This paper aims to solve the following general nonconvex nonsmooth low-rank minimization problem
where denotes the -th singular value of (we assume in this work). The penalty function
and loss functionsatisfy the following assumptions:
is continuous, concave and monotonically increasing on . It is possibly nonsmooth.
: is a smooth function of type , i.e., the gradient is Lipschitz continuous,
for any , is called Lipschitz constant of . is possibly nonconvex.
If is convex, it is the most widely used convex relaxation of the rank minimization problem:
The above low-rank minimization problem arises in many machine learning tasks such as multiple category classification , matrix completion , multi-task learning , and low-rank representation with squared loss for subspace segmentation . However, solving problem (4) is usually difficult, or even NP-hard. Most previous works solve the convex problem (3) instead. It has been proved that under certain incoherence assumptions on the singular values of the matrix, solving the convex nuclear norm regularized problem leads to a near optimal low-rank solution . However, such assumptions may be violated in real applications. The obtained solution by using nuclear norm may be suboptimal since it is not a perfect approximation of the rank function. A similar phenomenon has been observed in the convex -norm and nonconvex -norm for sparse vector recovery .
In order to achieve a better approximation of the -norm, many nonconvex surrogate functions of -norm have been proposed, including -norm , Smoothly Clipped Absolute Deviation (SCAD) , Logarithm , Minimax Concave Penalty (MCP) , Capped , Exponential-Type Penalty (ETP) , Geman , and Laplace . Table 1 tabulates these penalty functions and Figure 1 visualizes them. One may refer to  for more properties of these penalty functions. Some of these nonconvex penalties have been extended to approximate the rank function, e.g. the Schatten- norm . Another nonconvex surrogate of rank function is the truncated nuclear norm .
For nonconvex sparse minimization, several algorithms have been proposed to solve the problem with a nonconvex regularizer. A common method is DC (Difference of Convex functions) programming . It minimizes the nonconvex function based on the assumption that both and are convex. In each iteration, DC programming linearizes at , and minimizes the relaxed function as follows
where is a subgradient of at . DC programming may be not very efficient, since it requires some other iterative algorithm to solve (5). Note that the updating rule (5) of DC programming cannot be extended to solve the low-rank problem (1). The reason is that for concave , does not guarantee to be convex w.r.t. . DC programming also fails when is nonconvex in problem (1).
Another solver is to use the proximal gradient algorithm which is originally designed for convex problem . It requires computing the proximal operator of ,
in each iteration. However, for nonconvex , there may not exist a general solver for (6). Even if (6) is solvable, different from convex optimization, does not always hold. Thus we cannot perform on the singular values of directly for solving
The nonconvexity of makes the nonconvex low-rank minimization problem much more challenging than the nonconvex sparse minimization.
Another related work is the Iteratively Reweighted Least Squares (IRLS) algorihtm. It has been recently extended to handle the nonconvex Schatten- norm penalty . Actually it solves a relaxed smooth problem which may require many iterations to achieve a low-rank solution. It cannot solve the general nonsmooth problem (1). The alternative updating algorithm in  minimizes the truncated nuclear norm by using a special property of this penalty. It contains two loops, both of which require computing SVD. Thus it is not very efficient. It cannot be extended to solve the general problem (1) either.
In this work, all the existing nonconvex surrogate functions of -norm are extended on the singular values of a matrix to enhance low-rank recovery. In problem (1), can be any existing nonconvex penalty function shown in Table 1 or any other function which satisfies the assumption (A1). We observe that all the existing nonconvex surrogate functions are concave and monotonically increasing on . Thus their gradients (or supergradients at the nonsmooth points) are nonnegative and monotonically decreasing. Based on this key fact, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve problem (1). IRNN computes the proximal operator of the weighted nuclear norm, which has a closed form solution due to the nonnegative and monotonically decreasing supergradients. In theory, we prove that IRNN monotonically decreases the objective function value, and any limit point is a stationary point. To the best of our knowledge, IRNN is the first work which is able to solve the general problem (1) with convergence guarantee. Note that for nonconvex optmization, it is usually very difficult to prove that an algorithm converges to stationary points. At last, we test our algorithm with several nonconvex penalty functions on both synthetic data and real image data to show the effectiveness of the proposed algorithm.
2 Nonconvex Nonsmooth Low-Rank Minimization
In this section, we present a general algorithm to solve problem (1). To handle the case that is nonsmooth, e.g., Capped penalty, we need the concept of supergradient defined on the concave function.
2.1 Supergradient of a Concave Function
The subgradient of the convex function is an extension of gradient at a nonsmooth point. Similarly, the supergradient is an extension of gradient of the concave function at a nonsmooth point. If is concave and differentiable at , it is known that
Let be concave. A vector is a supergradient of at the point if for every , the following inequality holds
All supergradients of at are called the superdifferential of at , and are denoted as . If is differentiable at , is also a supergradient, i.e., . Figure 2 illustrates the supergradients of a concave function at both differentiable and nondifferentiable points.
For concave , is convex, and vice versa. From this fact, we have the following relationship between the supergradient of and the subgradient of .
Let be concave and . For any , , and vice versa.
The relationship of the supergradient and subgradient shown in Lemma 1 is useful for exploring some properties of the supergradient. It is known that the subdiffierential of a convex function is a monotone operator, i.e.,
for any , . The superdifferential of a concave function holds a similar property, which is called antimonotone operator in this work.
The superdifferential of a concave function is an antimonotone operator, i.e.,
for any , .
when . That is to say, the supergradient of is monotonically decreasing on . Table 1 shows some usual concave functions and their supergradients. We also visualize them in Figure 1. It can be seen that they all satisfy the assumption (A1). Note that for the penalty, we further define that . This will not affect our algorithm and convergence analysis as shown latter. The Capped penalty is nonsmooth at , with the superdifferential .
2.2 Iteratively Reweighted Nuclear Norm
In this subsection, we show how to solve the general nonconvex and possibly nonsmooth problem (1) based on the assumptions (A1)-(A2). For simplicity of notation, we denote and .
Since is concave on , by the definition of the supergradient, we have
Since , by the antimonotone property of supergradient (12), we have
This property is important in our algorithm shown latter. (13) motivates us to minimize its right hand side instead of . Thus we may solve the following relaxed problem
It seems that updating by solving the above weighted nuclear norm problem (16) is an extension of the weighted -norm problem in IRL1 algorithm  (IRL1 is a special DC programming algorithm). However, the weighted nuclear norm is nonconvex in (16) (it is convex if and only if ), while the weighted -norm is convex. Solving the nonconvex problem (16) is much more challenging than the convex weighted -norm problem. In fact, it is not easier than solving the original problem (1).
Input: - A Lipschitz constant of .
Initialize: , , and , .
while not converge do
Update by solving problem (18).
Update the weights , , by
Instead of updating by solving (16), we linearize at and add a proximal term:
where . Such a choice of guarantees the convergence of our algorithm as shown latter. Then we update by solving
[8, Theorem 2.3] For any , and , a globally optimal solution to the following problem
is given by the weighted singular value thresholding
where is the SVD of , and .
It is worth mentioning that for the penalty, if , . By the updating rule of in (18), we have . This guarantees that the rank of the sequence is nonincreasing.
3 Convergence Analysis
In this section, we give the convergence analysis for the IRNN algorithm. We will show that IRNN decreases the objective function value monotonically, and any limit point is a stationary point of problem (1). We first recall the following well-known and fundamental property for a smooth function in the class .
Proof. First, since is a global solution to problem (18), we get
It can be rewritten as
Second, since the gradient of is Lipschitz continuous, by using Lemma 4, we have
Third, since , by the definition of the supergradient, we have
Thus is monotonically decreasing. Summing all the inequalities in (25) for , we get
In particular, it implies that . The boundedness of is obtained based on the assumption (A3).
Proof. The sequence generated in Algorithm 1 is bounded as shown in Theorem 1. Thus there exists a matrix and a subsequence such that . From the fact that in Theorem 1, we have . Thus for . By the choice of and Lemma 1, we have . By the upper semi-continuous property of the subdifferential [9, Proposition 2.1.5], there exists such that . Again by Lemma 1, and .
4 Extension to Other Problems
Our proposed IRNN algorithm can solve a more general low-rank minimization problem as follows,
where , , are concave, and their supergradients satisfy , for any , . The truncated nuclear norm  satisfies the above assumption. Indeed, by letting
Their supergradients are
The convergence results in Theorem 1 and 2 also hold since (24) holds for each . Compared with the alternating updating algorithms in , which require double loops, our IRNN algorithm will be more efficient and with stronger convergence guarantee.
More generally, IRNN can solve the following problem
when is concave, and the following problem
can be cheaply solved. An interesting application of (33) is to extend the group sparsity on the singular values. By dividing the singular values into groups, i.e., , , , , where , we can define the group sparsity on the singular values as . This is exactly the first term in (33) by letting be the -norm of a vector. can be nonconvex functions satisfying the assumption (A1) or specially the convex absolute function.
In this section, we present several experiments on both synthetic data and real images to validate the effectiveness of the IRNN algorithm. We test our algorithm on the matrix completion problem
where is the set of indices of samples, and is a linear operator that keeps the entries in unchanged and those outside zeros. The gradient of squared loss function in (35) is Lipschitz continuous, with a Lipschitz constant . We set in Algorithm 1. For the choice of , we test all the penalty functions listed in Table 1 except for Capped and Geman, since we find that their recovery performances are sensitive to the choices of and in different cases. For the choice of in IRNN, we use a continuation technique to enhance the low-rank matrix recovery. The initial value of is set to a larger value , and dynamically decreased by with . It is stopped till reaching a predefined target .
is initialized as a zero matrix. For the choice of parameters (e.g.,and ) in the nonconvex penalty functions, we search it from a candidate set and use the one which obtains good performance in most cases 111Code of IRNN: https://sites.google.com/site/canyilu/..
5.1 Low-Rank Matrix Recovery
We first compare our nonconvex IRNN algorithm with state-of-the-art convex algorithms on synthetic data. We conduct two experiments. One is for the observed matrix without noise, and the other one is for with noise.
For the noise free case, we generate the rank matrix as , where , and are generated by the Matlab command randn. elements of are missing uniformly at random. We compare our algorithm with Augmented Lagrange Multiplier (ALM) 222Code: http://perception.csl.illinois.edu/matrix-rank/sample_code.html.  which solves the noise free problem
For this task, we set , , and in IRNN, and stop the algorithm when . For ALM, we use the default parameters in the released codes. We evaluate the recovery performance by the Relative Error defined as where is the recovered solution by a certain algorithm. If the Relative Error is smaller than , is regarded as a successful recovery of . We repeat the experiments 100 times with the underlying rank varying from 20 to 33 for each algorithm. The frequency of success is plotted in Figure 2(a). The legend IRNN- in Figure 2(a) denotes the penalty function used in problem (1) and solved by our proposed IRNN algorithm. It can be seen that IRNN with all the nonconvex penalty functions achieves much better recovery performance than the convex ALM algorithm. This is because the nonconvex penalty functions approximate the rank function better than the convex nuclear norm.
For the noisy case, the data are generated by +0.1randn. We compare our algorithm with convex Accelerated Proximal Gradient with Line search (APGL) 333Code: http://www.math.nus.edu.sg/~mattohkc/NNLS.html.  which solves the noisy problem
For this task, we set , and in IRNN. All the chosen algorithms are run 100 times with the underlying rank lying between 15 and 35. The relative errors can be ranging for each test, and the mean errors by different methods are plotted in Figure 2(b). It can be seen that IRNN for the nonconvex penalty outperforms the convex APGL for the noisy case. Note that we cannot conclude from Figure 3 that IRNN with , Logarithm and ETP penalty functions always perform better than SCAD and MCP, since the obtained solutions are not globally optimal.
5.2 Application to Image Recovery
In this section, we apply matrix completion for image recovery. As shown in Figure 4, the real image may be corrupted by different types of noises, e.g., Gaussian noise or unrelated text. Usually the real images are not of low-rank, but the top singular values dominate the main information . Thus the corrupted image can be recovered by low-rank approximation. For color images which have three channels, we simply apply matrix completion for each channel independently. The well known Peak Signal-to-Noise Ratio (PSNR) is employed to evaluate the recovery performance. We compare IRNN with some other matrix completion algorithms which have been applied for this task, including APGL, Low-Rank Matrix Fitting (LMaFit) 444Code: http://lmafit.blogs.rice.edu/..  and Truncated Nuclear Norm Regularization (TNNR) . We use the solver based on ADMM to solve a subproblem of TNNR in the released codes (denoted as TNNR-ADMM) 555Code: https://sites.google.com/site/zjuyaohu/.. We try to tune the parameters to be optimal of the chosen algorithms and report the best result.
In our test, we consider two types of noises on the real images. The first one replaces of pixels with random values (sample image (1) in Figure 4 (b)). The other one adds some unrelated texts on the image (sample image (2) in Figure 4 (b)). Figure 4 (c)-(g) show the recovered images by different methods. It can be observed that our IRNN method with different penalty functions achieves much better recovery performance than APGL and LMaFit. Only the results by IRNN- and IRNN-SCAD are plotted due to the limit of space. We further test on more images and plot the results in Figure 5. Figure 6 shows the PSNR values of different methods on all the test images. It can be seen that IRNN with all the evaluated nonconvex functions achieves higher PSNR values, which verifies that the nonconvex penalty functions are effective in this situation. The nonconvex truncated nuclear norm is close to our methods, but its running time is 35 times of that for ours.
6 Conclusions and Future Work
In this work, the nonconvex surrogate functions of -norm are extended on the singular values to approximate the rank function. It is observed that all the existing nonconvex surrogate functions are concave and monotonically increasing on . Then a general solver IRNN is proposed to solve problem (1) with such penalties. IRNN is the first algorithm which is able to solve the general nonconvex low-rank minimization problem (1) with convergence guarantee. The nonconvex penalty can be nonsmooth by using the supergradient at the nonsmooth point. In theory, we proved that any limit point is a local minimum. Experiments on both synthetic data and real images demonstrated that IRNN usually outperforms the state-of-the-art convex algorithms. An interesting future work is to solve the nonconvex low-rank minimization problem with affine constraint. A possible way is to combine IRNN with Alternating Direction Method of Multiplier (ADMM).
This research is supported by the Singapore National Research Foundation under its International Research Centre @Singapore Funding Initiative and administered by the IDM Programme Office. Z. Lin is supported by NSF of China (Grant nos. 61272341, 61231002, and 61121002) and MSRA.
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