Convolutional Dictionary Learning: Acceleration and Convergence

Convolutional dictionary learning (CDL or sparsifying CDL) has many applications in image processing and computer vision. There has been growing interest in developing efficient algorithms for CDL, mostly relying on the augmented Lagrangian (AL) method or the variant alternating direction method of multipliers (ADMM). When their parameters are properly tuned, AL methods have shown fast convergence in CDL. However, the parameter tuning process is not trivial due to its data dependence and, in practice, the convergence of AL methods depends on the AL parameters for nonconvex CDL problems. To moderate these problems, this paper proposes a new practically feasible and convergent Block Proximal Gradient method using a Majorizer (BPG-M) for CDL. The BPG-M-based CDL is investigated with different block updating schemes and majorization matrix designs, and further accelerated by incorporating some momentum coefficient formulas and restarting techniques. All of the methods investigated incorporate a boundary artifacts removal (or, more generally, sampling) operator in the learning model. Numerical experiments show that, without needing any parameter tuning process, the proposed BPG-M approach converges more stably to desirable solutions of lower objective values than the existing state-of-the-art ADMM algorithm and its memory-efficient variant do. Compared to the ADMM approaches, the BPG-M method using a multi-block updating scheme is particularly useful in single-threaded CDL algorithm handling large datasets, due to its lower memory requirement and no polynomial computational complexity. Image denoising experiments show that, for relatively strong additive white Gaussian noise, the filters learned by BPG-M-based CDL outperform those trained by the ADMM approach.Comment: 21 pages, 7 figures, submitted to IEEE Transactions on Image Processin

On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization

Conventional wisdom in deep learning states that increasing depth improves expressiveness but complicates optimization. This paper suggests that, sometimes, increasing depth can speed up optimization. The effect of depth on optimization is decoupled from expressiveness by focusing on settings where additional layers amount to overparameterization - linear neural networks, a well-studied model. Theoretical analysis, as well as experiments, show that here depth acts as a preconditioner which may accelerate convergence. Even on simple convex problems such as linear regression with $\ell_p$ loss, $p>2$, gradient descent can benefit from transitioning to a non-convex overparameterized objective, more than it would from some common acceleration schemes. We also prove that it is mathematically impossible to obtain the acceleration effect of overparametrization via gradients of any regularizer.Comment: Published at the International Conference on Machine Learning (ICML) 201

Deep BCD-Net Using Identical Encoding-Decoding CNN Structures for Iterative Image Recovery

In "extreme" computational imaging that collects extremely undersampled or noisy measurements, obtaining an accurate image within a reasonable computing time is challenging. Incorporating image mapping convolutional neural networks (CNN) into iterative image recovery has great potential to resolve this issue. This paper 1) incorporates image mapping CNN using identical convolutional kernels in both encoders and decoders into a block coordinate descent (BCD) signal recovery method and 2) applies alternating direction method of multipliers to train the aforementioned image mapping CNN. We refer to the proposed recurrent network as BCD-Net using identical encoding-decoding CNN structures. Numerical experiments show that, for a) denoising low signal-to-noise-ratio images and b) extremely undersampled magnetic resonance imaging, the proposed BCD-Net achieves significantly more accurate image recovery, compared to BCD-Net using distinct encoding-decoding structures and/or the conventional image recovery model using both wavelets and total variation.Comment: 5 pages, 3 figure