Biconvex sets and optimization with biconvex functions: a survey and extensions

By Jochen Gorski, Frank Pfeuffer, Kathrin Klamroth. 2007

751 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Organizational Behavior and Human Resource Management

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Citations in the corpus

Federated Multi-Task Learning (2017, background): “M OCHA is based on a bi-convex alternating approach, which is guaranteed to converge [16, 43] to a stationary solution of problem (1).”
Listening to Chaotic Whispers: A Deep Learning Framework for News-oriented Stock Trend Prediction (2017, methodology): “In this paper, we choose the linear regularizer proposed in [11], which linearly discriminates samples with respect to their loss, as: Similarto[11][13],weadoptAlternativeConvexSearch(ACS)[9] to solve Equation (1).”
MentorNet: Learning Data-Driven Curriculum for Very Deep Neural Networks on Corrupted Labels (2017, methodology): “eight with the model parameters [21], leading to a weighting scheme that assigns higher weights to examples of smaller training loss. After that, various weighting schemes have been proposed, e.g. in [38,16,26,34,6,28]. Alternative minimization is commonly used in existing methods to solve the joint”
Reduced rank stochastic regression with a sparse singular value decomposition (2012, background): “A multiconvex optimization problem may have multiple local minima, and its non-convex optimization requires more complicated algorithms (Gorski et al., 2007).”
Projective dictionary pair learning for pattern classification (2014, background): “The convergence of such a problem has already been intensively studied [23], and the proposed optimization algorithm is actually an alternate convex search (ACS) algorithm.”
Adaptive Decontamination of the Training Set: A Unified Formulation for Discriminative Visual Tracking (2016, methodology): “Therefore, we found 2Detailed results are presented in the supplementary material. a single (N = 1) ACS iteration to be sufficient to refine the estimate of θ and α from the previous frame.”
Convolutional Sparse Coding for Image Super-Resolution (2015, methodology): “We alternatively optimize the f and the W sub-problems, which is actually an alternate convex search (ACS) algorithm [10].”
Robust Markov Decision Processes (2013, neutral): “[13] for a detailed convergence analysis.”
Progressive Meta-Learning With Curriculum (2022, methodology): “This strategy is generally used in ACS (Alternative Convex Search) [38].”
Intelligent Reflecting Surface Aided Multigroup Multicast MISO Communication Systems (2019, background): “We note that R̃k(F, e|Fn, e) is biconcave of F and e [24], since R̃k(F|F) = R̃k(F, e|Fn, e) with given e is concave of F and R̃k(e|e) = R̃k(F, e|Fn, e) with given F is concave of e.”
Easy Samples First: Self-paced Reranking for Zero-Example Multimedia Search (2014, methodology): “However, it can be effectively optimized by Cyclic Coordinate Method (CCM) [5].”
MentorNet: Regularizing Very Deep Neural Networks on Corrupted Labels (2017, methodology): “The algorithm can be used to replace the commonly used alternative minimization [13] or majorize-minimization algorithm [42], in order to learn latent variables for deep CNNs.”
Efficient Multitask Feature and Relationship Learning (2017, background): “5) is not easy to solve directly. In this section we ﬁrst show that (5) can be converted into a multi-convex optimization problem. A multi-convex function is a generalization of a bi-convex function (Gorski et al., 2007) into multiple variables or blocks of variables. Deﬁnition 3.1 (Multi-convex fun”
Multi-task Sparse Structure Learning with Gaussian Copula Models (2016, methodology): “and Ω, thus a partial optimum will be found (Gorski et al., 2007).”
Penalized classification using Fisher’s linear discriminant (2011, background): “Moreover, (27) is a biconvex problem (see e.g. Gorski et al., 2007): that is, with β held ﬁxed, it is convex in u , and with u held ﬁxed, it is convex in β .”
Asymmetric multi-task learning based on task relatedness and loss (2016, background): “Further, the tasks that are associated through the regularization graph are mostly intuitively relevant ones.”
Self-Paced Curriculum Learning (2015, background): “(1) (Gorski, Pfeuffer, and Klamroth 2007).”
Distributionally Robust Co-Optimization of Energy and Reserve Dispatch (2016, background): “The convergence property of Algorithm 2 is addressed in [39].”
Scalable Multitask Representation Learning for Scene Classification (2014, background): “guarantees (sub)-convergence to a critical point of the objective of (1), see [7], which is the standard convergence result for non-convex problems.”