Estimators derived from the expectation-maximization (EM) algorithm are not robust since they are based on the maximization of the likelihood function. We propose an iterative proximal-point algorithm based on the EM ...
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Estimators derived from the expectation-maximization (EM) algorithm are not robust since they are based on the maximization of the likelihood function. We propose an iterative proximal-point algorithm based on the EM algorithm to minimize a divergence criterion between a mixture model and the unknown distribution that generates the data. The algorithm estimates in each iteration the proportions and the parameters of the mixture components in two separate steps. Resulting estimators are generally robust against outliers and misspecification of the model. Convergence properties of our algorithm are studied. The convergence of the introduced algorithm is discussed on a two-component Weibull mixture entailing a condition on the initialization of the EM algorithm in order for the latter to converge. Simulations on Gaussian and Weibull mixture models using different statistical divergences are provided to confirm the validity of our work and the robustness of the resulting estimators against outliers in comparison to the EM algorithm. An application to a dataset of velocities of galaxies is also presented. The Canadian Journal of Statistics 47: 392-408;2019 (c) 2019 Statistical Society of Canada
Banjac et al. (J Optim Theory Appl 183(2):490-519, 2019) recently showed that the Douglas-Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they show...
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Banjac et al. (J Optim Theory Appl 183(2):490-519, 2019) recently showed that the Douglas-Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility. Their result was shown in a finite-dimensional Euclidean setting and for a particular structure of the constraint set. In this paper, we extend the result to real Hilbert spaces and a general nonempty closed convex set. Moreover, we show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates.
Single-cell RNA sequencing (scRNA-seq) has emerged as a powerful tool for uncovering cellular heterogeneity. However, the high costs associated with this technique have rendered it impractical for studying large patie...
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Single-cell RNA sequencing (scRNA-seq) has emerged as a powerful tool for uncovering cellular heterogeneity. However, the high costs associated with this technique have rendered it impractical for studying large patient cohorts. We introduce ENIGMA (Deconvolution based on Regularized Matrix Completion), a method that addresses this limitation through accurately deconvoluting bulk tissue RNA-seq data into a readout with cell-type resolution by leveraging information from scRNA-seq data. By employing a matrix completion strategy, ENIGMA minimizes the distance between the mixture transcriptome obtained with bulk sequencing and a weighted combination of cell-type-specific expression. This allows the quantification of cell-type proportions and reconstruction of cell-type-specific transcriptomes. To validate its performance, ENIGMA was tested on both simulated and real datasets, including disease-related tissues, demonstrating its ability in uncovering novel biological insights.
For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro an...
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For solving strongly convex optimization problems, we propose and study the global convergence of variants of the accelerated hybrid proximal extragradient (A-HPE) and large-step A-HPE algorithms of R.D.C. Monteiro and B.F. Svaiter [An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods, SIAM J. Optim. 23 (2013), pp. 1092-1125.]. We prove linear and the superlinear O(k(-k(p-1/p+1))) global rates for the proposed variants of the A-HPE and large-step A-HPE methods, respectively. The parameter p >= 2 appears in the (high-order) large-step condition of the new large-step A-HPE algorithm. We apply our results to high-order tensor methods, obtaining a new inexact (relative-error) tensor method for (smooth) strongly convex optimization with iteration-complexity O(k(-k(p-1/p+1))). In particular, for p = 2, we obtain an inexact proximal-Newton algorithm with fast global O (k(-k/)(3)) convergence rate.
The main result of this paper is to prove the strong convergence of the sequence generated by the proximalpointalgorithm of Halpern type to a zero of a maximal monotone operator under the suitable assumptions on the...
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The main result of this paper is to prove the strong convergence of the sequence generated by the proximalpointalgorithm of Halpern type to a zero of a maximal monotone operator under the suitable assumptions on the parameters and error. The results extend some of the previous results or give some different conditions for convergence of the sequence. It is also indicated that when the maximal monotone operator is the subdifferential of a convex, proper, and lower semicontinuous function, the results extend all previous results in the literature. We also prove the boundedness of the sequence generated by the algorithm with a weak coercivity condition defined in the paper and without any additional assumptions on the parameters.
In this paper, we obtain some results on the boundedness and asymptotic behavior of the sequence generated by the proximalpointalgorithm without summability assumption on the error sequence. We also study the rate o...
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In this paper, we obtain some results on the boundedness and asymptotic behavior of the sequence generated by the proximalpointalgorithm without summability assumption on the error sequence. We also study the rate of convergence to minimum value of a proper, convex, and lower semicontinuous function. Finally, we consider the proximalpointalgorithm for solving equilibrium problems.
The main purpose of the present work is to introduce two parametric proximal-point type algorithms involving the gradient (or subdifferential) of a convex function. We take advantage of some properties of maximal mono...
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The main purpose of the present work is to introduce two parametric proximal-point type algorithms involving the gradient (or subdifferential) of a convex function. We take advantage of some properties of maximal monotone operators to prove monotonicity and convergence rate conditions. One example in Hilbert spaces and two numerical examples with program realizations are presented.
Macro-hybrid penalized finite element approximations are studied for steady filtration problems with seawater intrusion. On the basis of nonoverlapping domain decompositions with vertical interfaces, sections of coast...
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Macro-hybrid penalized finite element approximations are studied for steady filtration problems with seawater intrusion. On the basis of nonoverlapping domain decompositions with vertical interfaces, sections of coastal aquifers are decomposed into subsystems with simpler geometries and small scales, interconnected via transmission conditions of pressure and flux continuity. Corresponding local penalized formulations are derived from the global penalized variational formulation of the two-free boundary flow problem, with continuity transmission conditions modelled variationally in a dual sense. Then, macro-hybrid finite element approximations are derived for the system, defined on independent subdomain grids. Parallel relaxation penalty-duality algorithms are proposed from fixed-point problem characterizations. Numerical experiments exemplify the macro-hybrid penalized theory, showing a good agreement with previous primal conforming penalized finite element approximations.
The hybrid extragradient proximal-point method recently proposed by Solodov and Svaiter has the distinctive feature of allowing a relative error tolerance. We extend the error tolerance of this method, proving that it...
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The hybrid extragradient proximal-point method recently proposed by Solodov and Svaiter has the distinctive feature of allowing a relative error tolerance. We extend the error tolerance of this method, proving that it converges even if a summable error is added to the relative error. Furthermore, the extragradient step may be performed inexactly with a summable error. We present a convergence analysis, which encompasses other well-known variations of the proximal-point method, previously unrelated. We establish weak global convergence under mild assumptions.
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