Significant and Trivial Dependencies Separation in Data Tensor by the Projection Method

Yuriy Bunyak; Roman  Kvyetnyy; Olga Sofina

doi:10.7494/dmms.2025.19.7025

Authors

Yuriy Bunyak Spilna Sprava Company https://orcid.org/0000-0002-0862-880X
Roman Kvyetnyy Department of Automation and Intelligent Information Technologies, Vinnytsia National Technical University, Vinnytsia, Ukraine https://orcid.org/0000-0002-9192-9258
Olga Sofina Department of Automation and Intelligent Information Technologies, Vinnytsia National Technical University, Vinnytsia, Ukraine https://orcid.org/0000-0003-3774-9819

DOI:

https://doi.org/10.7494/dmms.2025.19.7025

Keywords:

multidimensional data tensor, higher-order singular value decomposition, singular values distribution entropy, singular projection, tensor projection, data pattern assessment

Abstract

The problem of data structure analysis through their multidimensional representation as a d-dimensional tensor is considered to assess dependencies on influencing factors in the decision-making process. The higher-order Singular Value Decomposition (SVD) is developed as a d-SVD schema to identify significant and trivial dependencies. The d-SVD includes the SVD of the tensor reshaped as a matrix and the SVDs of reduced size of the previous SVD vectors reshaped as matrices. The entropy of the distribution of the Singular Values (SVs) of the vectors’ decomposition is used for the separation of the significant and trivial vectors, in contrast to the commonly used approach based on the magnitude analysis of SVs. The singular projection in the significant vector space in selected dimensions gives the tensor’s low-rank approximation without loss of information in comparison with the truncated SVD. The tensor projection on a vector subspace of reduced dimension can be obtained by using a part of the SVs and the corresponding vectors as an alternative to the commonly used averaging. It was shown that data prediction in the subspace of the significant vectors allows stable assessments of the predicted values to be obtained.

References

Alvarez-Ramirez J. & Rodriguez E. (2021). A singular value decomposition entropy approach for testing stock market efficiency. Physica A: Statistical Mechanics and its Applications, 583, 126337. DOI: https://doi.org/10.1016/j.physa.2021.126337.

Bergqvist G. & Larsson E.G. (2010). The higher-order singular value decomposition: Theory and an application. IEEE Signal Processing Magazine, 27(3), pp. 151–154. DOI: https://doi.org/10.1109/MSP.2010.936030.

Caraiani P. (2014). The predictive power of singular value decomposition entropy for stock market dynamics. Physica A: Statistical Mechanics and its Applications, 393, pp. 571–578. DOI: https://doi.org/10.1016/j.physa.2013.08.071.

Caraiani P. (2018). Modeling the comovement of entropy between financial markets. Entropy, 20(6), 417. DOI: https://doi.org/10.3390/e20060417.

Cichocki A., Lee N., Oseledets I., Phan A.-H., Zhao Q. & Mandic D.P. (2016). Tensor networks for dimensionality reduction and large-scale optimization: Part 1. Low-rank tensor decompositions. Foundations and Trends in Machine Learning, 9(4–5), pp. 249–429. DOI: https://doi.org/10.1561/2200000059.

Cichocki A., Phan A.-H., Zhao Q., Lee N., Oseledets I. & Mandic D.P. (2017). Tensor networks for dimensionality reduction and large-scale optimization: Part 2. Applications and future perspectives. Foundations and Trends in Machine Learning, 9(6), pp. 431–673. DOI: https://doi.org/10.1561/2200000067.

Cheng D., Liu Y., Niu Z. & Zhang L. (2018). Modeling similarities among multi-dimensional financial time series. IEEE Access, 6, pp. 43404–43413. DOI: https://doi.org/10.1109/ACCESS.2018.2862908.

Comon P. (2002). Tensor decompositions: State of the art and applications. In: McWhirter J.G. & Proudler I.K. (Eds.), Mathematics in Signal Processing. Oxford, UK: Clarendon Press, pp. 1–24. DOI: https://doi.org/10.1093/oso/9780198507345.003.0001.

Czapkiewicz A. & Skalna I. (2014). Selected approaches for testing asset pricing models using polish stock market data. Decision Making in Manufacturing and Services, 8(1–2), pp. 25–38. DOI: https://doi.org/10.7494/dmms.2014.8.1.25.

De Lathauwer L., De Moor B. & Vandewalle J. (2000). A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21(4), pp. 1253–1278. DOI: https://doi.org/10.1137/S0895479896305696.

Dixon M.F., Halperin I. & Bilokon P. (2021). Machine learning in finance: From theory to practice. Cham: Springer International Publishing. DOI: https://doi.org/10.1007/978-3-030-41068-1.

Ganeshapillai G., Guttag J. & Lo A. (2013). Learning connections in financial time series. Proceedings of the 30th International Conference on Machine Learning, PMLR, 28(2), pp. 109–117.

Grasedyck L. (2010). Hierarchical singular value decomposition of tensors. SIAM Journal on Matrix Analysis and Applications, 31(4), pp. 2029–2054. DOI: https://doi.org/10.1137/090764189.

Kennet D.Y., Raddant M., Zatlavi L., Lux T. & Ben-Jacob E. (2012). Correlation and dependencies in the global financial village. International Journal of Modern Physics: Conference Series, 16, pp. 13–28. DOI: https://doi.org/10.1142/S201019451200774X.

Marple Jr. S.L. (1987). Digital Spectral Analysis with Applications. Englewood Cliffs: Prentice Hall.

Montagnon C.E. (2020). Forecasting by splitting a time series using singular value decomposition then using both ARMA and a Fokker Planck equation. Physica A: Statistical Mechanics and its Applications, 567, 125708. DOI: https://doi.org/10.1016/j.physa.2020.125708.

Oseledets I.V. (2011). Tensor-train decomposition. SIAM Journal on Scientific Computing, 33(5), pp. 2295–2317. DOI: https://doi.org/10.1137/090752286.

Phan A.-H. & Cichocki A. (2010). Tensor decompositions for feature extraction and classification of high dimensional datasets. Nonlinear Theory and Its Applications, IEICE, 1(1), pp. 37–68. DOI: https://doi.org/10.1587/nolta.1.37.

Savas V. & Eldén V. (2007). Handwritten digit classification using higher-order singular value decomposition. Pattern Recognition, 40(3), pp. 993–1003. DOI: https://doi.org/10.1016/j.patcog.2006.08.004.

Sofina O. & Bunyak Y. (2021). Internet bidding optimization by targeted advertising based on self learning database. In: Tatomyr I. & Kvasnii Z., The economics of postpandemics: prospects and challenges. Praha: OKTAN PRINT, pp. 123–139.

Spentzouris P., Koutsopoulos I., Madsen K.G & Hansen T.V. (2018). Advertiser bidding prediction and optimization in online advertising. In: Iliadis L., Maglogiannis I. & Plagianakos V. (Eds.), Artificial Intelligence Applications and Innovations. 14th IFIP WG 12.5 International Conference, AIAI 2018, Rhodes, Greece, May 25–27, 2018. Proceedings, pp. 413–424. DOI: https://doi.org/10.1007/978-3-319-92007-8_35.

Tucker L.R. (1966). Some mathematical notes on three mode factor analysis. Psychometrika, 31(3), pp. 279–311. DOI: https://doi.org/10.1007/BF02289464.

Vannieuwenhoven N., Vandebril R. & Meerbergen K. (2012). A new truncation strategy for the higher-order singular values decomposition. SIAM Journal on Scientific Computing, 34(2), pp. A1027–A1052. DOI: https://doi.org/10.1137/110836067.

Zhigljavsky A. (2010). Singular Spectrum Analysis for time series: Introduction to this special issue. Statistics and Its Interface, 3(3), pp. 255–258. DOI: https://doi.org/10.4310/SII.2010.v3.n3.a1.

Zniyed Y., Boyer R., de Almeida A.L.F. & Favier G. (2020). A TT-based hierarchical framework for decomposing high-order tensors. SIAM Journal on Scientific Computing, 42(2), pp. A822–A848. DOI: https://doi.org/10.1137/18M1229973.