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fixed-point algorithms for frequency estimation and structured low rank approximation

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2019年第1期46卷 40-65页

作者： Andersson, Fredrik Carlsson, Marcus Lund Univ Ctr Math Sci Box 118 S-22100 Lund Sweden

We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, i.e., frequency estimation. For the basic formulation of the fixed-point algorithm we show that it converges to the solution of a related minimization problem, namely the one obtained by replacing the original objective function with its convex envelope and keeping the structured matrix constraint unchanged. It often happens that this solution agrees with the solution to the original minimization problem, and we provide a simple criterion for when this is true. We also provide more general fixed-point algorithms that can be used to treat the problems of making weighted approximations by sums of exponentials given equally or unequally spaced sampling. We apply the method to the case of missing data, although the above mentioned convergence results do not hold in this case. However, it turns out that the method often gives perfect reconstruction (up to machine precision) in such cases. We also discuss multidimensional extensions, and illustrate how the proposed algorithms can be used to recover sums of exponentials in several variables, but when samples are available only along a curve. (C) 2017 Elsevier Inc. All rights reserved.

关键词： fixed-point algorithms Frequency estimation General domain Hankel matrices Fenchel conjugate Convex envelope

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fixed-point algorithms FOR INVERSE OF RESIDUAL RECTIFIER NEURAL NETWORKS

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MATHEMATICAL FOUNDATIONS OF COMPUTING 2021年第1期4卷 31-44页

作者： Wang, Ruhua An, Senjian Liu, Wanquan Li, Ling Curtin Univ Sch Elect Engn Comp & Math Sci Bentley WA Australia

A deep neural network with invertible hidden layers has a nice property of preserving all the information in the feature learning stage. In this paper, we analyse the hidden layers of residual rectifier neural networks, and investigate conditions for invertibility under which the hidden layers are invertible. A new fixed-point algorithm is developed to invert the hidden layers of residual networks. The proposed inverse algorithms are capable of inverting some residual networks which cannot be inverted by existing inverting algorithms. Furthermore, a special residual rectifier network is designed and trained on MNIST so that it can achieve comparable performance with the state-of-art performance while its hidden layers are invertible.

关键词： Invertible neural networks fixed-point algorithms invertible residual networks invertible rectifier neural networks inverse algorithm of neural networks

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Differential fast fixed-point algorithms for underdetermined instantaneous and convolutive partial blind source separation

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2007年第7期55卷 3717-3729页

作者： Thomas, Johan Deville, Yannick Hosseini, Shahram Univ Toulouse 3 CNRS Lab Astrophys Toulouse Tarbes F-31400 Toulouse France

This paper concerns underdetermined linear instantaneous and convolutive blind source separation (BSS), i.e., the case when the number P of observed mixed signals is lower than the number N of sources. We propose partial BSS methods, which separate P supposedly nonstationary sources of interest (while keeping residual components for the other N - P, supposedly stationary, "noise" sources). These methods are based on the general differential BSS concept that we introduced before. In the instantaneous case, the approach proposed in this paper consists of a differential extension of the FastICA method (which does not apply to underdetermined mixtures). In the convolutive case, we extend our recent-time-domain fast fixed-point C-FICA algorithm to underdetermined mixtures. Both proposed approaches thus keep the attractive features of the FastICA and C-FICA methods. Our approaches are based on differential sphering processes, followed by the optimization of the differential nonnormalized kurtosis that we introduce in this paper. Experimental tests show that these differential algorithms are much more robust to noise sources than the standard FastICA and C-FICA algorithms.

关键词： blind source separation (BSS) convolutive mixtures fixed-point algorithms independent component analysis (ICA) kurtosis underdetermined mixtures

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Time-domain fast fixed-point algorithms for convolutive ICA

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IEEE SIGNAL PROCESSING LETTERS 2006年第4期13卷 228-231页

作者： Thomas, J Deville, Y Hosseini, S Univ Toulouse 3 Lab Astrophys Toulouse Tarbes F-31400 Toulouse France

This letter presents new blind separation methods for moving average (MA) convolutive mixtures of independent MA processes. They consist of time-domain extensions of the FastICA algorithms developed by Hyvarinen and Oja for instantaneous mixtures. They perform a convolutive sphering in order to use parameter-free fast fixed-point algorithms associated with kurtotic or negentropic non-Gaussianity criteria for estimating the source innovation processes. We prove the relevance of this approach by mapping the mixtures into linear instantaneous ones. Test results are presented for artificial colored signals and speech signals.

关键词： convolutive mixtures fixed-point algorithms independent component analysis (ICA) non-Gaussian signals

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Monotonic convergence of fixed-point algorithms for ICA

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IEEE TRANSACTIONS ON NEURAL NETWORKS 2003年第4期14卷 943-949页

作者： Regalia, PA Kofidis, E Inst Natl Telecommun Dept Commun Image & Informat Proc F-91011 Evry France Univ Athens Dept Informat & Telecommun GR-15784 Athens Greece

We re-examine a fixed-point algorithm proposed recently by Hyvarinen for independent component analysis, wherein local convergence is proved subject to an ideal signal model using a square invertible mixing matrix. Here, we derive step-size bounds which ensure monotonic convergence to a local extremum for any initial condition. Our analysis does not assume an ideal signal model but appeals rather to properties of the contrast function itself, and so applies even with noisy data and/or more sources than sensors. The results help alleviate the guesswork that often surrounds step-size selection when the observed signal does not fit an idealized model.

关键词： fixed-point algorithms independent component analysis (ICA) monotonic convergence non-Gaussian signals

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Analysis of fixed-point and Coordinate Descent algorithms for Regularized Kernel Methods

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IEEE TRANSACTIONS ON NEURAL NETWORKS 2011年第10期22卷 1576-1587页

作者： Dinuzzo, Francesco Max Planck Inst Intelligent Syst D-72076 Tubingen Germany

In this paper, we analyze the convergence of two general classes of optimization algorithms for regularized kernel methods with convex loss function and quadratic norm regularization. The first methodology is a new class of algorithms based on fixed-point iterations that are well-suited for a parallel implementation and can be used with any convex loss function. The second methodology is based on coordinate descent, and generalizes some techniques previously proposed for linear support vector machines. It exploits the structure of additively separable loss functions to compute solutions of line searches in closed form. The two methodologies are both very easy to implement. In this paper, we also show how to remove non-differentiability of the objective functional by exactly reformulating a convex regularization problem as an unconstrained differentiable stabilization problem.

关键词： Convergence analysis coordinate descent decomposition methods fixed-point algorithms kernel methods support vector machines

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A QUADRATICALLY-CONVERGENT fixed-point ALGORITHM FOR ECONOMIC EQUILIBRIA AND LINEARLY CONSTRAINED OPTIMIZATION

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MATHEMATICAL PROGRAMMING 1980年第2期18卷 111-126页

作者： TODD, MJ Cornell University Ithaca NY USA

fixed-point algorithms for computing equilibria in economies with production or stationary points in constrained optimization generally use point-to-set mappings and therefore converge slowly. An alternative implementation uses continuous functions with a higher dimensionality corresponding to the inclusion of activity levels or dual variables. Here we develop algorithms that only increase the dimensionality implicitly. The solution path is piecewise-linear as in other algorithms. However, when viewed in the low-dimensional space, the path within each simplex can be piecewise-linear rather than linear. Asymptotically, these paths are linear and quadratic convergence is attained.

关键词： fixed-point algorithms Economic Equilibria Linearly Constrained Optimization Quadratic Convergence

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ADJOINTS OF fixed-point ITERATIONS 11

ADJOINTS OF FIXED-POINT ITERATIONS

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11th World Congress on Computational Mechanics (WCCM) / 5th European Conference on Computational Mechanics (ECCM) / 6th European Conference on Computational Fluid Dynamics (ECFD)

作者： Taftaf, A. Pascual, V. Hascoet, L. INRIA Sophia Antipolis France

ISBN: (纸本)9788494284472

Adjoint algorithms, and in particular those obtained through the adjoint mode of Automatic Differentiation (AD), are probably the most efficient way to obtain the gradient of a numerical simulation. This however needs to use the flow of data of the original simulation in reverse order, at a cost that increases with the length of the simulation. AD research looks for strategies to reduce this cost, taking advantage of the structure of the given program. One such frequent structure is fixed-point iterations, which occur e.g. in steady-state simulations, but not only. It is common wisdom that the first iterations of a fixed-point search operate on a meaningless state vector, and that reversing the corresponding data-flow may be suboptimal. An adapted adjoint strategy for this iterative process should consider only the last or the few last iterations. At least two authors, B. Christianson and A. Griewank, have studied mathematically fixed-point iterations with the goal of defining an efficient adjoint. In this paper, we describe and contrast these two strategies with the objective of implementing the best suited one into the AD tool that we are developing. We select a representative application to test the chosen strategy, to propose a set of user directives to trigger it, and to discuss the implementation implications in our tool.

关键词： Automatic Differentiation Adjoint fixed-point algorithms

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ON A fixed-point ALGORITHM FOR STRUCTURED LOW-RANK APPROXIMATION AND ESTIMATION OF HALF-LIFE PARAMETERS 24

ON A FIXED-POINT ALGORITHM FOR STRUCTURED LOW-RANK APPROXIMA...

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24th European Signal Processing Conference (EUSIPCO)

作者： Andersson, Fredrik Carlsson, Marcus Wendt, Herwig Lund Univ Ctr Math Sci Box 118 S-22100 Lund Sweden Univ Toulouse CNRS UMR 5505 IRIT Toulouse France

ISBN: (纸本)9780992862657

We study the problem of decomposing a measured signal as a sum of decaying exponentials. There is a direct connection to sums of these types and positive semi-definite (PSD) Hankel matrices, where the rank of these matrices equals the number of exponentials. We propose to solve the identification problem by forming an optimization problem with a misfit function combined with a rank penalty function that also ensures the PSD-constraint. This problem is non-convex, but we show that it is possible to compute the minimum of an explicit closely related convexified problem. Moreover, this minimum can be shown to often coincide with the minimum of the original non-convex problem, and we provide a simple criterion that enables to verify if this is the case.

关键词： Low rank approximation structured matrices fixed-point algorithms

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A Deep Neural Network Algorithm for Linear-Quadratic Portfolio Optimization With MGARCH and Small Transaction Costs

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IEEE ACCESS 2023年 11卷 16774-16792页

作者： Papanicolaou, Andrew Fu, Hao Krishnamurthy, Prashanth Khorrami, Farshad North Carolina State Univ Dept Math Raleigh NC 27695 USA NYU Tandon Sch Engn Elect & Comp Engn Brooklyn NY USA

We analyze a fixed-point algorithm for reinforcement learning (RL) of optimal portfolio mean-variance preferences in the setting of multivariate generalized autoregressive conditional-heteroskedasticity (MGARCH) with a small penalty on trading. A numerical solution is obtained using a neural network (NN) architecture within a recursive RL loop. A fixed-point theorem proves that NN approximation error has a big-oh bound that we can reduce by increasing the number of NN parameters. The functional form of the trading penalty has a parameter epsilon > 0 that controls the magnitude of transaction costs. When epsilon is small, we can implement an NN algorithm based on the expansion of the solution in powers of epsilon. This expansion has a base term equal to a myopic solution with an explicit form, and a first-order correction term that we compute in the RL loop. Our expansion-based algorithm is stable, allows for fast computation, and outputs a solution that shows positive testing performance.

关键词： Artificial neural networks Costs Approximation algorithms Optimization Portfolios Reinforcement learning Covariance matrices Deep learning Neural networks Hetereoskedasticity MGARCH fixed-point algorithms reinforcement learning deep neural networks

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