Fast and stable modification of the Gauss–Newton method for low‐rank signal estimation

Second-order optimization methods have desirable convergence properties. However, the exact Newton method requires expensive computation for the Hessian and its inverse. In this paper, we propose SPAN, a novel approximate and fast Newton method. SPAN computes the inverse of the Hessian matrix via low-rank approximation and stochastic Hessian-vector products. Our experiments on multiple benchmark datasets demonstrate that SPAN outperforms existing first-order and second-order optimization methods in terms of the convergence wall-clock time. Furthermore, we provide a theoretical analysis of the per-iteration complexity, the approximation error, and the convergence rate. Both the theoretical analysis and experimental results show that our proposed method achieves a better trade-off between the convergence rate and the per-iteration efficiency.

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Inexact Kleinman-Newton-ADI Method with Line Search to Solve Large-Scale Algebraic Riccati Equations

10.14293/p2199-8442.1.sop-math.psehos.v1 ◽

2015 ◽

Author(s):

Heiko K. Weichelt ◽

Matthias Heinkenschloss ◽

Jens Saak ◽

Peter Benner

Keyword(s):

Newton Method ◽

Large Scale ◽

Line Search ◽

Lyapunov Equation ◽

Riccati Equations ◽

Convergence Result ◽

Low Rank ◽

Algebraic Riccati Equations ◽

Newton Step ◽

The Difference

This poster shows recent improvements of the inexact Kleinman-Newton method for solving algebraic Riccati equations by incorporating a line search and by systematically integrating the low-rank structure resulting from ADI methods for the approximate solution of the Lyapunov equation that needs to be solved to compute the Kleinman-Newton step. A convergence result is pointed out that tailors the convergence proof for general inexact Newton methods to the structure of Riccati equations and avoids positive semi-definiteness assumptions on the difference between certain matrices and the Lyapunov equation residual, which in general do not hold for low-rank approaches. On a test example, the improved inexact Kleinman-Newton method demonstrates its advantages.

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Tensor Recovery via Nonconvex Low-Rank Approximation

2020 28th European Signal Processing Conference (EUSIPCO) ◽

10.23919/eusipco47968.2020.9287404 ◽

2021 ◽

Author(s):

Lin Chen ◽

Xue Jiang ◽

Xingzhao Liu ◽

Zhixin Zhou

Keyword(s):

Low Rank ◽

Low Rank Approximation ◽

Tensor Recovery ◽

Rank Approximation

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The Influences of the Rank of Coal on Methane Sorption Capacity in Coals/Wpływ Rzędu Węgla Na Pojemność Sorpcyjną Metanu W Węglach

Archives of Mining Sciences ◽

10.2478/amsc-2014-0037 ◽

2014 ◽

Vol 59 (2) ◽

pp. 509-516

Author(s):

Andrzej Olajossy

Keyword(s):

Sorption Capacity ◽

Coalbed Methane ◽

Vitrinite Reflectance ◽

Sorption Isotherms ◽

Volatile Matter ◽

Low Rank ◽

Hard Coal ◽

Petrographic Composition ◽

Methane Sorption ◽

Indian Coals

Abstract Methane sorption capacity is of significance in the issues of coalbed methane (CBM) and depends on various parameters, including mainly, on rank of coal and the maceral content in coals. However, in some of the World coals basins the influences of those parameters on methane sorption capacity is various and sometimes complicated. Usually the rank of coal is expressed by its vitrinite reflectance Ro. Moreover, in coals for which there is a high correlation between vitrinite reflectance and volatile matter Vdaf the rank of coal may also be represented by Vdaf. The influence of the rank of coal on methane sorption capacity for Polish coals is not well understood, hence the examination in the presented paper was undertaken. For the purpose of analysis there were chosen fourteen samples of hard coal originating from the Upper Silesian Basin and Lower Silesian Basin. The scope of the sorption capacity is: 15-42 cm3/g and the scope of vitrinite reflectance: 0,6-2,2%. Majority of those coals were of low rank, high volatile matter (HV), some were of middle rank, middle volatile matter (MV) and among them there was a small number of high rank, low volatile matter (LV) coals. The analysis was conducted on the basis of available from the literature results of research of petrographic composition and methane sorption isotherms. Some of those samples were in the form (shape) of grains and others - as cut out plates of coal. The high pressure isotherms previously obtained in the cited studies were analyzed here for the purpose of establishing their sorption capacity on the basis of Langmuire equation. As a result of this paper, it turned out that for low rank, HV coals the Langmuire volume VL slightly decreases with the increase of rank, reaching its minimum for the middle rank (MV) coal and then increases with the rise of the rank (LV). From the graphic illustrations presented with respect to this relation follows the similarity to the Indian coals and partially to the Australian coals.

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