To gain insight into complex systems it is a key challenge to infer nonlinear causal directional relations from observational time-series data. Specifically, estimating causal relationships between interacting components in large systems with only short recordings over few temporal observations remains an important, yet unresolved problem. Here, we introduce a large-scale Nonlinear Granger Causality (lsNGC) approach for inferring directional, nonlinear, multivariate causal interactions between system components from short high-dimensional time-series recordings. By modeling interactions with nonlinear state-space transformations from limited observational data, lsNGC identifies casual relations with no explicit a priori assumptions on functional interdependence between component time-series in a computationally efficient manner. Additionally, our method provides a mathematical formulation revealing statistical significance of inferred causal relations. We extensively study the ability of lsNGC to recovering network structure from two-node to thirty-four node chaotic time-series systems. Our results suggest that lsNGC captures meaningful interactions from limited observational data, where it performs favorably when compared to traditionally used methods. Finally, we demonstrate the applicability of lsNGC to estimating causality in large, real-world systems by inferring directional nonlinear, multivariate causal relationships among a large number of relatively short time-series acquired from functional Magnetic Resonance Imaging (fMRI) data of the human brain.

Spectral clustering is one of the most prominent clustering approaches. The distance-based similarity is the most widely used method for spectral clustering. However, people have already noticed that this is not suitable for multi-scale data, as the distance varies a lot for clusters with different densities. State of the art(ROSC and CAST ) addresses this limitation by taking the reachability similarity of objects into account. However, we observe that in real-world scenarios, data in the same cluster tend to present in a smooth manner, and previous algorithms never take this into account. Based on this observation, we propose a novel clustering algorithm, which con-siders the smoothness of data for the first time. We first divide objects into a great many tiny clusters. Our key idea is to cluster tiny clusters, whose centers constitute smooth graphs. Theoretical analysis and experimental results show that our clustering algorithm significantly outperforms state of the art. Although in this paper, we singly focus on multi-scale situations, the idea of data smoothness can certainly be extended to any clustering algorithms

Model selection is a fundamental part of Bayesian statistical inference; a widely used tool in the field of epidemiology. Simple methods such as Akaike Information Criterion are commonly used but they do not incorporate the uncertainty of the model's parameters, which can give misleading choices when comparing models with similar fit to the data. One approach to model selection in a more rigorous way that uses the full posterior distributions of the models is to compute the ratio of the normalising constants (or model evidence), known as Bayes factors. These normalising constants integrate the posterior distribution over all parameters and balance over and under fitting. However, normalising constants often come in the form of intractable, high-dimensional integrals, therefore special probabilistic techniques need to be applied to correctly estimate the Bayes factors. One such method is thermodynamic integration (TI), which can be used to estimate the ratio of two models' evidence by integrating over a continuous path between the two un-normalised densities. In this paper we introduce a variation of the TI method, here referred to as referenced TI, which computes a single model's evidence in an efficient way by using a reference density such as a multivariate normal - where the normalising constant is known. We show that referenced TI, an asymptotically exact Monte Carlo method of calculating the normalising constant of a single model, in practice converges to the correct result much faster than other competing approaches such as the method of power posteriors. We illustrate the implementation of the algorithm on informative 1- and 2-dimensional examples, and apply it to a popular linear regression problem, and use it to select parameters for a model of the COVID-19 epidemic in South Korea.

Optimal transport has been one of the most exciting subjects in mathematics, starting from the 18th century. As a powerful tool to transport between two probability measures, optimal transport methods have been reinvigorated nowadays in a remarkable proliferation of modern data science applications. To meet the big data challenges, various computational tools have been developed in the recent decade to accelerate the computation for optimal transport methods. In this review, we present some cutting-edge computational optimal transport methods with a focus on the regularization-based methods and the projection-based methods. We discuss their real-world applications in biomedical research.

A line of recent works established that when training linear predictors over separable data, using gradient methods and exponentially-tailed losses, the predictors asymptotically converge in direction to the max-margin predictor. As a consequence, the predictors asymptotically do not overfit. However, this does not address the question of whether overfitting might occur non-asymptotically, after some bounded number of iterations. In this paper, we formally show that standard gradient methods (in particular, gradient flow, gradient descent and stochastic gradient descent) never overfit on separable data: If we run these methods for $T$ iterations on a dataset of size $m$, both the empirical risk and the generalization error decrease at an essentially optimal rate of $\tilde{\mathcal{O}}(1/\gamma^2 T)$ up till $T\approx m$, at which point the generalization error remains fixed at an essentially optimal level of $\tilde{\mathcal{O}}(1/\gamma^2 m)$ regardless of how large $T$ is. Along the way, we present non-asymptotic bounds on the number of margin violations over the dataset, and prove their tightness.

We introduce Bootstrap Your Own Latent (BYOL), a new approach to self-supervised image representation learning. BYOL relies on two neural networks, referred to as online and target networks, that interact and learn from each other. From an augmented view of an image, we train the online network to predict the target network representation of the same image under a different augmented view. At the same time, we update the target network with a slow-moving average of the online network. While state-of-the art methods rely on negative pairs, BYOL achieves a new state of the art without them. BYOL reaches $74.3\%$ top-1 classification accuracy on ImageNet using a linear evaluation with a ResNet-50 architecture and $79.6\%$ with a larger ResNet. We show that BYOL performs on par or better than the current state of the art on both transfer and semi-supervised benchmarks. Our implementation and pretrained models are given on GitHub.

When it comes to machine learning classification tasks, the more data available to train algorithms, the better. In supervised learning, this data must be labeled with respect to the target class -- otherwise, these algorithms wouldn't be able to learn the relationships between the independent and target variables. So, what if we only have enough time and money to label some of a large data set, and choose to leave the rest unlabeled? Can this unlabeled data somehow be used in a classification algorithm? This is where semi-supervised learning comes in.

With the rise of digital marketing led by tools including Google Analytics, Google Adwords, and Facebook Ads, a key competitive advantage for businesses is using A/B testing to determine effects of digital marketing efforts. In short, small changes can have big effects. This is why A/B testing is a huge benefit. A/B Testing enables us to determine whether changes in landing pages, popup forms, article titles, and other digital marketing decisions improve conversion rates and ultimately customer purchasing behavior. A successful A/B Testing strategy can lead to massive gains - more satisfied users, more engagement, and more sales - Win-Win-Win. A major issue with traditional, statistical-inference approaches to A/B Testing is that it only compares 2 variables - an experiment/control to an outcome. The problem is that customer behavior is vastly more complex than this. Customers take different paths, spend different amounts of time on the site, come from different backgrounds (age, gender, interests), and more. This is where Machine Learning excels - generating insights from complex systems.

Today time series forecasting is ubiquitous, and decision-making processes in companies depend heavily on their ability to predict the future. Through a short series of articles I will present you with a possible approach to this kind of problems, combining state-space models with Bayesian statistics. In the initial articles, I will take some of the examples from the book An Introduction to State Space Time Series Analysis from Jacques J.F. Commandeur and Siem Jan Koopman [1]. It comprises a well-known introduction to the subject of state-space modeling applied to the time series domain. In classical regression analysis, it is assumed a linear relationship between a dependent variable y and a predictor variable x.

This article was reviewed by Renato Ambrósio, Jr, MD, PhD Ectasia is an intriguing and mysterious complication of laser-vision-correction (LVC) procedures. The potentially devastating problem underscores the importance of determining the susceptibility of the cornea for developing progressive ectasia, and of going beyond detecting just mild or subclinical keratoconus. The corneal structure as well as the potential impact of LVC should be considered to predict ectasia risk in every patient. "The LVC procedure and eye rubbing are the primary environmental culprits in the development of ectasia in any cornea," said Renato Ambrósio, Jr, MD, PhD. "So, a basic factor for avoiding ectasia is educating the patient not to rub the eye."