Cross-region dynamic connectivity, which describes the spatio-temporal dependence of neural activity among multiple brain regions of interest (ROIs), can provide important information for understanding cognition. For estimating such connectivity, magnetoencephalography (MEG) and electroencephalography (EEG) are well-suited tools because of their millisecond temporal resolution. However, localizing source activity in the brain requires solving an under-determined linear problem. In typical two-step approaches, researchers first solve the linear problem with generic priors assuming independence across ROIs, and secondly quantify cross-region connectivity. In this work, we propose a one-step state-space model to improve estimation of dynamic connectivity. The model treats the mean activity in individual ROIs as the state variable and describes non-stationary dynamic dependence across ROIs using time-varying auto-regression. Compared with a two-step method, which first obtains the commonly used minimum-norm estimates of source activity, and then fits the auto-regressive model, our state-space model yielded smaller estimation errors on simulated data where the model assumptions held. When applied on empirical MEG data from one participant in a scene-processing experiment, our state-space model also demonstrated intriguing preliminary results, indicating leading and lagged linear dependence between the early visual cortex and a higher-level scene-sensitive region, which could reflect feedforward and feedback information flow within the visual cortex during scene processing.

We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's

This paper studies the generalization performance of multi-class classification algorithms, for which we obtain--for the first time--a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis.

We improve a recent gurantee of Bach and Moulines on the linear convergence of SGD for smooth and strongly convex objectives, reducing a quadratic dependence on the strong convexity to a linear dependence. Furthermore, we show how reweighting the sampling distribution (i.e.

We improve a recent guarantee of Bach and Moulines on the linear convergence of SGD for smooth and strongly convex objectives, reducing a quadratic dependence on the strong convexity to a linear dependence. Furthermore, we show how reweighting the sampling distribution (i.e.

We improve a recent gurantee of Bach and Moulines on the linear convergence of SGD for smooth and strongly convex objectives, reducing a quadratic dependence on the strong convexity to a linear dependence. Furthermore, we show how reweighting the sampling distribution (i.e. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods.

Summary: Rademacher complexity is a powerful tool for producing generalization guarantees. The Rademacher complexity of a class H on a sample S is roughly the expected maximum gap between training and test performance if S were randomly partitioned into training and test sets, and we took the maximum over all h in H (eqn 11). This paper makes the observation that the core steps in the standard generalization bound proof will still go through if instead of taking the max over all h in H, you only look at the h's that your procedure can possibly output when given half of a double sample (Lemma 1). While unfortunately the usual final (or initial) high-probability step in this argument does not seem to go through directly, the paper shows (Theorem 2) that one can nonetheless get a useful generation bound from this using other means. The paper then shows how this generalization bound yields good sample complexity guarantees for a number of natural auction classes.

Causal discovery in time series is a rapidly evolving field with a wide variety of applications in other areas such as climate science and neuroscience. Traditional approaches assume a stationary causal graph, which can be adapted to nonstationary time series with time-dependent effects or heterogeneous noise. In this work we address nonstationarity via regime-dependent causal structures. We first establish identifiability for high-order Markov Switching Models, which provide the foundations for identifiable regime-dependent causal discovery. Our empirical studies demonstrate the scalability of our proposed approach for high-order regime-dependent structure estimation, and we illustrate its applicability on brain activity data.