Remarques sur un résultat non publié de B. Maurey

By G. Pisier. 1981

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Deep Neural Networks for Estimation and Inference: Application to Causal Effects and Other Semiparametric Estimands (2018, methodology): “In particular, Chen and White (1999) obtain sufficiently fast rates by following the approach of Barron (1993) in using Maurey’s method (Pisier, 1981) for approximation, but applying the refinement of Makovoz (1996).”
Spectrally-normalized margin bounds for neural networks (2017, methodology): “…b, (cid:15) and positive integer p be given, along with matrix X ∈ R n × d with The proof of Lemma 3.4 relies upon the Maurey sparsiﬁcation lemma (Pisier, 1980) which is stated in terms of sparsifying convex hulls, and in its use here is inspired by covering number bounds for linear predictors…”
Why ResNet Works? Residuals Generalize (2019, methodology): “This covering bound relies on Maurey sparsi cation lemma [43] and has been introduced in machine learning by previous works (see, e.”
Tight Bounds for Approximate Carathéodory and Beyond (2015, methodology): “When we approximate f ( y ) − f ( x ) ≈ ∇ f ( x ) ⊤ ( y − x ), the second term is a dot-product of a vector in the domain y − x , which we call the primal space and measure using k·k norm and a gradient vector, which we call the dual space and measure with dual norm k·k ∗ .”
Neural Networks for Control (1993, methodology): “The main tool is the following Lemma, which is often attributed to Maurey (see [29]): Let G be a bounded subset of a Hilbert space H, with ‖g‖ ≤ γ for all g ∈ G, and let Gn be the subset of H consisting of all convex combinations of at most n elements of G.”
Best subset selection, persistence in high-dimensional statistical learning and optimization under l1 constraint (2006, background): “It is a rephrasing of a result by Maurey (see [22]; a version of it may be found in [13], Lemma 4, and in [16], Proposition 2.”
The Convex Geometry of Linear Inverse Problems (2010, background): “Atomic norms have found fruitful applications in problems in approximation theory of various function classes [3, 24, 44, 59].”
Improved Rates and Asymptotic Normality for Nonparametric Neural Network Estimators (1999, methodology): “…rates by applying a fundamental approximation property of convex combinations of families of functions in a Hilbert space, i.e., Maurey's theorem in Pisier (1981). which can be stated loo~ely a\! follow~. whp.n the t~rget ftmction (gay I: belongs to the closure of the convex hull of a symmetric…”
Learning Halfspaces and Neural Networks with Random Initialization (2015, background): “Then for any point v is in the convex hull of G , there is a point v s in the convex hull of See the paper by Pisier (1980) for a proof.”
Discrete Fourier analysis (2013, unknown): “The result on short approximations can be vastly generalised; the proof given above essentially constitutes a proof of Maurey’s lemma in Banach space theory [8].”
Nonlinear Approximation with Dictionaries I. Direct Estimates (2004, unknown): “Maurey [Pis81] proved the Jackson inequality σm(f,D)X ≤ Cm|f |Kττ (D), m ≥ 1, α = 1/τ − 1/p (4) for D an arbitrary dictionary in X a Hilbert space, p = 2 and τ = 1 (α = 1/2), and Jones [Jon92] proved that the relaxed greedy algorithm reaches this rate of approximation.”