Matrix completion and extrapolation via kernel regression

Matrix completion and extrapolation (MCEX) are dealt with here over reproducing kernel Hilbert spaces (RKHSs) in order to account for prior information present in the available data. Aiming at a faster and low-complexity solver, the task is formulated as a kernel ridge regression. The resultant MCEX algorithm can also afford online implementation, while the class of kernel functions also encompasses several existing approaches to MC with prior information. Numerical tests on synthetic and real datasets show that the novel approach performs faster than widespread methods such as alternating least squares (ALS) or stochastic gradient descent (SGD), and that the recovery error is reduced, especially when dealing with noisy data

Sharp analysis of low-rank kernel matrix approximations

We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations

Machine Learning at the Edge: A Data-Driven Architecture with Applications to 5G Cellular Networks

The fifth generation of cellular networks (5G) will rely on edge cloud deployments to satisfy the ultra-low latency demand of future applications. In this paper, we argue that such deployments can also be used to enable advanced data-driven and Machine Learning (ML) applications in mobile networks. We propose an edge-controller-based architecture for cellular networks and evaluate its performance with real data from hundreds of base stations of a major U.S. operator. In this regard, we will provide insights on how to dynamically cluster and associate base stations and controllers, according to the global mobility patterns of the users. Then, we will describe how the controllers can be used to run ML algorithms to predict the number of users in each base station, and a use case in which these predictions are exploited by a higher-layer application to route vehicular traffic according to network Key Performance Indicators (KPIs). We show that the prediction accuracy improves when based on machine learning algorithms that rely on the controllers' view and, consequently, on the spatial correlation introduced by the user mobility, with respect to when the prediction is based only on the local data of each single base station.Comment: 15 pages, 10 figures, 5 tables. IEEE Transactions on Mobile Computin

Sparse Volterra and Polynomial Regression Models: Recoverability and Estimation

Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but unfortunately cannot be met by least-squares based or kernel regression methods. To this end, compressed sampling (CS) approaches, already successful in linear regression settings, can offer a viable alternative. The viability of CS for sparse Volterra and polynomial models is the core theme of this work. A common sparse regression task is initially posed for the two models. Building on (weighted) Lasso-based schemes, an adaptive RLS-type algorithm is developed for sparse polynomial regressions. The identifiability of polynomial models is critically challenged by dimensionality. However, following the CS principle, when these models are sparse, they could be recovered by far fewer measurements. To quantify the sufficient number of measurements for a given level of sparsity, restricted isometry properties (RIP) are investigated in commonly met polynomial regression settings, generalizing known results for their linear counterparts. The merits of the novel (weighted) adaptive CS algorithms to sparse polynomial modeling are verified through synthetic as well as real data tests for genotype-phenotype analysis.Comment: 20 pages, to appear in IEEE Trans. on Signal Processin