scholarly journals A Unified Study on L1 over L2 Minimization

2021 ◽  
Author(s):  
Min Tao ◽  
Xiao-Ping Zhang
Keyword(s):  
State Of The Art ◽  
Null Space ◽  
Linear Convergence ◽  
Computational Time ◽  
Measurement Matrix ◽  
Unconstrained Model ◽  
Local Linear Convergence ◽  
Alternating Direction ◽  
Unified Study ◽  
The Solution Existence

<div>In this paper, we carry out a unified study for L_1 over L_2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signal. First, we provide the exact recovery condition on both the constrained and the unconstrained models for a broad set of signals. Next, we prove the solution existence of these L_{1}/L_{2} models under the assumption that the null space of the measurement matrix satisfies the $s$-spherical section property. Then by deriving an analytical solution for the proximal operator of the L_{1} / L_{2} with nonnegative constraint, we develop a new alternating direction method of multipliers based method (ADMM$_p^+$) to solve the unconstrained model. We establish its global convergence to a d-stationary solution (sharpest stationary) and its local linear convergence under certain conditions. Numerical simulations on two specific applications confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.</div>

scholarly journals A Unified Study on L1 over L2 Minimization

2021 ◽  
Author(s):  
Min Tao ◽  
Xiao-Ping Zhang
Keyword(s):  
State Of The Art ◽  
Null Space ◽  
Linear Convergence ◽  
Computational Time ◽  
Measurement Matrix ◽  
Unconstrained Model ◽  
Local Linear Convergence ◽  
Alternating Direction ◽  
Unified Study ◽  
The Solution Existence

<div>In this paper, we carry out a unified study for L_1 over L_2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signal. First, we provide the exact recovery condition on both the constrained and the unconstrained models for a broad set of signals. Next, we prove the solution existence of these L_{1}/L_{2} models under the assumption that the null space of the measurement matrix satisfies the $s$-spherical section property. Then by deriving an analytical solution for the proximal operator of the L_{1} / L_{2} with nonnegative constraint, we develop a new alternating direction method of multipliers based method (ADMM$_p^+$) to solve the unconstrained model. We establish its global convergence to a d-stationary solution (sharpest stationary) and its local linear convergence under certain conditions. Numerical simulations on two specific applications confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.</div>

scholarly journals A Termination Criterion for Probabilistic Point Clouds Registration

Signals ◽  
2021 ◽  
Vol 2 (2) ◽  
pp. 159-173
Author(s):  
Simone Fontana ◽  
Domenico Giorgio Sorrenti
Keyword(s):  
State Of The Art ◽  
Point Clouds ◽  
Computational Time ◽  
Termination Criteria ◽  
Practical Usefulness ◽  
Local Point ◽  
Termination Criterion ◽  
Probabilistic Point ◽  
Excessive Number ◽  
Number Of Iterations

Probabilistic Point Clouds Registration (PPCR) is an algorithm that, in its multi-iteration version, outperformed state-of-the-art algorithms for local point clouds registration. However, its performances have been tested using a fixed high number of iterations. To be of practical usefulness, we think that the algorithm should decide by itself when to stop, on one hand to avoid an excessive number of iterations and waste computational time, on the other to avoid getting a sub-optimal registration. With this work, we compare different termination criteria on several datasets, and prove that the chosen one produces very good results that are comparable to those obtained using a very large number of iterations, while saving computational time.

scholarly journals A Sparse Optimization Approach for Energy-Efficient Timetabling in Metro Railway Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-19 ◽  
Author(s):  
Ziyan Luo ◽  
Xiaoyu Li ◽  
Naihua Xiu
Keyword(s):  
Optimization Model ◽  
Energy Efficient ◽  
Convex Relaxation ◽  
Sparse Optimization ◽  
Computational Time ◽  
Optimization Approach ◽  
Railway Systems ◽  
Alternating Direction ◽  
Np Hard Problem ◽  
Regenerative Energy

In this paper, we propose a sparse optimization approach to maximize the utilization of regenerative energy produced by braking trains for energy-efficient timetabling in metro railway systems. By introducing the cardinality function and the square of the Euclidean norm function as the objective function, the resulting sparse optimization model can characterize the utilization of the regenerative energy appropriately. A two-stage alternating direction method of multipliers is designed to efficiently solve the convex relaxation counterpart of the original NP-hard problem and then to produce an energy-efficient timetable of trains. The resulting approach is applied to Beijing Metro Yizhuang Line with different instances of service for case study. Comparison with the existing two-step linear program approach is also conducted which illustrates the effectiveness of our proposed sparse optimization model in terms of the energy saving rate and the efficiency of our numerical optimization algorithm in terms of computational time.

scholarly journals Land subsidence modelling for decision making on groundwater abstraction under emergency situation

2020 ◽  
Vol 382 ◽  
pp. 403-408
Author(s):  
Masaatsu Aichi
Keyword(s):  
Decision Making ◽  
Monte Carlo ◽  
Uncertainty Analysis ◽  
Real Time ◽  
Land Subsidence ◽  
Null Space ◽  
Emergency Situation ◽  
Computational Time ◽  
Groundwater Abstraction ◽  
Emergency Situations

Abstract. This study presents an inversion scheme with uncertainty analysis for a land subsidence modelling by a Monte Carlo filter in order to contribute to the decision-making on the groundwater abstraction. For real time prediction and uncertainty analysis under the limited computational resources and available information in emergency situations, one dimensional vertical land subsidence simulation was adopted for the forward modelling and the null-space Monte Carlo method was applied for the effective resampling. The proposed scheme was tested with the existing land subsidence monitoring data in Tokyo lowland, Japan. The results demonstrated that the prediction uncertainty converges and the prediction accuracy improves as the observed data increased with time. The computational time was also confirmed to be acceptable range for a real time execution with a laptop.

Global Bacteria Optimization Meta-Heuristic Algorithm for Jobshop Scheduling

2010 ◽  
Vol 1 (4) ◽  
pp. 47-58 ◽  
Author(s):  
Jairo R. Montoya-Torres ◽  
Libardo S. Gómez-Vizcaíno ◽  
Elyn L. Solano-Charris ◽  
Carlos D. Paternina-Arboleda
Keyword(s):  
Heuristic Algorithm ◽  
State Of The Art ◽  
Total Tardiness ◽  
Computational Time ◽  
Heuristic Procedure ◽  
Makespan Minimization ◽  
Light Stimulation ◽  
Jobshop Scheduling ◽  
Previous State

This paper examines the problem of jobshop scheduling with either makespan minimization or total tardiness minimization, which are both known to be NP-hard. The authors propose the use of a meta-heuristic procedure inspired from bacterial phototaxis. This procedure, called Global Bacteria Optimization (GBO), emulates the reaction of some organisms (bacteria) to light stimulation. Computational experiments are performed using well-known instances from literature. Results show that the algorithm equals and even outperforms previous state-of-the-art procedures in terms of quality of solution and requires very short computational time.

scholarly journals Fast and Simple Mixture of Softmaxes with BPE and Hybrid-LightRNN for Language Generation

2019 ◽  
Vol 33 ◽  
pp. 6626-6633
Author(s):  
Xiang Kong ◽  
Qizhe Xie ◽  
Zihang Dai ◽  
Eduard Hovy
Keyword(s):  
Machine Translation ◽  
State Of The Art ◽  
The State ◽  
Computational Time ◽  
Memory Consumption ◽  
Image Captioning ◽  
Vocabulary Size ◽  
Language Generation ◽  
Practical Applications ◽  
Coding Schemes

Mixture of Softmaxes (MoS) has been shown to be effective at addressing the expressiveness limitation of Softmax-based models. Despite the known advantage, MoS is practically sealed by its large consumption of memory and computational time due to the need of computing multiple Softmaxes. In this work, we set out to unleash the power of MoS in practical applications by investigating improved word coding schemes, which could effectively reduce the vocabulary size and hence relieve the memory and computation burden. We show both BPE and our proposed Hybrid-LightRNN lead to improved encoding mechanisms that can halve the time and memory consumption of MoS without performance losses. With MoS, we achieve an improvement of 1.5 BLEU scores on IWSLT 2014 German-to-English corpus and an improvement of 0.76 CIDEr score on image captioning. Moreover, on the larger WMT 2014 machine translation dataset, our MoSboosted Transformer yields 29.6 BLEU score for English-toGerman and 42.1 BLEU score for English-to-French, outperforming the single-Softmax Transformer by 0.9 and 0.4 BLEU scores respectively and achieving the state-of-the-art result on WMT 2014 English-to-German task.

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