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Research article Special Issues

A fast matrix completion method based on truncated L2,1 norm minimization

  • Received: 05 January 2024 Revised: 08 February 2024 Accepted: 29 February 2024 Published: 07 March 2024
  • In recent years, a truncated nuclear norm regularization (TNNR) method has obtained much attention from researchers in machine learning and image processing areas, because it is much more accurate on matrices with missing data than other traditional methods based on nuclear norm. However, the TNNR method is reported to be very slow, due to its large number of singular value decomposition (SVD) iterations. In this paper, a truncated L2,1 norm minimization method was presented for fast and accurate matrix completion, which is abbreviated as TLNM. In the proposed TLNM method, the truncated nuclear norm minimization model of TNNR was improved to a truncated L2,1 norm minimization model that aimed to optimize the truncated L2,1 Norm and a weighted noisy matrix simultaneously for improving the accuracy of TLNM. Using Qatar Riyal (QR) decomposition to calculate the orthogonal bases for reconstructing recovery results, the proposed TLNM method is much faster than the TNNR method. Adequate results for color images validate the effectiveness and efficiency of TLNM comparing with TNNR and other competing methods.

    Citation: Zhengyu Liu, Yufei Bao, Changhai Wang, Xiaoxiao Chen, Qing Liu. A fast matrix completion method based on truncated L2,1 norm minimization[J]. Electronic Research Archive, 2024, 32(3): 2099-2119. doi: 10.3934/era.2024095

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  • In recent years, a truncated nuclear norm regularization (TNNR) method has obtained much attention from researchers in machine learning and image processing areas, because it is much more accurate on matrices with missing data than other traditional methods based on nuclear norm. However, the TNNR method is reported to be very slow, due to its large number of singular value decomposition (SVD) iterations. In this paper, a truncated L2,1 norm minimization method was presented for fast and accurate matrix completion, which is abbreviated as TLNM. In the proposed TLNM method, the truncated nuclear norm minimization model of TNNR was improved to a truncated L2,1 norm minimization model that aimed to optimize the truncated L2,1 Norm and a weighted noisy matrix simultaneously for improving the accuracy of TLNM. Using Qatar Riyal (QR) decomposition to calculate the orthogonal bases for reconstructing recovery results, the proposed TLNM method is much faster than the TNNR method. Adequate results for color images validate the effectiveness and efficiency of TLNM comparing with TNNR and other competing methods.



    We consider the following family of nonlinear oscillators

    yzz+k(y)y3z+h(y)y2z+f(y)yz+g(y)=0, (1.1)

    where k, h, f0 and g0 are arbitrary sufficiently smooth functions. Particular members of (1.1) are used for the description of various processes in physics, mechanics and so on and they also appear as invariant reductions of nonlinear partial differential equations [1,2,3].

    Integrability of (1.1) was studied in a number of works [4,5,6,7,8,9,10,11,12,13,14,15,16]. In particular, in [15] linearization of (1.1) via the following generalized nonlocal transformations

    w=F(y),dζ=(G1(y)yz+G2(y))dz. (1.2)

    was considered. However, equivalence problems with respect to transformations (1.2) for (1.1) and its integrable nonlinear subcases have not been studied previously. Therefore, in this work we deal with the equivalence problem for (1.1) and its integrable subcase from the Painlevé-Gambier classification. Namely, we construct an equivalence criterion for (1.1) and a non-canonical form of Ince Ⅶ equation [17,18]. As a result, we obtain two new integrable subfamilies of (1.1). What is more, we demonstrate that for any equation from (1.1) that satisfy one of these equivalence criteria one can construct an autonomous first integral in the parametric form. Notice that we use Ince Ⅶ equation because it is one of the simplest integrable members of (1.1) with known general solution and known classification of invariant curves.

    Moreover, we show that transformations (1.2) preserve autonomous invariant curves for equations from (1.1). Since the considered non-canonical form of Ince Ⅶ equation admits two irreducible polynomial invariant curves, we obtain that any equation from (1.1), which is equivalent to it, also admits two invariant curves. These invariant curves can be used for constructing an integrating factor for equations from (1.1) that are equivalent to Ince Ⅶ equation. If this integrating factor is Darboux one, then the corresponding equation is Liouvillian integrable [19]. This demonstrates the connection between nonlocal equivalence approach and Darboux integrability theory and its generalizations, which has been recently discussed for a less general class of nonlocal transformations in [20,21,22].

    The rest of this work is organized as follows. In the next Section we present an equivalence criterion for (1.1) and a non-canonical form of the Ince Ⅶ equation. In addition, we show how to construct an autonomous first integral for an equation from (1.1) satisfying this equivalence criterion. We also demonstrate that transformations (1.2) preserve autonomous invariant curves for (1.1). In Section 3 we provide two examples of integrable equations from (1.1) and construct their parametric first integrals, invariant curves and integrating factors. In the last Section we briefly discuss and summarize our results.

    We begin with the equivalence criterion between (1.1) and a non-canonical form of the Ince Ⅶ equation, that is [17,18]

    wζζ+3wζ+ϵw3+2w=0. (2.1)

    Here ϵ0 is an arbitrary parameter, which can be set, without loss of generality, to be equal to ±1.

    The general solution of (1.1) is

    w=e(ζζ0)cn{ϵ(e(ζζ0)C1),12}. (2.2)

    Here ζ0 and C1 are arbitrary constants and cn is the Jacobian elliptic cosine. Expression (2.2) will be used below for constructing autonomous parametric first integrals for members of (1.1).

    The equivalence criterion between (1.1) and (2.1) can be formulated as follows:

    Theorem 2.1. Equation (1.1) is equivalent to (2.1) if and only if either

    (I)25515lgp2qy+2352980l10+(3430q6667920p3)l514580qp310q276545lgqppy=0, (2.3)

    or

    (II)343l5972p3=0, (2.4)

    holds. Here

    l=9(fgygfy+fgh3kg2)2f3,p=gly3lgy+l(f23gh),q=25515gylp25103lgppy+686l58505p2(f23gh)l+6561p3. (2.5)

    The expression for G2 in each case is either

    (I)G2=126l2qp2470596l10(1333584p3+1372q)l5+q2, (2.6)

    or

    (II)G22=49l3G2+9p2189pl. (2.7)

    In all cases the functions F and G1 are given by

    F2=l81ϵG32,G1=G2(f3G2)3g. (2.8)

    Proof. We begin with the necessary conditions. Substituting (1.2) into (2.1) we get

    yzz+k(y)y3z+h(y)y2z+f(y)yz+g(y)=0, (2.9)

    where

    k=FG31(ϵF2+2)+3G21Fy+G1FyyFyG1,yG2Fy,h=G2Fyy+(6G1G2G2,y)Fy+3FG2G21(ϵF2+2)G2Fy,f=3G2(Fy+FG1(ϵF2+2))Fy,g=FG22(ϵF2+2)Fy. (2.10)

    As a consequence, we obtain that (1.1) can be transformed into (2.1) if it is of the form (2.9) (or (1.1)).

    Conversely, if the functions F, G1 and G2 satisfy (2.10) for some values of k, h, f and g, then (1.1) can be mapped into (2.1) via (1.2). Thus, we see that the compatibility conditions for (2.10) as an overdertmined system of equations for F, G1 and G2 result in the necessary and sufficient conditions for (1.1) to be equivalent to (2.1) via (1.2).

    To obtain the compatibility conditions, we simplify system (2.10) as follows. Using the last two equations from (2.10) we find the expression for G1 given in (2.8). Then, with the help of this relation, from (2.10) we find that

    81ϵF2G32l=0, (2.11)

    and

    567lG32+(243lgh81lf281gly+243lgy)G27l2=0,243lgG2,y+324lG3281glyG2+2l2=0, (2.12)

    Here l is given by (2.5).

    As a result, we need to find compatibility conditions only for (2.12). In order to find the generic case of this compatibility conditions, we differentiate the first equation twice and find the expression for G22 and condition (2.3). Differentiating the first equation from (2.12) for the third time, we obtain (2.6). Further differentiation does not lead to any new compatibility conditions. Particular case (2.4) can be treated in the similar way.

    Finally, we remark that the cases of l=0, p=0 and q=0 result in the degeneration of transformations (1.2). This completes the proof.

    As an immediate corollary of Theorem 2.1 we get

    Corollary 2.1. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then an autonomous first integral of this equation can be presented in the parametric form as follows:

    y=F1(w),yz=G2wζFyG1wζ. (2.13)

    Here w is the general solution of (2.1) given by (2.2). Notice also that, formally, (2.13) contains two arbitrary constants, namely ζ0 and C1. However, without loss of generality, one of them can be set equal to zero.

    Now we demonstrate that transformations (1.2) preserve autonomous invariant curves for equations from (1.1).

    First, we need to introduce the definition of an invariant curve for (1.1). We recall that Eq (1.1) can be transformed into an equivalent dynamical system

    yz=P,uz=Q,P=u,Q=ku3hu2fug. (2.14)

    A smooth function H(y,u) is called an invariant curve of (2.14) (or, equivalently, of (1.1)), if it is a nontrivial solution of [19]

    PHy+QHu=λH, (2.15)

    for some value of the function λ, which is called the cofactor of H.

    Second, we need to introduce the equation that is equivalent to (1.1) via (1.2). Substituting (1.2) into (1.1) we get

    wζζ+˜kw3ζ+˜hw2ζ+˜fwζ+˜g=0, (2.16)

    where

    ˜k=kG32gG31+(G1,yhG1)G22+(fG1G2,y)G1G2F2yG22,˜h=(hFyFyy)G22(2fG1G2,y)G2Fy+3gG21FyF2yG22,˜f=fG23gG1G22,˜g=gFyG22. (2.17)

    An invariant curve for (2.16) can be defined in the same way as that for (1.1). Notice that, further, we will denote wζ as v.

    Theorem 2.2. Suppose that either (1.1) possess an invariant curve H(y,u) with the cofactor λ(y,u) or (2.16) possess an invariant curve ˜H(w,v) with the cofactor ˜λ(w,v). Then, the other equation also has an invariant curve and the corresponding invariant curves and cofactors are connected via

    H(y,u)=˜H(F,FyuG1u+G2),λ(y,u)=(G1u+G2)˜λ(F,FyuG1u+G2). (2.18)

    Proof. Suppose that ˜H(w,v) is an invariant curve for (2.16) with the cofactor ˜λ(w,v). Then it satisfies

    v˜Hw+(˜kv3˜hv2˜fv˜g)˜Hv=˜λ˜H. (2.19)

    Substituting (1.2) into (2.19) we get

    uHy+(ku3hu2fug)H=(G1u+G2)˜λ(F,FyuG1u+G2)H. (2.20)

    This completes the proof.

    As an immediate consequence of Theorem 2.2 we have that transformations (1.2) preserve autonomous first integrals admitted by members of (1.1), since they are invariant curves with zero cofactors.

    Another corollary of Theorem 2.2 is that any equation from (1.1) that is connected to (2.1) admits two invariant curves that correspond to irreducible polynomial invariant curves of (2.1). This invariant curves of (2.1) and the corresponding cofactors are the following (see, [23] formulas (3.18) and (3.19) taking into account scaling transformations)

    ˜H=±i2ϵ(v+w)+w2,˜λ=±2ϵw2. (2.21)

    Therefore, we have that the following statement holds:

    Corollary 2.2. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then is admits the following invariant curves with the corresponding cofactors

    H=±i2ϵ(FyuG1u+G2+F)+F2,λ=(G1u+G2)(±2ϵF2). (2.22)

    Let us remark that connections between (2.1) and non-autonomous variants of (1.1) can be considered via a non-autonomous generalization of transformations (1.2). However, one of two nonlocally related equations should be autonomous since otherwise nonlocal transformations do not map a differential equation into a differential equation [5].

    In this Section we have obtained the equivalence criterion between (1.1) and (2.1), that defines two new completely integrable subfamilies of (1.1). We have also demonstrated that members of these subfamilies posses an autonomous parametric first integral and two autonomous invariant curves.

    In this Section we provide two examples of integrable equations from (1.1) satisfying integrability conditions from Theorem 2.1.

    Example 1. One can show that the coefficients of the following cubic oscillator

    yzz12ϵμy(ϵμ2y4+2)2y3z6μyyz+2μ2y3(ϵμ2y4+2)=0, (3.1)

    satisfy condition (2.3) from Theorem 2.1. Consequently, Eq (3.1) is completely integrable and its general solution can be obtained from (2.2) by inverting transformations (1.2). However, it is more convenient to use Corollary 2.1 and present the autonomous first integral of (3.1) in the parametric form as follows:

    y=±wμ,yz=w(ϵw2+2)wζ2wζ+w(ϵw2+2), (3.2)

    where w is given by (2.2), ζ is considered as a parameter and ζ0, without loss of generality, can be set equal to zero. As a result, we see that (3.1) is integrable since it has an autonomous first integral.

    Moreover, using Corollary 2.2 one can find invariant curves admitted by (3.1)

    H1,2=y4[(2±ϵμy2)2(2ϵμy2)+2(ϵμy22ϵ)u]2μ2y2(ϵμ2y4+2)4u,λ1,2=±2(μy2(ϵμ2y4+2)2u)(2ϵμy22)y(ϵμ2y4+2) (3.3)

    With the help of the standard technique of the Darboux integrability theory [19], it is easy to find the corresponding Darboux integrating factor of (3.1)

    M=(ϵμ2y4+2)94(2ϵu2+(ϵμ2y4+2)2)34(μy2(ϵμ2y4+2)2u)32. (3.4)

    Consequently, equation is (3.1) Liouvillian integrable.

    Example 2. Consider the Liénard (1, 9) equation

    yzz+(biyi)yz+ajyj=0,i=0,4,j=0,,9. (3.5)

    Here summation over repeated indices is assumed. One can show that this equation is equivalent to (2.1) if it is of the form

    yzz9(y+μ)(y+3μ)3yz+2y(2y+3μ)(y+3μ)7=0, (3.6)

    where μ is an arbitrary constant.

    With the help of Corollary 2.1 one can present the first integral of (3.6) in the parametric form as follows:

    y=32ϵμw22ϵw,yz=77762ϵμ5wwζ(2ϵw2)5(2ϵwζ+(2ϵw+2ϵ)w), (3.7)

    where w is given by (2.2). Thus, one can see that (3.5) is completely integrable due to the existence of this parametric autonomous first integral.

    Using Corollary 2.2 we find two invariant curves of (3.6):

    H1=y2[(2y+3μ)(y+3μ)42u)](y+3μ)2[(y+3μ)4yu],λ1=6μ(uy(y+3μ)4)y(y+3μ), (3.8)

    and

    H2=y2(y+3μ)2y(y+3μ)4u,λ2=2(2y+3μ)(u2y(y+3μ)4)y(y+3μ). (3.9)

    The corresponding Darboux integrating factor is

    M=[y(y+3μ)4u]32[(2y+3μ)(y+3μ)42u]34. (3.10)

    As a consequence, we see that Eq (3.6) is Liouvillian integrable.

    Therefore, we see that equations considered in Examples 1 and 2 are completely integrable from two points of view. First, they possess autonomous parametric first integrals. Second, they have Darboux integrating factors.

    In this work we have considered the equivalence problem between family of Eqs (1.1) and its integrable member (2.1), with equivalence transformations given by generalized nonlocal transformations (1.2). We construct the corresponding equivalence criterion in the explicit form, which leads to two new integrable subfamilies of (1.1). We have demonstrated that one can explicitly construct a parametric autonomous first integral for each equation that is equivalent to (2.1) via (1.2). We have also shown that transformations (1.2) preserve autonomous invariant curves for (1.1). As a consequence, we have obtained that equations from the obtained integrable subfamilies posses two autonomous invariant curves, which corresponds to the irreducible polynomial invariant curves of (2.1). This fact demonstrate a connection between nonlocal equivalence approach and Darboux and Liouvillian integrability approach. We have illustrate our results by two examples of integrable equations from (1.1).

    The author was partially supported by Russian Science Foundation grant 19-71-10003.

    The author declares no conflict of interest in this paper.



    [1] J. Miao, K. I. Kou, Color image recovery using low-rank quaternion matrix completion algorithm, IEEE Trans. Image Process., 31 (2022), 190–201. https://doi.org/10.1109/TIP.2021.3128321 doi: 10.1109/TIP.2021.3128321
    [2] Z. Jia, Q. Jin, M. K. Ng, X. Zhao, Non-local robust quaternion matrix completion for large-scale color image and video inpainting, IEEE Trans. Image Process., 31 (2022), 3868–3883. https://doi.org/10.1109/TIP.2022.3176133 doi: 10.1109/TIP.2022.3176133
    [3] X. Li, H. Zhang, R. Zhang, Matrix completion via non-convex relaxation and adaptive correlation learning, IEEE Trans. Pattern Anal. Mach. Intell., 45 (2023), 1981–1991. https://doi.org/10.1109/TPAMI.2022.3157083 doi: 10.1109/TPAMI.2022.3157083
    [4] H. Cai, L. Huang, P. Li, D. Needell, Matrix completion with cross-concentrated sampling: Bridging uniform sampling and CUR sampling, IEEE Trans. Pattern Anal. Mach. Intell., 45 (2023), 10100–10113. https://doi.org/10.1109/TPAMI.2023.3261185 doi: 10.1109/TPAMI.2023.3261185
    [5] S. Bhattacharya, S. Chatterjee, Matrix completion with data-dependent missingness probabilities, IEEE Trans. Inf. Theory, 68 (2022), 6762–6773. https://doi.org/10.1109/TIT.2022.3170244 doi: 10.1109/TIT.2022.3170244
    [6] Y. Yang, C. Ma, Optimal tuning-free convex relaxation for noisy matrix completion, IEEE Trans. Inf. Theory, 69 (2023), 6571–6585. https://doi.org/10.1109/TIT.2023.3284341 doi: 10.1109/TIT.2023.3284341
    [7] K. F. Masood, J. Tong, J. Xi, J. Yuan, Y. Yu, Inductive matrix completion and root-MUSIC-based channel estimation for intelligent reflecting surface (IRS)-aided hybrid MIMO systems, IEEE Trans. Wireless Commun., 22 (2023), 7917–7931. https://doi.org/10.1109/TWC.2023.3257138 doi: 10.1109/TWC.2023.3257138
    [8] O. Elnahas, Y. Ma, Y. Jiang, Z. Quan, Clock synchronization in wireless networks using matrix completion-based maximum likelihood estimation, IEEE Trans. Wireless Commun., 19 (2020), 8220–8231. https://doi.org/10.1109/TWC.2020.3020191 doi: 10.1109/TWC.2020.3020191
    [9] Q. Liu, X. Li, H. Cao, Two-dimensional localization: Low-rank matrix completion with random sampling in massive MIMO system, IEEE Syst. J., 15 (2020), 3628–3631. https://doi.org/10.1109/JSYST.2020.3012775 doi: 10.1109/JSYST.2020.3012775
    [10] Z. Liu, X. Li, H. C. So, ℓ0-norm minimization based robust matrix completion approach for MIMO radar target localization, IEEE Trans. Aerosp. Electronic Syst., 59 (2023), 6759–6770. https://doi.org/10.1109/TAES.2023.3280462 doi: 10.1109/TAES.2023.3280462
    [11] M. Fazel, Matrix Rank Minimization with Applications, Ph.D thesis, Stanford University, 2002.
    [12] E. J. Candès, B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), 717–772. https://doi.org/10.1007/s10208-009-9045-5 doi: 10.1007/s10208-009-9045-5
    [13] J. Cai, E. Candès, Z. Shen, A singular value thresholding method for matrix completion, SIAM J. Optim., 20 (2010), 1956–1982. https://doi.org/10.1137/080738970 doi: 10.1137/080738970
    [14] S. Ma, D. Goldfarb, L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Math. Program., 128 (2011), 321–353. https://doi.org/10.1007/s10107-009-0306-5 doi: 10.1007/s10107-009-0306-5
    [15] K. C. Toh, S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems, Pacific J. Optim., 6 (2010), 615–640.
    [16] R. Meka, P. Jain, I. Dhillon, Guaranteed rank minimization via singular value projection, in Proceedings of Advances in Neural Information Processing Systems, (2009), 937–945.
    [17] Y. Hu, D. Zhang, J. Ye, X. Li, X. He, Fast and accurate matrix completion via truncated nuclear norm regularization, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 2117–2130. https://doi.org/10.1109/TPAMI.2012.271 doi: 10.1109/TPAMI.2012.271
    [18] Y. Liu, L. Jiao, F. Shang, F. Yin, F. Liu, An efficient matrix bi-factorization alternative optimization method for low-rank matrix recovery and completion, Neural Networks, 48 (2013), 8–18. https://doi.org/10.1016/j.neunet.2013.06.013 doi: 10.1016/j.neunet.2013.06.013
    [19] Y. Liu, L. Jiao, F. Shang, A fast tri-factorization method for low-rank matrix recovery and completion, Pattern Recognit., 46 (2013), 163–173. https://doi.org/10.1016/j.patcog.2012.07.003 doi: 10.1016/j.patcog.2012.07.003
    [20] Z. Wen, W. Yin, Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation, Math. Program. Comput., 4 (2012), 333–361. https://doi.org/10.1007/s12532-012-0044-1 doi: 10.1007/s12532-012-0044-1
    [21] S. Gu, L. Zhang, W. Zuo, X. Feng, Weighted nuclear norm minimization with application to image denoising, in Proceedings of 2014 IEEE Conference on Computer Vision and Pattern Recognition, (2014).
    [22] C. Wen, W. Qian, Q. Zhang, F. Cao, Algorithms of matrix recovery based on truncated Schatten p-norm, Int. J. Mach. Learn. Cybern., 12 (2021), 1557–1570. https://doi.org/10.1007/s13042-020-01256-7 doi: 10.1007/s13042-020-01256-7
    [23] G. Li, G. Guo, S. Peng, C. Wang, S. Yu, J. Niu, et al., Matrix completion via Schatten capped p norm, IEEE Trans. Knowl. Data Eng., 34 (2022), 394–404. https://doi.org/10.1109/TKDE.2020.2978465 doi: 10.1109/TKDE.2020.2978465
    [24] Q. Liu, Z. Lai, Z. Zhou, F. Kuang, Z. Jin, A truncated nuclear norm regularization method based on weighted residual error for matrix completion, IEEE Trans. Image Process., 25 (2016), 316–330. https://doi.org/10.1109/TIP.2015.2503238 doi: 10.1109/TIP.2015.2503238
    [25] W. Zeng, H. C. So, Outlier-robust matrix completion via ℓp-minimization, IEEE Trans. Signal Process., 66 (2018), 1125–1140. https://doi.org/10.1109/TSP.2017.2784361 doi: 10.1109/TSP.2017.2784361
    [26] X. P. Li, Q. Liu, H. C. So, Rank-one matrix approximation with ℓp-norm for image inpainting, IEEE Signal Process. Lett., 27 (2020), 680–684. https://doi.org/10.1109/LSP.2020.2988596 doi: 10.1109/LSP.2020.2988596
    [27] M. Muma, W. Zeng, A. M. Zoubir, Robust M-estimation based matrix completion, in ICASSP 2019–2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2019), 5476–5480. https://doi.org/10.1109/ICASSP.2019.8682657
    [28] Y. He, F. Wang, Y. Li, J. Qin, B. Chen, Robust matrix completion via maximum correntropy criterion and half-quadratic optimization, IEEE Trans. Signal Process., 68 (2019), 181–195. https://doi.org/10.1109/TSP.2019.2952057 doi: 10.1109/TSP.2019.2952057
    [29] Z. Shi, X. P. Li, W. Li, T. Yan, J. Wang, Y. Fu, Robust low-rank matrix recovery as mixed integer programming via ℓ0-norm optimization, IEEE Signal Process. Lett., 30 (2023), 1012–1016. https://doi.org/10.1109/LSP.2023.3301244 doi: 10.1109/LSP.2023.3301244
    [30] Q. Zhao, D. Meng, Z. Xu, W. Zuo, Y. Yan, L1-norm low rank matrix factorization by variational Bayesian method, IEEE Trans. Neural Network Learn. Syst., 26 (2015), 825–839. https://doi.org/10.1109/TNNLS.2014.2387376 doi: 10.1109/TNNLS.2014.2387376
    [31] X. P. Li, Z. Shi, Q. Liu, H. C. So, Fast robust matrix completion via entry-wise ℓ2-norm minimization, IEEE Trans. Cybern., 53 (2023), 7199–7212. https://doi.org/10.1109/TCYB.2022.3224070 doi: 10.1109/TCYB.2022.3224070
    [32] Q. Liu, F. Davoine, J. Yang, Y. Cui, Z. Jin, F. Han, A fast and accurate matrix completion method based on QR decomposition and L2, 1-norm minimization, IEEE Trans. Neural Networks Learn. Syst., 30 (2019), 803–817. https://doi.org/10.1109/TNNLS.2018.2851957
    [33] Q. Liu, C. Peng, P. Yang, X. Zhou, Z. Liu, A fast matrix completion method based on matrix bifactorization and QR decomposition, Wireless Commun. Mobile Comput., 2023 (2023), 2117876. https://doi.org/10.1155/2023/2117876 doi: 10.1155/2023/2117876
    [34] M. Yang, Y. Li, J. Wang, Feature and nuclear norm minimization for matrix completion, IEEE Trans. Knowl. Data Eng., 34 (2022), 2190–2199. https://doi.org/10.1109/TKDE.2020.3005978 doi: 10.1109/TKDE.2020.3005978
    [35] Q. Fan, Y. Liu, T. Yang, H. Peng, Fast and accurate spectrum estimation via virtual co-array interpolation based on truncated nuclear norm regularization, IEEE Signal Process. Lett., 29 (2021), 169–173. https://doi.org/10.1109/LSP.2021.3130018 doi: 10.1109/LSP.2021.3130018
    [36] G. Morison, SURE based truncated tensor nuclear norm regularization for low rank tensor completion, in Proceedings of the 28th European Signal Processing Conference (EUSIPCO) in 2020, (2021), 2001–2005. https://doi.org/10.23919/Eusipco47968.2020.9287726
    [37] J. Dong, Z. Xue, J. Guan, Z. F. Han, W. Wang, Low rank matrix completion using truncated nuclear norm and sparse regularizer, Signal Process. Image Commun., 68 (2018), 76–87. https://doi.org/10.1016/j.image.2018.06.007 doi: 10.1016/j.image.2018.06.007
    [38] M. Zhang, M. Zhang, F. Zhao, F. Zhang, Y. Liu, A. Evans, Truncated weighted nuclear norm regularization and sparsity for image denoising, in Proceedings of 2023 IEEE International Conference on Image Processing (ICIP), (2023), 1825–1829. https://doi.org/10.1109/ICIP49359.2023.10221971
    [39] Y. Song, J. Li, X. Chen, D. Zhang, Q. Tang, K. Yang, An efficient tensor completion method via truncated nuclear norm, J. Visual Commun. Image Representation, 70 (2020), 102791. https://doi.org/10.1016/j.jvcir.2020.102791 doi: 10.1016/j.jvcir.2020.102791
    [40] Y. Qiu, G. Zhou, J. Zeng, Q. Zhao, S. Xie, Imbalanced low-rank tensor completion via latent matrix factorization, Neural Networks, 155 (2022), 369. https://doi.org/10.1016/j.neunet.2022.08.023 doi: 10.1016/j.neunet.2022.08.023
    [41] Z. Hu, F. Nie, R. Wang, X. Li, Low rank regularization: A review, Neural Networks, 136 (2021), 218. https://doi.org/10.1016/j.neunet.2020.09.021 doi: 10.1016/j.neunet.2020.09.021
    [42] Q. Shen, S. Yi, Y. Liang, Y. Chen, W. Liu, Bilateral fast low-rank representation with equivalent transformation for subspace clustering, IEEE Trans. Multimedia, 25 (2023), 6371–6383. https://doi.org/10.1109/TMM.2022.3207922 doi: 10.1109/TMM.2022.3207922
    [43] Y. Yang, J. Zhang, S. Song, C. Zhang, D. Liu, Low-rank and sparse matrix decomposition with orthogonal subspace projection-based background suppression for hyperspectral anomaly detection, IEEE Geosci. Remote Sens. Lett., 17 (2020), 1378–1382. https://doi.org/10.1109/LGRS.2019.2948675 doi: 10.1109/LGRS.2019.2948675
    [44] X. Wang, X. P. Li, H. C. So, Truncated quadratic norm minimization for bilinear factorization-based matrix completion, Signal Process., 214 (2024), 109219. https://doi.org/10.1016/j.sigpro.2023.109219 doi: 10.1016/j.sigpro.2023.109219
    [45] J. Wen, H. He, Z. He, F. Zhu, A pseudo-inverse-based hard thresholding algorithm for sparse signal recovery, IEEE Trans. Intell. Transp. Syst., 24 (2023), 7621–7630. https://doi.org/10.1109/TITS.2022.3172868 doi: 10.1109/TITS.2022.3172868
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