We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Sphere Packings, Lattices and Groups

The compressive sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices provide measurement mappings that, with overwhelming probability, achieve nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the binary e-stable embedding property, which characterizes the robustness of the measurement process to sign changes. We show that the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the binary iterative hard thresholding algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.

Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erdős-Renyi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification of online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.

/pdf/discrete-signal-processing-on-graphs-sampling-theory-54wtrrynr1.pdf

Discrete Signal Processing on Graphs: Sampling Theory

Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

/pdf/solving-inverse-problems-using-data-driven-models-3jeyo4a896.pdf

Solving inverse problems using data-driven models

We present a new image reconstruction method that replaces the projector in a projected gradient descent (PGD) with a convolutional neural network (CNN). Recently, CNNs trained as image-to-image regressors have been successfully used to solve inverse problems in imaging. However, unlike existing iterative image reconstruction algorithms, these CNN-based approaches usually lack a feedback mechanism to enforce that the reconstructed image is consistent with the measurements. We propose a relaxed version of PGD wherein gradient descent enforces measurement consistency, while a CNN recursively projects the solution closer to the space of desired reconstruction images. We show that this algorithm is guaranteed to converge and, under certain conditions, converges to a local minimum of a non-convex inverse problem. Finally, we propose a simple scheme to train the CNN to act like a projector. Our experiments on sparse-view computed-tomography reconstruction show an improvement over total variation-based regularization, dictionary learning, and a state-of-the-art deep learning-based direct reconstruction technique.

CNN-Based Projected Gradient Descent for Consistent CT Image Reconstruction

We present a new method for image reconstruction which replaces the projector in a projected gradient descent (PGD) with a convolutional neural network (CNN). CNNs trained as high-dimensional (image-to-image) regressors have recently been used to efficiently solve inverse problems in imaging. However, these approaches lack a feedback mechanism to enforce that the reconstructed image is consistent with the measurements. This is crucial for inverse problems, and more so in biomedical imaging, where the reconstructions are used for diagnosis. In our scheme, the gradient descent enforces measurement consistency, while the CNN recursively projects the solution closer to the space of desired reconstruction images. We provide a formal framework to ensure that the classical PGD converges to a local minimizer of a non-convex constrained least-squares problem. When the projector is replaced with a CNN, we propose a relaxed PGD, which always converges. Finally, we propose a simple scheme to train a CNN to act like a projector. Our experiments on sparse view Computed Tomography (CT) reconstruction for both noiseless and noisy measurements show an improvement over the total-variation (TV) method and a recent CNN-based technique.

CNN-Based Projected Gradient Descent for Consistent Image Reconstruction

Downsampling of signals living on a general weighted graph is not as trivial as of regular signals where we can simply keep every other samples. In this paper we propose a simple, yet effective downsampling scheme in which the underlying graph is approximated by a maximum spanning tree (MST) that naturally defines a graph multiresolution. This MST-based method significantly outperforms the two previous downsampling schemes, coloring-based and SVD-based, on both random and specific graphs in terms of computations and partition efficiency quantified by the graph cuts. The benefit of using MST-based downsampling for recently developed critical-sampling graph wavelet transforms in compression of graph signals is demonstrated.

Downsampling of Signals on Graphs Via Maximum Spanning Trees

Interpreting chest X-rays via CNNs that exploit hierarchical disease dependencies and uncertainty labels

The chest X-rays (CXRs) is one of the views most commonly ordered by radiologists (NHS),which is critical for diagnosis of many different thoracic diseases. Accurately detecting thepresence of multiple diseases from CXRs is still a challenging task. We present a multi-labelclassification framework based on deep convolutional neural networks (CNNs) for diagnos-ing the presence of 14 common thoracic diseases and observations. Specifically, we trained astrong set of CNNs that exploit dependencies among abnormality labels and used the labelsmoothing regularization (LSR) for a better handling of uncertain samples. Our deep net-works were trained on over 200,000 CXRs of the recently released CheXpert dataset (Irvinandal., 2019) and the final model, which was an ensemble of the best performing networks,achieved a mean area under the curve (AUC) of 0.940 in predicting 5 selected pathologiesfrom the validation set. To the best of our knowledge, this is the highest AUC score yetreported to date. More importantly, the proposed method was also evaluated on an inde-pendent test set of the CheXpert competition, containing 500 CXR studies annotated by apanel of 5 experienced radiologists. The reported performance was on average better than2.6 out of 3 other individual radiologists with a mean AUC of 0.930, which had led to thecurrent state-of-the-art performance on the CheXpert test set.

/pdf/interpreting-chest-x-rays-via-cnns-that-exploit-hierarchical-4uadlok3bn.pdf

Ha Q. Nguyen

Papers

CNN-Based Projected Gradient Descent for Consistent CT Image Reconstruction

CNN-Based Projected Gradient Descent for Consistent Image Reconstruction

Downsampling of Signals on Graphs Via Maximum Spanning Trees

Interpreting chest X-rays via CNNs that exploit hierarchical disease dependencies and uncertainty labels

Interpreting Chest X-rays via CNNs that Exploit Hierarchical Disease Dependencies and Uncertainty Labels