We propose a new majorization-minimization (MM) method for non-smooth and non-convex programs, which is general enough to include the existing MM methods. Besides the local majorization condition, we only require that the difference between the directional derivatives of the objective function and its surrogate function vanishes when the number of iterations approaches infinity, which is a very weak condition. So our method can use a surrogate function that directly approximates the non-smooth objective function. In comparison, all the existing MM methods construct the surrogate function by approximating the smooth component of the objective function. We apply our relaxed MM methods to the robust matrix factorization (RMF) problem with different regularizations, where our locally majorant algorithm shows advantages over the state-of-the-art approaches for RMF. This is the first algorithm for RMF ensuring, without extra assumptions, that any limit point of the iterates is a stationary point.
Zhejiang University City College（浙大城市学院）；Southwest Jiaotong University（西南交通大学）；Zhejiang University（浙江大学）
In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex tuning-free robust regression problems. The basic idea is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). We must emphasize that the main difficulty in the design of the algorithm lies in how to overcome the singular difficulty of the inner subproblem. Furthermore, we also prove that the PPMM algorithm converges to a d-stationary point. …
In this paper, we consider the problem of distributed pose graph optimization (PGO) that has extensive applications in multi-robot simultaneous localization and mapping (SLAM). We propose majorization minimization methods to distributed PGO and show that our proposed methods are guaranteed to converge to first-order critical points under mild conditions. Furthermore, since our proposed methods rely a proximal operator of distributed PGO, the convergence rate can be significantly accelerated with Nesterov's method, and more importantly, the acceleration induces no compromise of theoretical guar…
Google Research（谷歌研究院）；Facebook Reality Labs（Facebook现实实验室）；University of California San Diego（加州大学圣地亚哥分校）；Boston University（波士顿大学）；Brown University（布朗大学）
Non-convex optimization is ubiquitous in machine learning. Majorization-Minimization (MM) is a powerful iterative procedure for optimizing non-convex functions that works by optimizing a sequence of bounds on the function. In MM, the bound at each iteration is required to \emph{touch} the objective function at the optimizer of the previous bound. We show that this touching constraint is unnecessary and overly restrictive. We generalize MM by relaxing this constraint, and propose a new optimization framework, named Generalized Majorization-Minimization (G-MM), that is more flexible. For instanc…
Stochastic majorization-minimization (SMM) is a class of stochastic optimization algorithms that proceed by sampling new data points and minimizing a recursive average of surrogate functions of an objective function. The surrogates are required to be strongly convex and convergence rate analysis for the general non-convex setting was not available. In this paper, we propose an extension of SMM where surrogates are allowed to be only weakly convex or block multi-convex, and the averaged surrogates are approximately minimized with proximal regularization or block-minimized within diminishing rad…
Zhejiang University City College（浙大城市学院）；Southwest Jiaotong University（西南交通大学）；The Hong Kong Polytechnic University（香港理工大学）；National University of Singapore（新加坡国立大学）
In this paper, we consider high-dimensional nonconvex square-root-loss regression problems and introduce a proximal majorization-minimization (PMM) algorithm for these problems. Our key idea for making the proposed PMM to be efficient is to develop a sparse semismooth Newton method to solve the corresponding subproblems. By using the Kurdyka-Łojasiewicz property exhibited in the underlining problems, we prove that the PMM algorithm converges to a d-stationary point. We also analyze the oracle property of the initial subproblem used in our algorithm. Extensive numerical experiments are presente…
Block Majorization Minimization with Extrapolation and Application to $β$-NMF
Acapela Group（Acapela集团）；Institute of data sciences（数据科学研究所）；Faculty of Engineering（工程学院）；University of Mons（蒙斯大学）；Department of Mathematics and Operational Research（数学与运筹学系）
In this paper, we consider a class of nonsmooth nonconvex optimization problems whose objective is the sum of a block relative smooth function and a proper and lower semicontinuous block separable function. Although the analysis of block proximal gradient (BPG) methods for the class of block $L$-smooth functions have been successfully extended to Bregman BPG methods that deal with the class of block relative smooth functions, accelerated Bregman BPG methods are scarce and challenging to design. Taking our inspiration from Nesterov-type acceleration and the majorization-minimization scheme, we …
We consider the problem of minimizing a block separable convex function (possibly nondifferentiable, and including constraints) plus Laplacian regularization, a problem that arises in applications including model fitting, regularizing stratified models, and multi-period portfolio optimization. We develop a distributed majorization-minimization method for this general problem, and derive a complete, self-contained, general, and simple proof of convergence. Our method is able to scale to very large problems, and we illustrate our approach on two applications, demonstrating its scalability and ac…
Huawei Belgium Research Center（华为比利时研究中心）；HCMC University of Education（胡志明市教育大学）；University of Mons（蒙斯大学）
In this paper, we introduce TITAN, a novel inerTIal block majorizaTion minimizAtioN framework for non-smooth non-convex optimization problems. To the best of our knowledge, TITAN is the first framework of block-coordinate update method that relies on the majorization-minimization framework while embedding inertial force to each step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavy-ball and Nesterov-type accelerations for block proximal gradient methods as special cases. By choosing various surrogate functions, such as proximal, Lipschitz gr…
We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The majorization-minimization approach is naturally introduced to develop the smoothness and convexity of the SBAL function. Our method only depends on a factorization of the constant matrix independent of iterations and does not need any computation on step sizes, thus can be expected to be particularly appropriate for large-scale linear programs. The method shares some similar pro…
We consider the problem of localizing the source using range and range-difference measurements. Both the problems are non-convex and non-smooth and are challenging to solve. In this paper, we develop an iterative algorithm - Source Localization Via an Iterative technique (SOLVIT) to localize the source using all the distinct range-difference measurements, i.e., without choosing a reference sensor. SOLVIT is based on the Majorization Minimization approach - in which a novel upper bound is formulated and minimized to get a closed-form solution at every iteration. We also solve the source localiz…
Graz University of Technology（格拉茨工业大学）；University of Bath（巴斯大学）；University of Trieste（的里雅斯特大学）；Inria（法国国家信息与自动化研究所）；CentraleSupélec（巴黎中央理工-高等电力学院）；University Paris Saclay（巴黎萨克雷大学）
Variable Bregman Majorization-Minimization algorithms for nonconvex nonsmooth optimization, with application to Poisson imaging
Université Paris-Saclay, CentraleSupélec, Inria, Centre pour la Vision Numérique（巴黎萨克雷大学、中央苏佩莱克学院、法国国家信息与自动化研究所、数字视觉中心）；Université Paris-Saclay, Laboratoire BioMaps, CEA（巴黎萨克雷大学、BioMaps实验室、法国原子能和替代能源委员会）
Matrix decomposition is ubiquitous and has applications in various fields like speech processing, data mining and image processing to name a few. Under matrix decomposition, nonnegative matrix factorization is used to decompose a nonnegative matrix into a product of two nonnegative matrices which gives some meaningful interpretation of the data. Thus, nonnegative matrix factorization has an edge over the other decomposition techniques. In this paper, we propose two novel iterative algorithms based on Majorization Minimization (MM)-in which we formulate a novel upper bound and minimize it to ge…
Energy minimization methods are a classical tool in a multitude of computer vision applications. While they are interpretable and well-studied, their regularity assumptions are difficult to design by hand. Deep learning techniques on the other hand are purely data-driven, often provide excellent results, but are very difficult to constrain to predefined physical or safety-critical models. A possible combination between the two approaches is to design a parametric energy and train the free parameters in such a way that minimizers of the energy correspond to desired solution on a set of training…
In this paper, we introduce a novel iterative algorithm for the problem of phase-retrieval where the measurements consist of only the magnitude of linear function of the unknown signal, and the noise in the measurements follow Poisson distribution. The proposed algorithm is based on the principle of majorization-minimization (MM); however, the application of MM here is very novel and distinct from the way MM has been usually used to solve optimization problems in the literature. More precisely, we reformulate the original minimization problem into a saddle point problem by invoking Fenchel dua…
We propose a generalized formulation of direction of arrival estimation that includes many existing methods such as steered response power, subspace, coherent and incoherent, as well as speech sparsity-based methods. Unlike most conventional methods that rely exclusively on grid search, we introduce a continuous optimization algorithm to refine DOA estimates beyond the resolution of the initial grid. The algorithm is derived from the majorization-minimization (MM) technique. We derive two surrogate functions, one quadratic and one linear. Both lead to efficient iterative algorithms that do not…
We consider the problem of distributed pose graph optimization (PGO) that has important applications in multi-robot simultaneous localization and mapping (SLAM). We propose the majorization minimization (MM) method for distributed PGO ($\mathsf{MM-PGO}$) that applies to a broad class of robust loss kernels. The $\mathsf{MM-PGO}$ method is guaranteed to converge to first-order critical points under mild conditions. Furthermore, noting that the $\mathsf{MM-PGO}$ method is reminiscent of proximal methods, we leverage Nesterov's method and adopt adaptive restarts to accelerate convergence. The res…
Automated methods for facial animation are a necessary tool in the modern industry since the standard blendshape head models consist of hundreds of controllers and a manual approach is painfully slow. Different solutions have been proposed that produce output in real-time or generalize well for different face topologies. However, all these prior works consider a linear approximation of the blendshape function and hence do not provide a high-enough level of details for modern realistic human face reconstruction. We build a method for solving the inverse rig in blendshape animation using quadrat…
Majorization-minimization Bregman proximal gradient algorithms for NMF with the Kullback--Leibler divergence
Graduate School of Information Science and Technology, The University of Tokyo（东京大学大学院信息理工学系研究科）；Department of Statistical Inference and Mathematics, The Institute of Statistical Mathematics（统计数学研究所统计推断与数学系）；The Graduate University for Advanced Studies（日本高级研究院）
A Majorization-Minimization Approach to Design of Power Transmission Networks
我们提出了一种优化方法来设计具有成本效益的电力传输网络。也就是说，我们的目标是选择网络结构和线路电导（线路尺寸），以优化网络效率（传输网络内的低功耗）和构建网络的成本之间的权衡。我们从基于论文“Minimizing Effective Resistance of a Graph”[Ghosh，Boyd \& Saberi]的凸优化方法开始。我们表明，这种（直流）电阻网络方法可以适应的情况下，交流功率流。然而，这并没有解决选择网络结构的组合方面。我们将这个问题视为在过完备网络中选择一个子图，假设最小化（凸）网络功耗加上线路电导的非凸成本，这鼓励稀疏网络，其中许多线路电导被设置为零。我们开发了一种启发式方法来解决这个非凸优化问题，使用：（1）连续的方法来插值从光滑，凸问题（非光滑，非凸）组合问题，（2）优化最小化算法执行必要的中间光滑，但非凸优化步骤。最终，这涉及到解决一系列的凸优化问题，其中我们迭代地重新加权线电导的线性成本，以适应实际的非凸成本。几个例子表明，整体方法是一个很好的启发式网络设计。我们还考虑如何获得稀疏网络，仍然是强大的线路和/或发电机故障。
The alternating direction method of multipliers (ADMM) is a popular method for solving convex separable minimization problems with linear equality constraints. The generalization of the two-block ADMM to the three-block ADMM is not trivial since the three-block ADMM is not convergence in general. Many variants of three-block ADMM have been developed with guarantee convergence. Besides the ADMM, the alternating minimization algorithm (AMA) is also an important algorithm for solving the convex separable minimization problem with linear equality constraints. The AMA is first proposed by Tseng, an…
In this paper, we introduce a class of nonsmooth nonconvex least square optimization problem using convex analysis tools and we propose to use the iterative minimization-majorization (MM) algorithm on a convex set with initializer away from the origin to find an optimal point for the optimization problem. For this, first we use an approach to construct a class of convex majorizers which approximate the value of non-convex cost function on a convex set. The convergence of the iterative algorithm is guaranteed when the initial point $x^{(0)}$ is away from the origin and the iterative points $x^{…
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced and some basic properties of this map, including continuity, concavity, and differentiability with respect to the parameter in the diffusibility matrix, are established. For maximization problem, the admissible control set is convexified to get the existence of an optimal convexified relaxed solution. Whereas, for minimization problem, the relaxation of the pr…
