Paper:
Hierarchical Sparse Coding (HSC) is a powerful model to efficiently represent multi-dimensional, structured data such as images. The simplest solution to solve this computationally hard problem is to decompose it into independent layerwise subproblems. However, neuroscientific evidence would suggest inter-connecting these subproblems as in the Predictive Coding (PC) theory, which adds top-down connections between consecutive layers. In this study, a new model called 2-Layers Sparse Predictive Coding (2L-SPC) is introduced to assess the impact of this inter-layer feedback connection. In particular, the 2L-SPC is compared with a Hierarchical Lasso (Hi-La) network made out of a sequence of Lasso layers. 2L-SPC and a 2-layers Hi-La networks are trained on 4 different databases and with different sparsity parameters on each layer. First, we show that the overall prediction error generated by 2L-SPC is lower thanks to the feedback mechanism as it transfers prediction error between layers. Second, we demonstrate that the inference stage of the 2L-SPC is faster to converge than for the Hi-La model. Third, we show that the 2L-SPC also accelerates the learning process. Finally, the qualitative analysis of both models dictionaries, supported by their activation probability, show that the 2L-SPC features are more generic and informative.

Review:
This work proposed a new model called Sparse Deep Predictive Coding, which introduced top-down connections between consecutive layers, to improve the solutions to hierarchical sparse coding (HSC) problems. Instead of decomposing the HSC problem into independent subproblems, the proposed model added a new term to the loss function, which represents the influence of the latter-layer on the current layer.
#Pros:
-- The proposed model adopted the idea from predictive coding and came up with a relatively novel idea for HSC problems.
-- The experiments are solid. The experiments evaluated the proposed methods with different hyper-parameter settings and three real-world datasets.
-- The figures in the result sections are well designed and concise.
#Cons:
-- The mathematical description of the main problem and the proposed model is not clear. For example, the dimensionality for the variables in Eq.(1) is not clarified.
-- The test procedure is not clear. How the internal state variables are obtained for the test set is not clarified.
-- The proposed model was only compared with a basic Hierarchical Lasso network. There are not any state-of-art methods included as baseline methods.
#Detailed comments:
(1) The proposed model is named as sparse DEEP predictive coding, however, the experiments only considered the SDPC and Hi-La networks with 2 layers. I am wondering if a deeper structure will improve the performance?
(2) For the structure shown in Fig.1, the decoding dictionaries are $D^T_i$, but I am confused why the encoding dictionaries are reciprocal to encoding dictionaries. Does it come from the optimization updates shown in Eq.(3)?
(3) According to Eq.(1), $x$ is a vector and $D$ is a 2d matrix. However, the real inputs in the experiments are images and $D$ is a convolutional filter with 4 dimensions. How are the matrices reshaped?
(4) For section 2.2 and 2.3, the number/index of samples is not shown in the loss function for training. The loss should be over the whole training set. Besides, the test procedure is not clarified.
(4) The number of iterations using FISTA of SDPC and Hi-La networks is shown to compare the rate of convergence. However, considering both models are solving a lasso-type regression problem, I would suggest using coordinate descent for optimization.
(5) For the main result of prediction error, why is the “global prediction error” more important than the reconstruction error? Is the first-layer prediction error the reconstruction error? If yes, Fig.2 shows that Hi-La has a lower prediction error compared to SDPC for the first layer.
(6) Two minor comments on writing:
(a) It would be better to have a separate section for 2.5 since it describes the dataset and is not related to the proposed model.
(b) A typo of “neuronal implementation” exists in the introduction section.