A randomized Kaczmarz algorithm with exponential convergence

The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system, but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling

Uncovering protein interaction in abstracts and text using a novel linear model and word proximity networks

We participated in three of the protein-protein interaction subtasks of the Second BioCreative Challenge: classification of abstracts relevant for protein-protein interaction (IAS), discovery of protein pairs (IPS) and text passages characterizing protein interaction (ISS) in full text documents. We approached the abstract classification task with a novel, lightweight linear model inspired by spam-detection techniques, as well as an uncertainty-based integration scheme. We also used a Support Vector Machine and the Singular Value Decomposition on the same features for comparison purposes. Our approach to the full text subtasks (protein pair and passage identification) includes a feature expansion method based on word-proximity networks. Our approach to the abstract classification task (IAS) was among the top submissions for this task in terms of the measures of performance used in the challenge evaluation (accuracy, F-score and AUC). We also report on a web-tool we produced using our approach: the Protein Interaction Abstract Relevance Evaluator (PIARE). Our approach to the full text tasks resulted in one of the highest recall rates as well as mean reciprocal rank of correct passages. Our approach to abstract classification shows that a simple linear model, using relatively few features, is capable of generalizing and uncovering the conceptual nature of protein-protein interaction from the bibliome. Since the novel approach is based on a very lightweight linear model, it can be easily ported and applied to similar problems. In full text problems, the expansion of word features with word-proximity networks is shown to be useful, though the need for some improvements is discussed

CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters

The rise of graph-structured data such as social networks, regulatory networks, citation graphs, and functional brain networks, in combination with resounding success of deep learning in various applications, has brought the interest in generalizing deep learning models to non-Euclidean domains. In this paper, we introduce a new spectral domain convolutional architecture for deep learning on graphs. The core ingredient of our model is a new class of parametric rational complex functions (Cayley polynomials) allowing to efficiently compute spectral filters on graphs that specialize on frequency bands of interest. Our model generates rich spectral filters that are localized in space, scales linearly with the size of the input data for sparsely-connected graphs, and can handle different constructions of Laplacian operators. Extensive experimental results show the superior performance of our approach, in comparison to other spectral domain convolutional architectures, on spectral image classification, community detection, vertex classification and matrix completion tasks

Joint Optimization Framework for Operational Cost Minimization in Green Coverage-Constrained Wireless Networks

In this work, we investigate the joint optimization of base station (BS) location, its density, and transmit power allocation to minimize the overall network operational cost required to meet an underlying coverage constraint at each user equipment (UE), which is randomly deployed following the binomial point process (BPP). As this joint optimization problem is nonconvex and combinatorial in nature, we propose a non-trivial solution methodology that effectively decouples it into three individual optimization problems. Firstly, by using the distance distribution of the farthest UE from the BS, we present novel insights on optimal BS location in an optimal sectoring type for a given number of BSs. After that we provide a tight approximation for the optimal transmit power allocation to each BS. Lastly, using the latter two results, the optimal number of BSs that minimize the operational cost is obtained. Also, we have investigated both circular and square field deployments. Numerical results validate the analysis and provide practical insights on optimal BS deployment. We observe that the proposed joint optimization framework, that solves the coverage probability versus operational cost tradeoff, can yield a significant reduction of about $65\%$ in the operational cost as compared to the benchmark fixed allocation scheme.Comment: 30 pages, 15 figures, submitted to IEEE Transactions on Green Communications and Networkin