization with Convex Objectives (2016, background): “In many cases, such tradeoffs require either conditions such as the Mammen-Tsybakov noise condition [15, 6] or localization results [3, 2, 17] made possible by curvature conditions that relate the risk and variance.”
Local Rademacher complexities (2005, methodology): “1, is that under the assumptions of the theorem, the minimizer of the empirical loss and of the true loss are close with respect to the L2(P ) and the L2(Pn) distances (this has also been used in [23] and [33, 34]).”
PERFORMANCE GUARANTEES FOR INDIVIDUALIZED TREATMENT RULES (2011, background): “3) is similar to the margin condition in classification [18, 24, 32]; in classification this assumption is often used to obtain sharp upper bounds on the excess 0–1 risk in terms of other surrogate risks [2].”
Woodroofe's One-Armed Bandit Problem Revisited (2009, unknown): “This behavior can be quantified by a condition similar to the so-called Tsybakov margin condition in classification [cf. Mammen and Tsybakov (1999), Tsybakov (2004a)].”
Theory of classification : a survey of some recent advances (2005, methodology): “For regression, Birgé and Massart [36] and Lee, Bartlett and Williamson [134] proved exponential inequalities The fact that this phenomenon also occurs in the context of discriminant analysis and classification, under conditions on the noise (sometimes called margin) has been pointed out by Mammen a”
Empirical Processes in M-Estimation (2000, unknown): “(Mammen and Tsybakov(1999)) Let Ĝn = arg min G∈G Rn(G), where G is a class of subsets of X , satisfying for some constants c > 0 and 0 < ρ < 1, HB(δ,G, Q) ≤ cδ−ρ, δ > 0.”
User-friendly introduction to PAC-Bayes bounds (2021, background): “Mammen and Tsybakov [131] proved that, if |η(Xi)−1/2| ≥ τ almost surely for some τ > 0, then Bernstein assumption holds for some K that depends on τ .”
Bandwidth choice for nonparametric classification (2005, background): “If Bayes risk-based classifiers use kernel estimators, or related nonparametric methods based on places where densities cross, then they converge at slower rates than n−1, which are nevertheless minimax-optimal [e.g., Marron (1983) and Mammen and Tsybakov (1999)].”
Empirical Processes in M-Estimation (2001, unknown): “(Mammen and Tsybakov(1999)) Let Ĝn = argmin G∈G Rn(G), where G is a class of subsets of X , satisfying for some constants c > 0 and 0 < ρ < 1, HB(δ,G, Q) ≤ cδ−ρ, δ > 0.”
Optimal rates for plug-in estimators of density level sets (2006, methodology): “When Q is the Lebesgue measure on IRd, density level sets typically correspond to minimum volume sets of given P -probability mass, as shown in Polonik (1997). Remark 1.”
A high-dimensional Wilks phenomenon (2011, background): “The possibility to derive interesting bounds (fast rates) for the excess risk critically relies on the next assumption (Mammen and Tsybakov 1999; Massart 2000b; Koltchinskii 2006; Bartlett and Mendelson 2006; Massart 2007; Boucheron et al. 2005a).”
Adapting to Unknown Smoothness by Aggregation of Thresholded Wavelet Estimators (2006, background): “This assumption has been first introduced by [43], for the problem of discriminant analysis, and [50], for the classification problem.”
Nonparametric Bandits with Covariates (2010, methodology): “It has been used in Rigollet and Vert (2009) and Goldenshluger and Zeevi (2009).”
Estimating conditional quantiles with the help of the pinball loss (2011, background): “Consequently, our new self-calibration inequality is more general and, modulo the constant √ 2, also sharper It is well-known that self-calibration inequalities for Lipschitz continuous losses lead to variance bounds, which in turn are important for the statistical analysis of ERM approaches, see [1”