First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper proposes projective dictionary pair learning (DPL) to jointly learn a synthesis and analysis dictionary so as to apply the dictionary learning approach to the classification setting while also enabling more efficient computation. Overall the paper proposes an interesting method. Even though the proposed method is a straight-forward extension of DL, the empirical results (esp the computational gains) looks quite promising. It would have been great to see: i) an accompanying theoretical analysis esp around convergence ii) analysis of the complexity of method as sample size increases (and hence with increasing m) It is interesting how the method imposes sparsity indirectly (as seen in Sec 2.4) by training $p^*_k$ to produce small coefficients.

--In this paper, we introduce a Deep Convolutional Analysis Dictionary Model (DeepCAM) by learning convolutional dictionaries instead of unstructured dictionaries as in the case of deep analysis dictionary model introduced in the companion paper . Convolutional dictionaries are more suitable for processing high-dimensional signals like for example images and have only a small number of free parameters. By exploiting the properties of a convolutional dictionary, we present an efficient convolutional analysis dictionary learning approach. A L-layer DeepCAM consists of L layers of convolutional analysis dictionary and element-wise soft-thresholding pairs and a single layer of convolutional synthesis dictionary. Similar to DeepAM, each convolutional analysis dictionary is composed of a convolutional Information Preserving Analysis Dictionary (IPAD) and a con-volutional Clustering Analysis Dictionary (CAD). The IPAD and the CAD are learned using variations of the proposed learning algorithm. We demonstrate that DeepCAM is an effective multi-layer convolutional model and, on single image super-resolution, achieves performance comparable with other methods while also showing good generalization capabilities. ONVOLUTIONAL dictionary learning has attracted increasing interests in signal and image processing communities as it leads to a more elegant framework for high-dimensional signal analysis. An advantage of convolutional dictionaries [1]-[13] is that they can take the high-dimensional signal as input for sparse representation and processing, whereas traditional approaches [14]-[20] have to divide the high-dimensional signal into overlapping low-dimensional patches and perform sparse representation on each patch independently. It is a structured dictionary and can be represented as a concatenation of Toeplitz matrices where each Toeplitz matrix is constructed using the taps of a filter and the usual assumption is that the filters are with compact support. So a convolutional dictionary is effective for processing high-dimensional signals while also restraining the number of free parameters. To achieve efficient convolutional dictionary learning, the convolutional dictionary is usually modelled as a concatenation of circulant matrices [1]-[6] by assuming a periodic boundary condition on the signals. As all circulant matrices share the same set of eigenvectors which is the Discrete Fourier Transform (DFT) matrix, a circular convolution can be therefore represented as a multiplication in Fourier domain and can be efficiently implemented using Fast Fourier Transform (FFT). However, using a circulant matrix to approximate a general Toeplitz matrix may lead to boundary artifacts [3], [21], [22] especially when the boundary region is large. A multi-layer convolutional dictionary model is able to represent multiple levels of abstraction of the input signal.

Inspired by the recent success of Deep Neural Networks and the recent efforts to develop multi-layer dictionary models, we propose a Deep Analysis dictionary Model (DeepAM) which is optimized to address a specific regression task known as single image super-resolution. Contrary to other multi-layer dictionary models, our architecture contains L layers of analysis dictionary and soft-thresholding operators to gradually extract high-level features and a layer of synthesis dictionary which is designed to optimize the regression task at hand. In our approach, each analysis dictionary is partitioned into two sub-dictionaries: an Information Preserving Analysis Dictionary (IPAD) and a Clustering Analysis Dictionary (CAD). The IPAD together with the corresponding soft-thresholds is designed to pass the key information from the previous layer to the next layer, while the CAD together with the corresponding soft-thresholding operator is designed to produce a sparse feature representation of its input data that facilitates discrimination of key features. Simulation results show that the proposed deep analysis dictionary model achieves comparable performance with a Deep Neural Network which has the same structure and is optimized using back-propagation.

Dictionary learning (DL) is an effective feature learning technique, and has led to interesting results in many classification tasks. Recently, by combining DL with multiple kernel learning (which is a crucial and effective technique for combining different feature representation information), a few multi-kernel DL methods have been presented to solve the multiple feature representations based classification problem. However, how to improve the representation capability and discriminability of multi-kernel dictionary has not been well studied. In this paper, we propose a novel multi-kernel DL approach, named multi-kernel low-rank dictionary pair learning (MKLDPL). Specifically, MKLDPL jointly learns a kernel synthesis dictionary and a kernel analysis dictionary by exploiting the class label information. The learned synthesis and analysis dictionaries work together to implement the coding and reconstruction of samples in the kernel space. To enhance the discriminability of the learned multi-kernel dictionaries, MKLDPL imposes the low-rank regularization on the analysis dictionary, which can make samples from the same class have similar representations. We apply MKLDPL for multiple features based image classification task. Experimental results demonstrate the effectiveness of the proposed approach.

Unsupervised feature selection (UFS) aims to reduce the time complexity and storage burden, as well as improve the generalization performance. Most existing methods convert UFS to supervised learning problem by generating labels with specific techniques (e.g., spectral analysis, matrix factorization and linear predictor). Instead, we proposed a novel coupled analysis-synthesis dictionary learning method, which is free of generating labels. The representation coefficients are used to model the cluster structure and data distribution. Specifically, the synthesis dictionary is used to reconstruct samples, while the analysis dictionary analytically codes the samples and assigns probabilities to the samples. Afterwards, the analysis dictionary is used to select features that can well preserve the data distribution. The effective L2p-norm (0 < p <1) regularization is imposed on the analysis dictionary to get much sparse solution and is more effective in feature selection.We proposed an iterative reweighted least squares algorithm to solve the L2p-norm optimization problem and proved it can converge to a fixed point. Experiments on benchmark datasets validated the effectiveness of the proposed method

Dictionary learning has played an important role in the success of sparse representation. Although synthesis dictionary learning for sparse representation has been well studied for universality representation (i.e., the dictionary is universal to all classes) and particularity representation (i.e., the dictionary is class-particular), jointly learning an analysis dictionary and a synthesis dictionary is still in its infant stage. Universality-particularity representation can well match the intrinsic characteristics of data (i.e., different classes share commonality and distinctness), while analysis-synthesis dictionary can give a more complete view of data representation (i.e., analysis dictionary is a dual-viewpoint of synthesis dictionary). In this paper, we proposed a novel model of analysis-synthesis dictionary learning for universality-particularity (ASDL-UP) representation based classification. The discrimination of universality and particularity representation is jointly exploited by simultaneously learning a pair of analysis dictionary and synthesis dictionary. More specifically, we impose a label preserving term to analysis coding coefficients for universality representation. Fisher-like regularizations for analysis coding coefficients and the subsequent synthesis representation are introduced to particularity representation. Compared with other state-of-the-art dictionary learning methods, ASDL-UP has shown better or competitive performance in various classification tasks.