In this paper, a new method is proposed for low-rank matrix completion which is based on the least squares approximating to the known elements in the manifold formed by the singular vectors of the partial singular value decomposition alternatively. The method can achieve a reduction of the rank of the manifold by gradually reducing the number of the singular value of the thresholding and get the optimal low-rank matrix. It is proven that the manifold-alternative approximating method is convergent under some conditions. Furthermore, compared with the augmented Lagrange multiplier and the orthogonal rank-one matrix pursuit algorithms by random experiments, it is more effective as regards the CPU time and the low-rank property.
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1 Introduction
Matrix completion, proposed by Candès and Recht [7] in 2009, is a challenging problem. There has been a lot of study (see [1‐8, 11‐19, 23‐28, 30, 33‐35]) both in theoretical and algorithmic aspects on this problem. Explicitly seeking the lowest-rank matrix consistent with the known entries is mathematically expressed as
where the matrix \(M\in\mathbb{R}^{n\times n}\) is the unknown matrix, Ω is a random subset of indices for the known entries. The problem occurs in many areas of engineering and applied science, such as model reduction [20], machine learning [1, 2], control [22], pattern recognition [10], imaging inpainting [3] and computer vision [29].
As is well known, Candés and Rechat [7] replaced the rank objective in (1.1) with its convex relaxation, and they showed that the lowest-rank matrices could be recovered exactly from most sufficiently large sets of sampled entries by computing the matrix of minimum nuclear norm that agreed with the provided entries, i.e., the exact matrix completion via convex optimization, as follows:
$$\begin{aligned} &\min_{X\in\mathbb{R}^{n\times n}} \Vert X \Vert _{*} \\ &\quad \mbox{subject to } X_{ij}=M_{ij},\quad (i,j)\in \varOmega, \end{aligned}$$
(1.2)
where the functional \(\|X\|_{*}\) is the nuclear norm of the matrix X, the unknown matrix \(M\in\mathbb{R}^{n\times n}\) of r-rank is square, and one has available m sampled entries \(\{M_{ij}:(i,j)\in\varOmega\}\) with Ω a random subset of cardinality m.
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There have been many algorithms which were designed to attempt to solve the global minimum of (1.2) directly. For example, the hard thresholding algorithms [4, 15, 17, 26], the singular value theresholding (SVT) method [6], the accelerated singular values thresholding method (ASVT [14]), the proximal forward–backward splitting [9], the augmented Lagrange multiplier (ALM [19]) method, the interior point methods [7, 28], and the new gradient projection (NGP [34]) method.
Based on the bi-linear decomposition of an r-rank matrix, some algorithms have been presented to solve (1.1) under the r-rank that is known or can be estimated [20, 21]. We mention the Riemannian geometry method [30] and the Riemannian trust-region method [5, 23], the alternating minimization method [16] and the alternating steepest descent method [26]. The rank of many completion matrices, however, is unknown, so that one has to estimate it ahead of time or approximate it from a lower rank, which causes the difficulty of solving the matrix completion problem. Wen et al. [33] presented the two-stage iteration algorithms for the unknown-rank problem. To decrease the computational cost, based on extending the orthogonal matching pursuit (OMP) procedure from the vector to matrix level, Wang et al. [31] presented an orthogonal rank-one matrix pursuit (OR1MP) method, in which only the top singular vector pair was calculated at each iteration step and an ϵ-feasible solution can be obtained in only \(O(\log(\frac{1}{\epsilon}))\) iterations with less computational cost. However, the method converges to a feasible point rather than the optimal one with minimization rank such that the accuracy is poor and cannot be improved if the rank is reached. In this study, we come up with a manifold-alternative approximating method for solving the problem (1.2) motivated by the above. In an outer iteration, the approximated process can be done in the left-singular vector subspace and the approximation will be alternatively carried out in the right-singular vector subspace in an inner iteration. In a whole iteration, the reduction of the rank results in an alternating optimization, while the completed matrix satisfies \(M_{ij}=(UV^{T})_{ij}\), for \((i,j)\in\varOmega\).
Here are some notations and preliminaries. Let \(\varOmega\subset\{ 1,2,\ldots,n\}\times\{1,2,\ldots,n\}\) denote the indices of the observed entries of the matrix \(X\in\mathbb{R}^{n\times n}\), Ω̄ denote the indices of the missing entries. \(\|X\|_{*}\) represents the nuclear norm (also called Schatten 1-norm) of X, that is, the sum of the singular values of X, \(\|X\|_{2}, \|X\|_{F}\) denote 2-norm and F-norm of X, respectively. We denote by \(\langle X,Y\rangle=\operatorname{trace}(X^{*},Y)\) the inner product between two matrices \((\|X\|_{F}^{2}=\langle X,X\rangle)\). The Cauchy–Schwartz inequality gives \(\langle X,Y\rangle\leq\|X\|_{F}\cdot\|Y\|_{F}\) and it is well known that \(\langle X,Y\rangle\leq\|X\|_{2}\cdot\|Y\|_{*}\) [7, 32].
For a matrix \(A\in\mathbb{R}^{n\times n}\), \(\operatorname{vec}(A)=(a_{1}^{T},a_{2}^{T},\ldots ,a_{n}^{T})^{T}\) denotes a vector reshaped from matrix A by concatenating all its column vectors, dim\((A)\) is always used to represent the dimensions of A and \(r(A)\) stands for the rank of A.
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The rest of the paper is organized as follows. After we provide a brief review of the ALM and the OR1MP methods, a manifold-alternative approximating method is proposed in Sect. 2. The convergence results of the new method are discussed in Sect. 3. Finally, numerical experiments are shown with comparison to other methods in Sect. 4. We end the paper with a concluding remark in Sect. 5.
2 Methods
2.1 The method of augmented Lagrange multipliers
The method of augmented Lagrange multipliers (ALMs) was proposed in [19] for solving a convex optimization (1.2). It should be described subsequently.
Since the matrix completion problem is closely connected to the robust principal component analysis (RPCA) problem, it can be formulated in the same way as RPCA, an equivalent problem of (1.2) can be considered as follows.
As E will compensate for the unknown entries of M, the unknown entries of M are simply set as zeros. Suppose that the given data are arranged as the columns of a large matrix \(M\in\mathbb{R}^{m\times n}\). The mathematical model for estimating the low-dimensional subspace is to find a low-rank matrix \(X\in\mathbb{R}^{m\times n}\), such that the discrepancy between X and M is minimized, leading to the following constrained optimization:
$$\begin{aligned} &\min_{X,E\in\mathbb{R}^{m\times n}} \Vert X \Vert _{*} \\ &\quad \mbox{subject to } X+E=M,\quad \pi_{\varOmega}(E)=0, \end{aligned}$$
(2.1)
where \(\pi_{\varOmega}: \mathbb{R}^{m\times n}\rightarrow\mathbb{R}^{m\times n}\) is a linear operator that keeps the entries in Ω unchanged and sets those outside Ω (say, in Ω̅) zeros. Then the partial augmented Lagrange function is
$$L(X,E,Y,\mu)= \Vert X \Vert _{*}+\langle Y, M-X-E\rangle+\frac{\mu}{2} \Vert M-X-E \Vert _{F}^{2}. $$
The augmented Lagrange multipliers method is summarized in the following:
It is reported that the method of augmented Lagrange multipliers has been applied to the problem (1.2). It is of much better numerical behavior, and it is also of much higher accuracy. However, the method has the disadvantage of the penalty function: the matrix sequences \(\{X_{k}\}\) generated by the method are not feasible. Hence, the accepted solutions are not feasible.
2.2 The method of the orthogonal rank-one matrix pursuit (OR1MP)
We proceed based on the expression of the matrix \(X\in\mathbb {R}^{m\times n}\),
where \(\{M_{i}: i\in\varLambda\}\) is the set of all \(m\times n\) rank-one matrices with unit Frobenius norm.
The original low-rank matrix approximation problem aims to minimize the zero-norm of the vector \(\theta=(\theta_{i})_{i\in\varLambda}\) subject to the equality constraint
where \(\|\theta\|_{0}\) represents the number of nonzero elements of the vector θ, and \(P_{\varOmega}\) is the orthogonal projector onto the span of matrices vanishing outside of Ω.
The authors in [31] reformulate further the problem as
they could solve it by an orthogonal matching pursuit (OMP) type algorithm using rank-one matrices as the basis. It is implemented by two steps alternatively: one is to pursue the basis \(M_{k}\), and the other is to learn the weight of the basis \(\theta_{k}\).
Initialize: Set \(X_{0}=0; \theta^{0}=0\) and \(k=1\).
repeat
Step 1:
Find a pair of top left- and right-singular vectors \((u_{k},v_{k})\) of the observed residual matrix \(R_{k}=Y_{\varOmega}-X_{k-1}\) and set \(M_{k}=u_{k}v_{k}^{T}\).
Step 2:
Compute the weight vector \(\theta^{k}\) using the closed form least squares solution \(\theta^{k}=(\bar{M}_{k}^{T}\bar {M}_{k})^{-1}\bar{M}_{k}^{T}\dot{y}\).
Step 3:
Set \(X_{k}=\sum_{i=1}^{k}\theta _{i}^{k}(M_{i})_{\varOmega}\) and \(k\leftarrow k+1\).
To decrease the computational cost, based on extending the orthogonal matching pursuit (OMP) procedure from the vector to matrix level, Wang et al. [31] presented an orthogonal rank-one matrix pursuit (OR1MP) method, in which only the top singular vector pair was calculated at each iteration step and an ϵ-feasible solution can be obtained in only \(O(\log(\frac{1}{\epsilon }))\) iterations with less computational cost. However, the method converges to a feasible point rather than the optimal one with minimization rank such that the accuracy is poor and cannot be improved if the rank is reached.
2.3 The method of a manifold-alternative approximating (MAA)
For convenience, \([U_{k},\varSigma_{k},V_{k}]_{\tau_{k}}=\operatorname{lansvd}(Y_{k})\) denotes the top-\(\tau_{k}\) singular pairs of the matrix \(Y_{k}\) by using the Lanczos method, where \(U_{k}=(u_{1},u_{2},\ldots,u_{\tau_{k}}), V_{k}=(v_{1},v_{2},\ldots,v_{\tau_{k}})\) and \(\varSigma_{k}=\operatorname{diag}(\sigma _{1k},\sigma_{2k},\ldots,\sigma_{\tau_{k},k}), \sigma_{1k}\geq\sigma _{2k}\geq\cdots\geq\sigma_{\tau_{k},k}>0\).
where \(St_{k}^{m}\) is the Stiefel manifold of \(m\times k\) real, orthogonal matrices, and \(\operatorname{diag}(\sigma_{i})\) denotes a diagonal matrix with \(\sigma_{i}\), \(i=1,2 ,\ldots, k\) on the diagonal.
Method 2.3
(MAA)
Input:\(D=P_{\varOmega}(M)\), \(\operatorname{vec}(D)=D(i,j)\), \((i,j)\in\varOmega\), \(\tau_{0}>0\) (\(\tau_{k}\in{N}^{+}\)), \(0< c_{1},c_{2}<1\), a tolerance \(\epsilon>0\).
Initialize: Set \(Y_{0}=D\) and \(k=0\).
repeat
Step 1:
Compute the partial SVD of the matrix \(Y_{k}: [U_{k},\varSigma_{k},V_{k}]_{\tau_{k}}=\operatorname{lansvd}(Y_{k})\).
Step 2:
Solve the following optimization models, \(\min\|\operatorname{vec}(D)-\operatorname{vec}(P_{\varOmega}(U_{k}X_{k}))\|_{F}\), set \(Y_{k+1}=U_{k}X_{k}\).
Step 3:
When \(\frac{\|Y_{k+1}-Y_{k}\|_{F}}{\|D\| _{F}}<\epsilon\), stop; otherwise, go to the next step.
Step 4:
For \(k>0\), if \(\|\operatorname{vec}(D)-\operatorname{vec}(P_{\varOmega}(Y_{k+1}))\|_{F}< c_{2}\|\operatorname{vec}(D)-\operatorname{vec}(P_{\varOmega}(Y_{k}))\|_{F}, \tau_{k+1}=[c_{1}\tau_{k}]\) go to the next step; otherwise, do
(1):
Set \(Z_{k}=D+P_{\overline{\varOmega}}(Y_{k+1})\), compute the partial SVD of the matrix \(Z_{k}\): \([U_{k},\varSigma_{k},V_{k}]_{\tau_{k}}=\operatorname{lansvd}(Z_{k})\). Let \(W_{K}=U_{k}\varSigma_{k}V_{k}^{T}, \alpha_{k}=\|\operatorname{vec}(D)-\operatorname{vec}(P_{\varOmega}(W_{k}))\|_{F}\).
Set \(Z_{k+\frac{1}{2}}=D+P_{\overline{\varOmega}}(W_{k})\).
Then \(W_{k+\frac{1}{2}}=U_{k+\frac{1}{2}}\varSigma_{k+\frac{1}{2}}V_{k+\frac{1}{2}}^{T}\).
(3):
Solve the following minimum problems, yielding \(Y_{k+\frac{1}{2}}\) and \(\alpha_{k+\frac{1}{2}}\), \(\min \|\operatorname{vec}(D)-\operatorname{vec}(P_{\varOmega}(X_{k+\frac {1}{2}}V_{k+\frac{1}{2}}^{T}))\|_{F}\), set \(Y_{k+\frac{1}{2}}=X_{k+\frac{1}{2}}V_{k+\frac{1}{2}}^{T}\), \(\alpha_{k+\frac {1}{2}}=\|\operatorname{vec}(D)-\operatorname{vec}(P_{\varOmega}(Y_{k+\frac{1}{2}}))\| _{F}\).
Set \(Z_{k+1}=D+P_{\overline{\varOmega}}(Y_{k+\frac{1}{2}})\).
(4):
If \(\alpha_{k+\frac{1}{2}}\leq c_{2}\alpha_{k}\), \(\tau_{k+1}=\tau_{k}-1\); if \(\alpha_{k+\frac{1}{2}}\geq\alpha_{k}\), \(\tau_{k+1}=\tau_{k}+1\), go to Step 1. Otherwise, if \(c_{2}\alpha_{k}\leq \alpha_{k+\frac{1}{2}}<\alpha_{k}\), \(\tau_{k+1}=\tau_{k}\), go to the next step.
Step 5:
\(k:=k+1\), go to Step 2.
until stopping criterion is satisfied
Output: Constructed matrix \(Y_{k}\).
3 Convergence analysis
Now, the convergence theory will be discussed in the following.
Lemma 3.1
Let\(Y^{*}\)be the optimal solution of (1.1). Then there exists a nonnegative number\(\varepsilon_{0}\)such that
Assume that\(\{Y_{k}\}\)is the feasible matrix sequence generated by Method2.3, \(\{W_{k}\}\)is the low-dimensional matrix sequence formed by partial singular pairs, then
then the iteration matrices sequence\(\{Y_{k}\}\)generated by Method2.3converges to the optimal solution\(Y^{*}\)of (1.2) when the terminated rule\(\epsilon\rightarrow0\)is satisfied.
At that time, the procedure can be transferred into Step 4 of Method 2.3, and then \(\tau_{k_{0}+1}=\tau_{k_{0}}+1\); repeat it, there exists an index \(k_{1}\) such that \(r(W_{k_{1}})=r(Y^{*})\).
Case II. We assume that there exists an index \(k_{2}\) such that the inequality (3.1) holds false but \(r(W_{k_{2}})>r(Y^{*})\), and then the procedure can be transferred into the Step 4 of Method 2.3. Because of the assumption (3.1) and Lemma 3.2, we know that there exists an index \(k_{3}\) such that the following holds true:
At that time, \(\tau_{k_{3}+1}=\tau_{k_{3}}-1\), say, the number of dimensionality is decreasing. Repeat the above again and again until there exists an index \(k_{4}\) such that \(r(W_{k_{4}})=r(Y^{*})\).
It is well known that the OR1MP methd is the most simple and efficient for solving problem (1.1) and the ALM method is one of the most popular and efficient methods for solving problem (1.2). In this section we test several experiments to analyze the performance of our Method 2.3, and compare with the ALM and OR1MP methods.
We compare the methods using general matrix completion problem. In the experiments, \(p=m/n^{2}\) denotes the observation ratio, where m is the number of observed entries. Here, \(p=0.1,0.2,0.3,0.5\) are the different choices of the above ratio. The relative error is \(\mathrm{RES}=\frac{\|Y_{k}-D\| _{F}}{\|D\|_{F}}\). The values of the parameters are: \(\tau _{0}=100\), \(c_{1}=0.8\), \(c_{2}=0.9\) and \(\epsilon=5e{-}6\).
The results of the experiments are presented in Tables 1–4. From Tables 1–4 we can see that Method 2.3 takes much fewer iterations (denoted by “IT”)) and requires much less computational time (denoted by CPU) than the ALM and OR1MP methods. Thus, Method 2.3 is much more efficient than the other two methods.
Table 1
Comparison results of three methods for \(p=0.1\)
Size
\(r(Y_{0})\)
Method
RES
IT
CPU
2000 × 2000
20
MAA
6.3989e−05
19
61.0538
ALM
1.3289e−05
147
814.4656
OR1MP
1.5920e−02
100
80.2022
3000 × 3000
30
MAA
2.4436e−04
15
113.8439
ALM
1.2810e−05
155
3448.8579
OR1MP
1.5641e−02
100
177.3123
4000 × 4000
40
MAA
1.2204e−04
13
163.4071
ALM
1.1951e−05
166
9876.8939
OR1MP
1.8042e−02
100
318.5400
5000 × 5000
50
MAA
5.3731e−05
11
210.7015
ALM
9.6254e−06
173
22,641.7724
OR1MP
2.0254e−02
100
505.3112
Table 2
Comparison results of three methods for \(p=0.2\)
Size
\(r(Y_{0})\)
Method
RES
IT
CPU
2000 × 2000
20
MAA
2.4308e−04
10
54.8639
ALM
9.2238e−06
70
237.4327
OR1MP
4.3432e−03
100
95.9820
3000 × 3000
30
MAA
5.0593e−05
8
94.3904
ALM
5.6067e−05
72
863.4068
OR1MP
5.9270e−02
100
213.9196
4000 × 4000
40
MAA
1.3172e−04
8
166.8769
ALM
5.4632e−06
72
2336.2629
OR1MP
8.5351e−03
100
382.8628
5000 × 5000
50
MAA
9.1096e−06
8
248.6944
ALM
1.0802e−05
64
5141.8507
OR1MP
1.1188e−02
100
603.1532
Table 3
Comparison results of three methods for \(p=0.3\)
Size
\(r(Y_{0})\)
Method
RES
IT
CPU
2000 × 2000
20
MAA
6.3561e−05
7
53.9095
ALM
6.8401e−06
44
74.5572
OR1MP
2.0841e−03
100
112.3723
3000 × 3000
30
MAA
6.8760e−06
7
112.3313
ALM
7.8787e−06
46
157.1458
OR1MP
3.3314e−03
100
251.2893
4000 × 4000
40
MAA
1.4268e−05
6
170.5440
ALM
8.7585e−06
45
258.4726
OR1MP
5.5726e−03
100
447.9783
5000 × 5000
50
MAA
1.2876e−05
6
279.8556
ALM
8.7935e−06
43
420.4459
OR1MP
8.0070e−03
100
724.6392
Table 4
Comparison results of three methods for \(p=0.5\)
Size
\(r(Y_{0})\)
Method
RES
IT
CPU
2000 × 2000
20
MAA
1.2636e−05
5
65.7244
ALM
8.7149e−06
25
44.0239
OR1MP
7.3980e−04
100
163.9605
3000 × 3000
30
MAA
7.1438e−06
5
149.9019
ALM
2.7670e−06
24
95.7395
OR1MP
1.4161e−03
100
338.6274
4000 × 4000
40
MAA
5.7499e−06
5
285.8564
ALM
3.6098e−06
25
205.6450
OR1MP
3.1864e−03
100
601.8650
5000 × 5000
50
MAA
5.1190e−06
5
443.8769
ALM
8.8482e−06
25
245.6650
OR1MP
4.9806e−03
100
938.3361
In order to display the effectiveness of our method further, we conduct an experiment on a \(3000\times3000\) matrix with three different ranks 10, 30, 50 for three methods with the observation ratios ranging from 0.1 to 0.9, as shown in Fig. 1.
Figure 1
Comparison of completion performance of the MAA, ALM, OR1MP methods with different percentages of observations: the figures correspond to the results on three \(3000 \times 3000\) random matrices of rank 10 (top figure), rank 30 (middle figure), rank 50 (bottom figure)
×
5 Concluding remark
Based on the least squares approximation to the known elements, we proposed a manifold-alternative approximating method for the low matrix completion problem. Compared with the ALM and OR1MP methods, shown in Tables 1–4, our method performs better as regards the computing time and the low-rank property. The method can achieve a reduction of the rank of the manifold by gradually reducing the number of the singular value of the thresholding and get the optimal low-rank matrix each iteration step.
Acknowledgements
The authors are very much indebted to the editor and anonymous referees for their helpful comments and suggestions. The authors are thankful for the support from the NSF of Shanxi Province (201601D011004).
Availability of data and materials
Please contact author for data requests.
Competing interests
The authors declare that they have no competing interests.
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