On the Best Constants in the Khintchine Inequality

By Johannes Wissel. 2007

229 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/04b4e9a13b24e2a3d7d3e8b28bd5e818af039d97

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Citations in the corpus

Hubs in Space: Popular Nearest Neighbors in High-Dimensional Data (2010, unknown): “The equalityxΓ(x) = Γ(x+1) and the convexity of logΓ(x) yield (Haagerup, 1982, p. 237): Γ ( d+1 2 )2 ≤ Γ ( d 2 ) Γ ( d+2 2 ) = d 2 Γ ( d 2 )2 , 13.”
Smallest singular value of random matrices and geometry of random polytopes (2005, background): “Proof: By the symmetry of »i’s and Khinchine’s inequality ([ H ]),”
Zaremba’s conjecture (2011, methodology): “1=p ˝ 0 @ X !2SL 2(q) 1 2T X "2f 1gT p 1 T XT j=1 " jf !( j;˘ j) 1 A 1=p ˝ 0 @ X !2SL 2(q) pp=2 XT j=1 ! f ( j;˘) T 2 ! p=2 1 A 1=p ˝ q3=pp1=2T 1=2; where we applied Khintchine’s inequality [Haa81] (the implied constant is absolute). Now we choose p= logq, so that D q˝(logq)1=2T 1=2 (logQ)”
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals (2009, background): “The following sharp version is due to Haagerup [60].”
Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers (2017, background): “Here (i) is Khintchine’s inequality (Haagerup, 1981) and (ii) is the assumed first moment bound. It remains to bound (11). The key is the following inequality asserting that the dual norm ‖ · ‖∗ is strongly smooth whenever ‖ · ‖ is strongly convex (c.f. Lemma 17 of Shalev-Shwartz (2007)): Lemma 14 I”
Norm-Based Capacity Control in Neural Networks (2015, unknown): “The sharp value of the constant C p was found by Haagerup (1981) but for our analysis, it is enough to note that if p ≥ 1 we have C p ≤ √ p .”
Low-Rank Approximation and Regression in Input Sparsity Time (2012, methodology): “We will need the upper bound part of Khintchine’s inequality, using a constant factor following from Haagerup’s bound [Haagerup 1981].”
A NEW METHOD OF NORMAL APPROXIMATION (2006, unknown): “Finally, by Khintchine’s inequality [26], we get”
Prediction, learning, and games (2006, background): “The form of Khinchine’s inequality cited here is due to Szarek [281].”
The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{\frac{\log n}{n}} (2013, background): “The best constant AR,p and AC,p are known (see [14] and [17]).”
Grothendieck's Theorem, past and present (2011, background): “[150, 48, 145]), they are respectively cp = 2 1 p − 1 2 for 1 ≤ p ≤ p0 and cp = ‖gR 1 ‖p = for p0 ≤ p <∞, where 1 < p0 < 2 is determined by the equality 2 1 p0 − 1 2 = ‖gR 1 ‖p0 .”
An Improvement to Zaremba’s Conjecture (2013, background): “SL2(q) !!0 (mod q0) T j=1 f q;!(s j;s 0 j ) p 1 CCC CCC CC CCC CA 1 : By the concativity of x1=p, the symmetric property of f q;!(s;s0), and Kahane contraction principle with Khintchine’s inequality [Haa82] , (A.16) becomes E s(D s) + 1 T E sE s0 0 BBB BBB BBB BBB BBB @ 0 BBB BBB BBB BBB B@ X q2D !2”
Graphons, cut norm and distance, couplings and rearrangements (2010, unknown): “See also [32] for related results.”
Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators (2013, background): “The best constants AR,p and AC,p are known (see [15] and [18]): • AR,p = ”
A Vector-Contraction Inequality for Rademacher Complexities (2016, background): “For Rademacher variables the best constant is C = √ 2 ([19], see also inequality (4.3) in [10] or Theorem 5.20 in [ ?”
Implicit Regularization in Deep Learning (2017, methodology): “The sharp value of the constant Cp was found by Haagerup [54] but for our analysis, it is enough to note that if p ≥ 1 we have Cp ≤ √p.”