DETECTING DELAY ERRORS IN OPTICAL FIBER NETWORKS BASED ON NON-CONVEX PENALIZED REGRESSION

In 5G networks, the accuracy of time synchronization directly impacts the performance and stability of communication systems. Precision Time Protocol (PTP) based on network precision timing is one of the key technologies ensuring high-precision time synchronization among network nodes. In large-scale networks, factors such as environmental variations and equipment aging can lead to time desynchronization among fibers.

Traditional fault detection methods involve manually deploying Global Navigation Satellite System (GNSS) receivers throughout the network. However, GNSS equipment is costly and challenging to deploy on a large scale, making manual deployment time-consuming and labor-intensive. To address this issue, this thesis proposes a method based on non-convex regression capable of effectively identifying faulty fibers within network topologies. In simulation experiments, this method achieves an accuracy rate of nearly 100% and is already being deployed in the industry.

Initially, this study models the complex network fiber structure using a tree structure, transforming the identification of faulty fibers into a problem of solving underdetermined linear equation systems. Considering the sparsity of faults within the network fibers, this thesis employs various regularization techniques, including (L_0) normalization and Smoothly Clipped Absolute Deviation (SCAD). Experimental results demonstrate that the SCAD method outperforms all other regularization methods. With a fiber delay edge ratio of 10%, the SCAD method maintains an overall accuracy of 99.77% on the total dataset and a precision of 94.84% for time-delayed edges.

Furthermore, this study introduces a tree fusion method that merges consecutive unbranched edges within the network, reducing the solution time by 32.31% and enhancing algorithm accuracy. Through the analysis of confusion matrices and Receiver Operating Characteristic (ROC) curve diagrams from simulation experiments, this thesis recommends adopting the standard of relative thresholds to identify time-delayed edges.

In the final analysis, for the 5% of time-delayed edges incorrectly solved by SCAD, this research proposes an innovative partitioning algorithm. Through localized topological visualization, samples predicted as time-delay edges are divided into three groups: sub-high confidence group, high confidence group, and ultra-high confidence group. Notably, edges in the ultra-high confidence partition reach a 100% confidence level. To improve the confidence levels of the sub-high and high confidence groups, incorporation of GNSS observation points at critical sub-high confidence nodes is utilized. After adding observation points and iterating multiple times, all three partitions achieves a 100% confidence level, offering robust technical support for the precise identification and localization of time delays within intricate networks.

[1] LECHNER W, BAUMANN S. Global navigation satellite systems[J]. Computers and Electronics in Agriculture, 2000, 25(1-2): 67-85.
[2] SAND S, DAMMANN A, MENSING C. Positioning in wireless communications systems[M].John Wiley & Sons, 2014.
[3] WATT S T, ACHANTA S, ABUBAKARI H, et al. Understanding and applying precision timeprotocol[C]//2015 Saudi Arabia Smart Grid (SASG). IEEE, 2015: 1-7.
[4] WESTON N D, SCHWIEGER V. Cost effective GNSS positioning techniques[M]. FIG, 2014.
[5] LÉVESQUE M, TIPPER D. A survey of clock synchronization over packet-switched networks[J]. IEEE Communications Surveys & Tutorials, 2016, 18(4): 2926-2947.
[6] PAPADIMITRIOU C H, STEIGLITZ K. Combinatorial optimization: algorithms and complexity[M]. Courier Corporation, 1998.
[7] EIDSON J C, FISCHER M, WHITE J. IEEE 1588 standard for a precision clock synchronization protocol for networked measurement and control systems[J]. 2nd ISA/IEEE Sensors forIndustry Conference, 2002: 98-105.
[8] MATEOS G, BAZERQUE J A, GIANNAKIS G B. Distributed sparse linear regression[J].IEEE Transactions on Signal Processing, 2010, 58(10): 5262-5276.
[9] ELAD M. Sparse and redundant representations: from theory to applications in signal andimage processing: Vol. 2[M]. Springer, 2010.
[10] SCOTT J, TUMA M. Solving large linear least squares problems with linear equality constraints[J]. BIT Numerical Mathematics, 2022, 62(4): 1765-1787.
[11] BENNING M, BURGER M. Modern regularization methods for inverse problems[J]. Actanumerica, 2018, 27: 1-111.
[12] DOKMANIC I, PARHIZKAR R, RANIERI J, et al. Euclidean distance matrices: essentialtheory, algorithms, and applications[J]. IEEE Signal Processing Magazine, 2015, 32(6): 12-30.
[13] MATSUURA K, OKABE Y. Selective minimum-norm solution of the biomagnetic inverseproblem[J]. IEEE Transactions on Biomedical Engineering, 1995, 42(6): 608-615.
[14] RUDER S. An overview of gradient descent optimization algorithms[A]. 2016.
[15] HAJI S H, ABDULAZEEZ A M. Comparison of optimization techniques based on gradientdescent algorithm: A review[J]. PalArch’s Journal of Archaeology of Egypt/Egyptology, 2021,18(4): 2715-2743.
[16] MILLER A. Subset selection in regression[M]. chapman and hall/CRC, 2002.
[17] BERTSIMAS D, KING A, MAZUMDER R. Best subset selection via a modern optimizationlens[Z]. 2016.
[18] HASTIE T, TIBSHIRANI R, TIBSHIRANI R J. Extended comparisons of best subset selection,forward stepwise selection, and the lasso[A]. 2017.
[19] ZHU J, WEN C, ZHU J, et al. A polynomial algorithm for best-subset selection problem[J].Proceedings of the National Academy of Sciences, 2020, 117(52): 33117-33123.
[20] BLONDEL V, TSITSIKLIS J N. NP-hardness of some linear control design problems[J]. SIAMjournal on control and optimization, 1997, 35(6): 2118-2127.
[21] WOLSEY L A, NEMHAUSER G L. Integer and combinatorial optimization: Vol. 55[M]. JohnWiley & Sons, 1999.
[22] BERTSIMAS D, KING A, MAZUMDER R. Best subset selection via a modern optimizationlens[Z]. 2016.
[23] BERTSIMAS D, VAN PARYS B. Sparse high-dimensional regression: Exact scalable algorithms and phase transitions[Z]. 2020.
[24] HAZIMEH H, MAZUMDER R, SAAB A. Sparse regression at scale: Branch-and-boundrooted in first-order optimization[J]. Mathematical Programming, 2022, 196(1-2): 347-388.
[25] HAZIMEH H, MAZUMDER R. Fast best subset selection: Coordinate descent and local combinatorial optimization algorithms[J]. Operations Research, 2020, 68(5): 1517-1537.
[26] DEDIEU A, HAZIMEH H, MAZUMDER R. Learning sparse classifiers: Continuous andmixed integer optimization perspectives[J]. The Journal of Machine Learning Research, 2021,22(1): 6008-6054.
[27] WRIGHT S J. Coordinate descent algorithms[J]. Mathematical programming, 2015, 151(1):3-34.
[28] HAZIMEH H, MAZUMDER R, NONET T. L0learn: A scalable package for sparse learningusing l0 regularization[J]. Journal of Machine Learning Research, 2023, 24(205): 1-8.
[29] TIBSHIRANI R. Regression shrinkage and selection via the lasso[J]. Journal of the RoyalStatistical Society Series B: Statistical Methodology, 1996, 58(1): 267-288.
[30] CANDES E J, ROMBERG J K, TAO T. Stable signal recovery from incomplete and inaccuratemeasurements[J]. Communications on Pure and Applied Mathematics: A Journal Issued by theCourant Institute of Mathematical Sciences, 2006, 59(8): 1207-1223.

[31] HOERL A E, KENNARD R W. Ridge regression: Biased estimation for nonorthogonal problems[J]. Technometrics, 1970, 12(1): 55-67.

[32] ZOU H, HASTIE T. Regularization and variable selection via the elastic net[J]. Journal of theRoyal Statistical Society Series B: Statistical Methodology, 2005, 67(2): 301-320.

[33] WEN F, CHU L, LIU P, et al. A Survey on Nonconvex Regularization-Based Sparse and LowRank Recovery in Signal Processing, Statistics, and Machine Learning[J]. IEEE Access, 2018,6: 69883-69906.

[34] FAN J, LI R. Variable selection via nonconcave penalized likelihood and its oracle properties[J]. Journal of the American statistical Association, 2001, 96(456): 1348-1360.

[35] ZHANG C H. Nearly unbiased variable selection under minimax concave penalty[J]. TheAnnals of Statistics, 2010, 38(2): 894-942.

[36] BREHENY P, HUANG J. Coordinate descent algorithms for nonconvex penalized regression,with applications to biological feature selection[J]. The annals of applied statistics, 2011, 5(1):232.

[37] TARJAN R. Depth-first search and linear graph algorithms[J]. SIAM journal on computing,1972, 1(2): 146-160.

[38] DOMINGOS P. The role of Occam’s razor in knowledge discovery[J]. Data mining and knowledge discovery, 1999, 3: 409-425.

[39] KOYEJO O O, NATARAJAN N, RAVIKUMAR P K, et al. Consistent binary classificationwith generalized performance metrics[J]. Advances in neural information processing systems,2014, 27.

[40] BERTRAND Q, KLOPFENSTEIN Q, BLONDEL M, et al. Implicit differentiation of lasso-typemodels for hyperparameter optimization[C]//International Conference on Machine Learning.PMLR, 2020: 810-821.

[41] FAWCETT T. An introduction to ROC analysis[J]. Pattern recognition letters, 2006, 27(8):861-874.