A Method for Finding Projections onto the Intersection of Convex Sets in Hilbert Spaces

By James P. Boyle, R. Dykstra. 1986

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Citations in the corpus

The Ideal Continual Learner: An Agent That Never Forgets (2023, methodology): “…of Karczmarz (1937) and Cimmino (1938) for the case where Gt’s are finitely many affine hyperplanes (as Combettes (1993) reviewed); the method of Boyle & Dykstra (1986) for computing a point in the intersection of two convex sets; as well as many extensions of these methods for finitely many…”
The Singular Values of Convolutional Layers (2018, methodology): “, a k × k convolution with bounded operator norm (Cheney and Goldstein (1959) Theorem 4, Boyd and Dattorro (2003) Section 2), and the projection onto that intersection could be found using the more complicated Dykstra’s projection algorithm (Boyle and Dykstra, 1986).”
A convex relaxation approach for computing minimal partitions (2009, methodology): “In order to perform the projection, we use the iterative projection algorithm of Dykstra [6].”
Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory (2017, background): “Projection methods have a long history and are well studied [13, 4].”
Construction of best Bregman approximations in reflexive Banach spaces (2003, background): “3) [4, 7, 8, 11] are based on Dykstra’s algorithm [6, 14] for the metric best approximation problems in Hilbert spaces.”
An introduction to Total Variation for Image Analysis (2009, methodology): “This is performed by alternate projections, following Dykstra’s algorithm for projecting onto intersection of convex sets [17].”
Anomaly Detection with Robust Deep Autoencoders (2017, methodology): “In particular, we are inspired by ideas such as the Alternating Direction Method of Multipiers [4] and R. L. Dykstra’s alternating projection method [5] 2 .”
A Primer on Monotone Operator Methods (2015, methodology): “which resembles but is not the same as Dykstra’s method of alternating projections [8, 11, 21].”
Dykstra's Algorithm, ADMM, and Coordinate Descent: Connections, Insights, and Extensions (2017, methodology): “Meanwhile, Dykstra’s algorithm converges in general (for any closed, convex sets C 1 , . . . , C d with nonempty intersection, see, e.g., Boyle and Dykstra (1986); Han (1988); Gaffke and Mathar (1989)).”
Convex Optimization & Euclidean Distance Geometry (2004, background): “The first successful application of alternating projection to this problem is attributed to Dykstra [73] [43] who in 1983 provided an elegant algorithm that prevails today.”
A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging (2011, methodology): “This can for example be performed by Dykstra’s algorithm [3].”
Subspace Robust Wasserstein distances (2019, methodology): “Indeed, we can project onto the set of constraints R = { Ω ∈ R d × d | 0 Ω I ; trace ( Ω ) = k } using Dykstra’s projection al-gorithm (Boyle & Dykstra, 1986).”
A Dykstra-like algorithm for two monotone operators (2007, background): “2 (Dykstra’s algorithm) [7] Let z ∈ H, let U and V be closed convex subsets of H such that U ∩ V 6= ∅, and set”
Large-Scale Semantic 3D Reconstruction: An Adaptive Multi-resolution Model for Multi-class Volumetric Labeling (2016, background): “In contrast, a Dykstra-like projection scheme [6] avoids storing additional variables altogether, at the cost of a small increase in computation time.”
An algorithm for minimizing the Mumford-Shah functional (2009, methodology): “We therefore use Dykstra’s iterative projection algorithm [6] to compute the projection.”