An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets

By R. Dykstra. 1985

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In Mathematical Optimization

Citations in the corpus

Mixture-of-Experts with Expert Choice Routing (2022, methodology): “We use Dykstra’s algorithm [9] that alternatively projects the intermediate solution onto one of the convex sets.”
Iterative Bregman Projections for Regularized Transportation Problems (2014, background): “For references in probability and statistics that also address the case of continuous distributions see [25, 30, 51, 9].”
Entropic Approximation of Wasserstein Gradient Flows (2015, background): “The natural extension of iterative projections to generic closed convex sets is the so-called Dykstra’s algorithm [38, 32], which can be interpreted as a block-coordinate optimization on the dual problem.”
Belief Propagation, Dykstra's Algorithm, and Iterated Information Projections (2010, methodology): “An example from [32], however, shows that this need not converge to the correct solution.”
A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps (2016, background): “…in Brown, Chase, and Pittenger 1993: The extension of linear families of probability distributions is still covered by Csiszár 1975, while finding the I-projection for closed, convex but nonlinear constraints requires different means such as Dykstra’s iterative fitting procedure (cf. Dykstra 1985).”
Probabilistic graph and hypergraph matching (2008, background): “For example, if Cj are linear subspaces and f() is the L2 distance then the matrices λj are redundant (also known as ”deflections”) and the scheme is attributed to VonNeumann [13] and for general convex sets it is attributed to Dykstra [9, 4].”
Fast global convergence rates of gradient methods for high-dimensional statistical recovery (2010, methodology): “This is equivalent to projection on the intersection of l1-ball with Ω′ in the updates (3) and (4) and can be done efficiently with Dykstra’s algorithm [15], for instance, as long as the individual projections are efficient.”
Fast global convergence of gradient methods for high-dimensional statistical recovery (2011, methodology): “where I ′(θ) is zero when θ ∈ Ω′ and ∞ otherwise. This is equivalent to projection on the intersection of ℓ1-ball with Ω′ in the updates (3) and (4) and can be done eﬃciently with Dykstra’s algorithm [15], for instance, as long as the individual projections are eﬃcient. 10 In the special case of lin”