This article studies the problem of reconstructing the topology of a network of interacting agents via observations of the state-evolution of the agents. We focus on the large-scale network setting with the additional constraint of @math observations, where only a small fraction of the agents can be feasibly observed. The goal is to infer the underlying subnetwork of interactions and we refer to this problem as @math @math . In order to study the large-scale setting, we adopt a proper stochastic formulation where the unobserved part of the network is modeled as an Erd " o s-R 'enyi random graph, while the observable subnetwork is left arbitrary. The main result of this work is establishing that, under this setting, local tomography is actually possible with high probability, provided that certain conditions on the network model are met (such as stability and symmetry of the network combination matrix). Remarkably, such conclusion is established under the @math - @math @math , where the cardinality of the observable subnetwork is fixed, while the size of the overall network can increase without bound.
In summary, contrasted with recent results about topology recovery on graphical models, the results obtained in the present work constitute an advance because: @math we deal with a dynamical system, see ); @math the partial observations setting considered in the present work relies on assumptions different from those used in @cite_6 @cite_15 : in our case the unobserved component is Erd "os-R 'enyi, but the subnetwork of observable nodes is deterministic and arbitrary, and the combination matrix obeys transparent conditions borrowed from the adaptive networks literature. We believe that the possibility of working with dynamical models, the arbitrariness of the monitored subnetwork, as well as the direct physical meaning of the conditions on the combination matrix, provide useful novel insights on the problem of topology inference under partial observations.
@cite_15 @cite_6
2770205545 2110355775 2927933146 2951999827
Recovering images from undersampled linear measurements typically leads to an ill-posed linear inverse problem, that asks for proper statistical priors. Building effective priors is however challenged by the low train and test overhead dictated by real-time tasks; and the need for retrieving visually "plausible" and physically "feasible" images with minimal hallucination. To cope with these challenges, we design a cascaded network architecture that unrolls the proximal gradient iterations by permeating benefits from generative residual networks (ResNet) to modeling the proximal operator. A mixture of pixel-wise and perceptual costs is then deployed to train proximals. The overall architecture resembles back-and-forth projection onto the intersection of feasible and plausible images. Extensive computational experiments are examined for a global task of reconstructing MR images of pediatric patients, and a more local task of superresolving CelebA faces, that are insightful to design efficient architectures. Our observations indicate that for MRI reconstruction, a recurrent ResNet with a single residual block effectively learns the proximal. This simple architecture appears to significantly outperform the alternative deep ResNet architecture by 2dB SNR, and the conventional compressed-sensing MRI by 4dB SNR with 100x faster inference. For image superresolution, our preliminary results indicate that modeling the denoising proximal demands deep ResNets. Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that promote the individual structures are known, and allow for recovery using an orderwise optimal number of measurements (e.g., @math norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, using multiobjective optimization with these norms can do no better, orderwise, than exploiting only one of the structures, thus revealing a fundamental limitation in sample complexity. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation. Further, specializing our results to the case of sparse and low-rank matrices, we show that a nonconvex formulation recovers the model from very few measurements (on the order of the degrees of freedom), whereas the convex problem combining the @math and nuclear norms requires many more measurements, illustrating a gap between the performance of the convex and nonconvex recovery problems. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give sample complexity bounds for problems such as sparse phase retrieval and low-rank tensor completion. The current superresolution (SR) methods based on deep learning have shown remarkable comparative advantages but remain unsatisfactory in recovering the high-frequency edge details of the images in noise-contaminated imaging conditions, e.g., remote sensing satellite imaging. In this paper, we propose a generative adversarial network (GAN)-based edge-enhancement network (EEGAN) for robust satellite image SR reconstruction along with the adversarial learning strategy that is insensitive to noise. In particular, EEGAN consists of two main subnetworks: an ultradense subnetwork (UDSN) and an edge-enhancement subnetwork (EESN). In UDSN, a group of 2-D dense blocks is assembled for feature extraction and to obtain an intermediate high-resolution result that looks sharp but is eroded with artifacts and noises as previous GAN-based methods do. Then, EESN is constructed to extract and enhance the image contours by purifying the noise-contaminated components with mask processing. The recovered intermediate image and enhanced edges can be combined to generate the result that enjoys high credibility and clear contents. Extensive experiments on Kaggle Open Source Data set , Jilin-1 video satellite images, and Digitalglobe show superior reconstruction performance compared to the state-of-the-art SR approaches. In this paper, we consider the matrix completion problem when the observations are one-bit measurements of some underlying matrix M, and in particular the observed samples consist only of ones and no zeros. This problem is motivated by modern applications such as recommender systems and social networks where only "likes" or "friendships" are observed. The problem of learning from only positive and unlabeled examples, called PU (positive-unlabeled) learning, has been studied in the context of binary classification. We consider the PU matrix completion problem, where an underlying real-valued matrix M is first quantized to generate one-bit observations and then a subset of positive entries is revealed. Under the assumption that M has bounded nuclear norm, we provide recovery guarantees for two different observation models: 1) M parameterizes a distribution that generates a binary matrix, 2) M is thresholded to obtain a binary matrix. For the first case, we propose a "shifted matrix completion" method that recovers M using only a subset of indices corresponding to ones, while for the second case, we propose a "biased matrix completion" method that recovers the (thresholded) binary matrix. Both methods yield strong error bounds --- if M is n by n, the Frobenius error is bounded as O(1 ((1-rho)n), where 1-rho denotes the fraction of ones observed. This implies a sample complexity of O(n n) ones to achieve a small error, when M is dense and n is large. We extend our methods and guarantees to the inductive matrix completion problem, where rows and columns of M have associated features. We provide efficient and scalable optimization procedures for both the methods and demonstrate the effectiveness of the proposed methods for link prediction (on real-world networks consisting of over 2 million nodes and 90 million links) and semi-supervised clustering tasks.