An Elementary Introduction to Modern Convex Geometry

By Keith Ball. 1997

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Citations in the corpus

Differentially Private Empirical Risk Minimization (2009, neutral): “The volume of the unit ball is Vol(X ) = πd/2 Γ(d 2 +1) (see Ball (1997) for more details).”
Satisfying Real-world Goals with Dataset Constraints (2016, methodology): “This is the Brunn-Minkowski inequality [e.g. Ball, 1997].”
Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems (2012, neutral): “Basically, the theorem states that E has many contact points with K, and these contact points are “nicely” distributed, in the sense that they almost form an orthonormal basis —see Ball [1997] for a proof.”
Sample Complexity Bounds for Differentially Private Learning (2011, methodology): “, 2007; Barak et al., 2007; McSherry and Mironov, 2009). Some work that is relevant to ours include Blum et al. (2008), which provides a general method for publishing datasets on discrete data domains while preserving differential privacy so that the answers to queries from a function class with bou”
Information-theoretic lower bounds for distributed statistical estimation with communication constraints (2013, methodology): “[3] K.”
Bandit Algorithms (2020, background): “A proof of John’s theorem may be found in the short book by Ball [1997].”
Mixed Integer Linear Programming Formulation Techniques (2015, methodology): “For more information on extended formulations in combinatorial optimization we refer the reader to [35, 154, 95] and for more on polyhedral approximations of convex sets we refer the reader to [14, 16, 152] and [93, Chapters 8 and 17].”
On the Sensitivity of Adversarial Robustness to Input Data Distributions (2018, unknown): “Considering robust accuracy as the clean accuracy with a smaller gap between the spheres, the next theorem provides a theoretical guarantee in relating together the difﬁculty of attaining good accuracy under attack and the model capacity (Ball, 1997), verifying our intuition above.”
Thrifty approximations of convex bodies by polytopes (2012, background): “…(cid:9) where k · k is the Euclidean norm, the contact points x i ∈ C ∩ ∂B provide a certain decomposition of the identity operator I , called the John decomposition (recall that x ⊗ x for x ∈ R d is interpreted as a d × d symmetric matrix). result for We need the following , see example, [Ba97].”
Expander flows, geometric embeddings and graph partitioning (2004, unknown): “Levy’s isoperimetric inequality ([5]) states that μ(Aγ)/μ(A) is minimized for spherical caps 2 The”
Geometry of Quantum States: Index (2007, methodology): “The approach of Aubrun, Szarek and Werner [68] is based on Dvoretzky’s theorem [279, 87], which implies that a generic section of a convex body in high dimensions is close to spherical.”
An asymptotic formula for the number of non-negative integer matrices with prescribed row and column sums (2009, unknown): “in y∈A kx− yk. Let A⊂ Bbe a closed set such that P(A) ≥ 1/2. Then, for any r≥ 0 we have P n x∈ B: dist(x,A) ≥ r o ≤ 2e−cr2. Proof. See, for example, Section 2.2of [Le01]or Theorem 8.1and its proof in [B97a], which, although stated for the Gaussian measure is adapted in a straightforward way to our s”
John's Decomposition in the General Case and Applications (2004, methodology): “We refer to [1] and [2] for detailed discussion on John’s ellipsoid and its unicity and to [19] for the case when L is a parallelepiped.”