We consider sparse reconstruction problems which attempt to find sparse solutions to over-determined systems of equations. A basic example of such a problem is to recover a sparse vectorfrom measurements , where with , and captures corruption by noise. Attempting to find maximally sparse solutions is known to be NP-hard, so convex relaxations involving -norms have gained unprecedented popularity. Basis pursuit (or LASSO in statistics literature) minimizes the following objective:
Here is a parameter that balances sparsity versus the norm of the residual error. There is truly a myriad of algorithms for solving (1) (see e.g. [1, 2, 3]), and for large-scale instances, variations of iterative soft thresholding have become very popular:
where applies soft-thresholding for each entry:
Based on operator splitting and proximal projection theories, the algorithm in (2) converges if the spectral norm [4, 5]. This can be achieved simply by rescaling . Accelerated versions of iterative thresholding have appeared .
An exciting albeit simple improvement over -norms for approximating sparsity involves weighting the -norm: with . Ideal weights require knowledge of the sparse solution, but a practical idea is to use weights based on solutions of previous iterations [7, 8]:
This approach can be motivated as a local linearization of the
-heuristic for sparsity. There is strong empirical  and recent theoreical evidence that reweighted approaches improve recovery of sparse signals, in the sense of enabling recovery from fewer measurements [9, 10].
In this paper, we consider the log-regularized formulation that gives rise to the re-weighting schemes mentioned above, and propose a simple prox-like optimization algorithm for its optimization. We derive a closed-form solution for the proximal step, which we call -thresholding. We establish monotone convergence of iterative -thresholding (ILT) to its fixed points, and derive conditions under which these fixed points are local minima of the log-regularized objective. Sparse recovery performance of the method on numerical examples surpasses both soft and hard iterative thresholding (IST and IHT). We also extend the approach to minimizing rank for matrix functions via singular value -thresholding.
To put this into context of related work,  has considered iterative thresholding based on non-convex -norm penalties for sparsity. However, these penalties do not have a connection to re-weighted optimization. Also,  have investigated coordinate descent based solutions for non-convex penalties including the log-penalized penalty, but their approach does not use closed form log-thresholding. Finally, general classes of non-convex penalties, their benefits for sparse recovery, and reweighed convex-type methods for their optimization are studied in . This class of methods is different from the log-thresholding we propose.
2 ISTA as proximal splitting
We briefly review how soft-thresholding can be used to solve the sparse reconstruction problem in (1). Functions of the form where is convex differentiable with a Lipschitz gradient, and is general convex can be solved by a general proximal splitting method :
The prox-operation is a generalization of projection onto a set to general convex functions:
If is an indicator function for a convex set, then the prox-operation is equivalent to the projection onto the set, and ISTA itself is equivalent to the projected gradient approach.
The reweighted- approach can be justified as an iterative upper bounding by a linear approximation to the concave -heuristic for sparsity (here is a small positive constant) :
While the -penalty is concave rather than convex, we still consider the scalar proximal objective around a fixed :
We note that for small enough, the global minimum of over (with held constant) is always at . However, when , the function also exhibits a local minimum, which disappears for small . We show that it is the local, rather than the global minimum, that provides the link to re-weighted minimization and is key to the log-prox algorithm we propose. Indeed, it is also the local (and not global) minimum that provides the link to iteratively re-weighteld algorithms.
Our algorithm arises directly from first order necessary conditions for optimality. For , we solve the equation to find the local minimum in closed-form. We call this operation log-thresholding , :
where . We illustrate log-thresholding in Figure 1. The left plot shows as a function of for several values of . For large the function has a local minimum, but for small the local minimum disappears. For -thresholding we are specifically interested in the the local minimum: an iterative re-weighted approach with small enough step size starting at , i.e. beyond the local minimum, will converge to the local minimum, avoiding the global one. The right plot in Figure 1 shows the -thresholding operation with as a function of . It can be seen as a smooth alternative falling between hard and soft thresholding.
In analogy to ISTA, we can now formally define the iterative -thresholding algorithm:
where applied the element-wise -thresholding operation we obtained in (9). We establish its convergence next.
3.1 Convergence of iterative log-thresholding
The theory of forward-backward splitting does not allow an analysis of -thresholding, because the log is non-convex, and log-thresholding is not a contraction (in particular, it is not firmly non-expansive). Therefore, for the analysis we use an approach based on optimization transfer using surrogate functions  to prove convergence of ILT to its fixed points. At a high-level the analysis follows the program for IHT in , but some of the steps are notably different, and in particular some assumptions on the operator action are necessary to establish that fixed points correspond to local minima of our formulation. In the appendix we establish:
Under the assumption , the ILT algorithm in (10) monotonically decreases the objective in (7), and converges to fixed points. A sufficient condition for these fixed points to be local minima is that restricted to the non-zero coefficients is well-conditioned, specifically that the lowest singular values of the restriction are greater than .
4 Singular Value Log-thresholding
A closely related problem to finding sparse solutions to systems of linear equations is finding low-rank matrices from sparse observations, known as matrix completion:
Similar to sparsity, rank is a combinatorial objective which is typically intractable to optimize directly. However, the nuclear norm , where are the singular values of X, serves as the tightest convex relaxation of rank, analogous to -norm being the convex relaxation of the -norm. In fact, the nuclear norm is exactly the -norm of the singular value spectrum of a matrix. This connection enables the application of various singular value thresholding algorithms: for instance, the SVT algorithm of  alternates soft-thresholding of the singular value spectrum with gradient descent steps. In the experimental section we investigate a simplified singular-value log-thresholding algorithm for matrix completion, where we replace soft thresholding with hard and log-thresholdings. We present very promising empirical results of singular value log-thresholding in Section 5, and a full convergence analysis will appear in a later publication.
We investigate the performance of iterative log thresholding via numerical experiments on noiseless and noisy sparse recovery. Intuitively we expect ILT to recover sparser solution than soft-thresholding due to the connection to re-weighted- norms, and also to behave better than the non-smooth iterative hard thresholding.
First we consider sparse recovery without noise, i.e. we would like to find the sparsest solution that satisfies exactly. One could in principle solve a sequence of problems (1) with decreasing , i.e. increasing penalty on via IST, IHT, ILT. Howeveer, when we know an upper bound on the desired number of non-zero coefficients, a more successful approach is to adaptively change to eliminate all except the top- coefficients in each iteration111 This is easy for IST and IHT by sorting in descending order: let then . For ILT we have from (9). as used e.g. in . We compare the performance of IST, IHT, and the proposed ILT in Figure 2. We have and we vary . Apart from changing the thresholding operator, all the algorithms are exactly the same. The top plot shows the average reconstruction error from the true sparse solution . It is averaged over trials allowing IST, IHT and ILT to run for up-to iterations. The bottom plot shows probability of recovering the true sparse solution. We can see that ILT is superior in both probability of recovery (higher probability of recovery) and in reconstruction error (lower reconstruction error) over both IST and IHT.
Our next experiment compares the three iterative thresholding algorithms on noisy data. Since regularization parameters have a different meaning for the different penalties, we plot the whole solution path of squared residual error vs. sparsity for the three algorithms in Figure 3. We compute the average residual norm for a given level of sparsity for all three algorithms, averaged over runs. We have , and a small amount of noise is added. We can see that the iterative log thresholding consistently achieves the smallest error for each level of sparsity.
In our final experiment we consider singular value log-thresholding for matrix completion. We study a simplified algorithm that parallels the noiseless sparse recovery algorithm with known number of nonzero-elements . We alternate gradient steps with steps of eliminating all but the first singular values by soft, hard and log-thresholding. We have an matrix with observed entries, and rank, . We show the average error in Frobenius norm from the true underlying solution as a function of iteration number over random runs in Figure 4. We see that the convergence of log-SV-thresholding to the correct solution is consistently faster. We expect similar improvements to hold for other algorithms involving soft-thresholding, and to other problems beyond matrix completion, e.g. robust PCA.
6 Convergence of ILT
Note that . Simplifying (6) we have
where and contains terms independent of . The optimization over is now separable, i.e. can be done independently for each coordinate. We can see that finding local minima over of corresponds to iterative log-thresholding.
Using this motivation for ILT, we can now prove convergence to fixed points of . First, we have:
and , are monotonically decreasing with iterations as long as the spectral norm .
The proof parallels the IHT proof of  using
the fact that ,
which is independent of the thresholding used. The main difference for ILT is that
is not the global minimum of but it still holds
Next, we have:
Proof: Given a fixed point of (10) define
Suppose first that . Explicitly writing (10),
squaring both sides, and simplifying, we have
which is precisely equivalent to local optimality of (7) with respect to the th coordinate. Otherwise, suppose . Then we have , and so .
For any fixed point of the ILT algorithm (7) and any small perturbation , if is small enough, for small we have
where and denote the projections onto the zero and nonzero indices of . The precise condition on is as follows:
We now consider lower bounds for each of these two sums taking all WLOG:
Given that (15) holds, the quantity on the last line is positive for small enough. For , we have
where the last inequality comes from the fact that for any , we have .
ILT converges to its fixed points if . Moreover, if the singular values of the restriction of to the columns are greater than , these fixed points must be the local minima of (7).
The result follows from the proof technique of [Theorem 3]; in particular the sums are monotonically increasing and bounded, so the iterates must converge. Finally, we have
This quantity is non-negative provided that the singular values of the restriction of to are greater than .
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