Image restoration, aiming to restore high-quality images from their degraded counterparts, is a fundamental computer vision problem and has been studied for many years. Due to its ill-posed nature, early research efforts [1,2,3]typically focus on developing single-task models, with each model handling only one specific degradation.

In this work, we demonstrate that these two seemingly orthogonal concepts are remarkably well-suited for each other.

We explore combining dropout with robust training methods and obtain better generalization.

's leaf nodes to form Given the definition of our attention in Eq. 9 in the main text, the highest polynomial order is Before providing the proof of Theorem 4, we establish Lemma 1 as its foundation. We follow the principle of Y an et al's work [ Figure 1, we consider two kinds of value functions, i.e., In P AM-Net, we set the number of levels to 2. A grid search is performed over different configurations We conduct grid searches on the dropout rate over {0, 0.1, 0.2} and the initial

Learning feature interactions can be the key for multivariate predictive modeling. ReLU-activated neural networks create piecewise linear prediction models.

First we provide Lemma 8, which will be used in the proof of Proposition 2 and 4. Lemma 8. Given n non-negative feature vectors fi =[fi0,fi1,...,fim], where i=1,...,n, there exists n matrices Qi with shape nm m and n vector ˆfi =QifTi, s.t. Proposition 2. A factor graph G =(V,C,E) with variable log potentialsθi(xi) and factor log potentials ϕc(xc) can be converted to a factor graph G0 with the same variable potentials and the decomposed log-potentials ϕic(xi,zc) using a one-layer FGNN. Without loss of generality, we assume that logφc(xc)>1. Then for each i the item θic(xi,zc) in (9) have kn+1 entries, and each entry is either a scaled entry of the vectorgc or arbitrary negative number less than maxxcθc(xc). Thusifweorganize θic(xi,zc) asalength-kn+1 vector fic, thenwedefinea kn+1 kn matrix Qci, where if and only if thelth entry of fic is set to the mth entry of gc multiplied by 12 1/|s(c)|, the entry of Qci in lth row, mth column will be set to 1/|s(c)|; all the other entries of Qci is set to some negative number smaller than maxxcθc(xc).

Most of the successful deep neural network architectures are structured, often consisting of elements like convolutional neural networks and gated recurrent neural networks. Recently, graph neural networks (GNNs) have been successfully applied to graph-structureddata such as point cloud and molecular data. These networks often only consider pairwise dependencies, as they operate on a graph structure.