We study an extention of total variation denoising over images to over Cartesian power graphs and its applications to estimating non-parametric network models. The power graph fused lasso (PGFL) segments a matrix by exploiting a known graphical structure, @math , over the rows and columns. Our main results shows that for any connected graph, under subGaussian noise, the PGFL achieves the same mean-square error rate as 2D total variation denoising for signals of bounded variation. We study the use of the PGFL for denoising an observed network @math , where we learn the graph @math as the @math -nearest neighborhood graph of an estimated metric over the vertices. We provide theoretical and empirical results for estimating graphons, a non-parametric exchangeable network model, and compare to the state of the art graphon estimation methods.
We next turn our attention to graphon estimation using the PGFL on a learned graph. The statistical limits of graphon estimation have been well characterized for smooth graphons, and it was found that computationally intractable profile likelihood maximization is minimax optimal for H "older graphons, @cite_27 @cite_26 . One tractable approach to graphon estimation is to order the vertices according to some graph statistics, such as the degrees of @math , and then treat the resulting re-ordered matrix @math as an image and applying image segmentation tools ( @math is the permutation associated with this sorting). This methodology is called sorting and smoothing (SAS), and in @cite_23 they use TV denoising to perform the image segmentation. The implicit assumption is that the degree is a decent proxy for the latent variable, @math , which does not hold for most graphons.
@cite_27 @cite_26 @cite_23
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We consider the task of learning a classifier for semantic segmentation using weak supervision in the form of image labels which specify the object classes present in the image. Our method uses deep convolutional neural networks (CNNs) and adopts an Expectation-Maximization (EM) based approach. We focus on the following three aspects of EM: (i) initialization; (ii) latent posterior estimation (E-step) and (iii) the parameter update (M-step). We show that saliency and attention maps, our bottom-up and top-down cues respectively, of simple images provide very good cues to learn an initialization for the EM-based algorithm. Intuitively, we show that before trying to learn to segment complex images, it is much easier and highly effective to first learn to segment a set of simple images and then move towards the complex ones. Next, in order to update the parameters, we propose minimizing the combination of the standard softmax loss and the KL divergence between the true latent posterior and the likelihood given by the CNN. We argue that this combination is more robust to wrong predictions made by the expectation step of the EM method. We support this argument with empirical and visual results. Extensive experiments and discussions show that: (i) our method is very simple and intuitive; (ii) requires only image-level labels; and (iii) consistently outperforms other weakly-supervised state-of-the-art methods with a very high margin on the PASCAL VOC 2012 dataset. We consider the task of learning a classifier for semantic segmentation using weak supervision in the form of image labels specifying objects present in the image. Our method uses deep convolutional neural networks (cnns) and adopts an Expectation-Maximization (EM) based approach. We focus on the following three aspects of EM: (i) initialization; (ii) latent posterior estimation (E-step) and (iii) the parameter update (M-step). We show that saliency and attention maps, bottom-up and top-down cues respectively, of images with single objects (simple images) provide highly reliable cues to learn an initialization for the EM. Intuitively, given weak supervisions, we first learn to segment simple images and then move towards the complex ones. Next, for updating the parameters (M step), we propose to minimize the combination of the standard softmax loss and the KL divergence between the latent posterior distribution (obtained using the E-step) and the likelihood given by the cnn. This combination is more robust to wrong predictions made by the E step of the EM algorithm. Extensive experiments and discussions show that our method is very simple and intuitive, and outperforms the state-of-the-art method with a very high margin of 3.7 and 3.9 on the PASCAL VOC12 train and test sets respectively, thus setting new state-of-the-art results. The estimation of probabilities of network edges from the observed adjacency matrix has important applications to predicting missing links and network denoising. It has usually been addressed by estimating the graphon, a function that determines the matrix of edge probabilities, but this is ill-defined without strong assumptions on the network structure. Here we propose a novel computationally efficient method, based on neighborhood smoothing to estimate the expectation of the adjacency matrix directly, without making the structural assumptions that graphon estimation requires. The neighborhood smoothing method requires little tuning, has a competitive mean-squared error rate, and outperforms many benchmark methods on link prediction in simulated and real networks. SummaryThe estimation of probabilities of network edges from the observed adjacency matrix has important applications to the prediction of missing links and to network denoising. It is usually addressed by estimating the graphon, a function that determines the matrix of edge probabilities, but this is ill-defined without strong assumptions on the network structure. Here we propose a novel computationally efficient method, based on neighbourhood smoothing, to estimate the expectation of the adjacency matrix directly, without making the structural assumptions that graphon estimation requires. The neighbourhood smoothing method requires little tuning, has a competitive mean squared error rate and outperforms many benchmark methods for link prediction in simulated and real networks.