Recently, linear formulations and convex optimization methods have been proposed to predict diffusion-weighted Magnetic Resonance Imaging (dMRI) data given estimates of brain connections generated using tractography algorithms. The size of the linear models comprising such methods grows with both dMRI data and connectome resolution, and can become very large when applied to modern data. In this paper, we introduce a method to encode dMRI signals and large connectomes, i.e., those that range from hundreds of thousands to millions of fascicles (bundles of neuronal axons), by using a sparse tensor decomposition. We show that this tensor decomposition accurately approximates the Linear Fascicle Evaluation (LiFE) model, one of the recently developed linear models. We provide a theoretical analysis of the accuracy of the sparse decomposed model, LiFESD, and demonstrate that it can reduce the size of the model significantly. Also, we develop algorithms to implement the optimisation solver using the tensor representation in an efficient way.

Contextual bandits have become popular as they offer a middle ground between very simple approaches based on multi-armed bandits and very complex approaches using the full power of reinforcement learning. They have demonstrated success in web applications and have a rich body of associated theoretical guarantees. Linear models are well understood theoretically and preferred by practitioners because they are not only easily interpretable but also simple to implement and debug. Furthermore, if the linear model is true, we get very strong performance guarantees. Unfortunately, in emerging applications in mobile health, the time-invariant linear model assumption is untenable.

In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply. We first extend the half-trek criterion to cover a broader class of models and apply our extension to finding testable constraints implied by the model. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification and constraint discovery power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification and constraint discovery algorithms for non-parametric models.

Multi-response linear models aggregate a set of vanilla linear models by assuming correlated noise across them, which has an unknown covariance structure. To find the coefficient vector, estimators with a joint approximation of the noise covariance are often preferred than the simple linear regression in view of their superior empirical performance, which can be generally solved by alternating-minimization type procedures. Due to the non-convex nature of such joint estimators, the theoretical justification of their efficiency is typically challenging. The existing analyses fail to fully explain the empirical observations due to the assumption of resampling on the alternating procedures, which requires access to fresh samples in each iteration. In this work, we present a resampling-free analysis for the alternating minimization algorithm applied to the multi-response regression. In particular, we focus on the high-dimensional setting of multi-response linear models with structured coefficient parameter, and the statistical error of the parameter can be expressed by the complexity measure, Gaussian width, which is related to the assumed structure. More importantly, to the best of our knowledge, our result reveals for the first time that the alternating minimization with random initialization can achieve the same performance as the well-initialized one when solving this multi-response regression problem. Experimental results support our theoretical developments.

Neural Information Processing Systems http://nips.cc/

Recently, there has been a marked increase in interest regarding the generalization capabilities of unregularized gradient-based learning methods.