The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming

By L. Bregman. 1967

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Citations in the corpus

Adaptive Gradient-Based Meta-Learning Methods (2019, positive): “The task-similarity will be measured using the Bregman divergence BR(θ||φ) = R(θ) − R(φ) − 〈∇R(φ), θ − φ〉 of a 1-strongly-convex function R : Θ 7→ R [10], a generalized notion of distance.”
Proper local scoring rules (2011, positive): “It follows that H(P ) = ∫ dµ(x)φ{p(x)} and so, assuming H(P ) is finite, the corresponding (separable) Bregman divergence [Bregman (1967), Csiszár (1991)]—also termed U -divergence [Eguchi (2008)]—is d(P,Q) = ∫ dµ(x) ([φ{q(x)}+ {p(x)− q(x)}φ′{q(x)}]− φ{p(x)}).”
A Comprehensive Survey on Transfer Learning (2019, neutral): “Measurement Related Algorithms Maximum Mean Discrepancy [28] [29] [30] [31] [32]· · · Kullback-Leibler Divergence [33] [34] [35] [36] [37]· · · Jensen-Shannon Divergence [38] [39] [40] [41] [42]· · · Bregman Divergence [43] [44] [45] [46] [47]· · · Hilbert-Schmidt Independence Criterion [48] [29] [4”
Computational Optimal Transport (2018, methodology): “For discrete vectors in Rn, a Bregman divergence [Bregman, 1967] associated to a smooth strictly convex function ψ : Rn → R is defined as”
A Survey on Metric Learning for Feature Vectors and Structured Data (2013, background): “Wu et al. (2009, 2012) propose to learn Bregman distances (or Bregman divergences), a family of metrics that do not necessarily satisfy the triangle inequality or symmetry (Bregman, 1967).”
Piecewise Flat Embedding for Image Segmentation (2018, methodology): “Following the derivation of the SOC algorithm, we deﬁne P = D 1 / 2 Y , and rewrite (8) as, which can be iteratively solved using Bregman iterations [60] as follows: (a) Y ( k +1) = arg min (b) P ( k +1) = arg min where B is an auxiliary matrix, and B (0) = 0 .”
Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems (2009, methodology): “This notion of distance goes back to Bregman [6] and has successfully been used in investigations of problems in Banach space settings, see, e.”
Iterative Bregman Projections for Regularized Transportation Problems (2014, background): “This problem can then be solved eﬃciently using either Bregman iterative projection (for intersection of aﬃne spaces) or a more general Dykstra-like algorithm—these being well-known ﬁrst order non-smooth optimization schemes.”
The geometry of proper scoring rules (2007, unknown): “The corresponding Bregman divergence (Bregman, 1967; Csiszár, 1991) is d(P, Q) = ∫”
Tracking the Best Linear Predictor (2001, methodology): “As is the case with Euclidean distance, the projection of a point onto a hyperplane allows a Pythagorean-type theorem (Bregman, 1967) to be proven.”
Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers (2011, unknown): “A considerable body of recent research considers replacing the quadratic penalty term in the standard method of multipliers with a more general deviation penalty, such as one derived from a Bregman divergence [30, 58]; see [22] for background material.”
Entropic Approximation of Wasserstein Gradient Flows (2015, background): “It has been extended to the general setting of Bregman’s divergences by Bregman [20].”
Proximal Reinforcement Learning: A New Theory of Sequential Decision Making in Primal-Dual Spaces (2014, methodology): “This problem originally motivated the development of Bregman divergences [4].”