Time series that record temporal changes in sets of system variables are ubiquitous across many scientific disciplines [1], from physics and engineering [2] to biomedicine [3, 4], climate science [5, 6], economics [7, 8] or online human behaviour [9, 10]. Many real-world systems are thus described as multivariate time series of (possibly) interlinked processes tracking the temporal evolution (deterministic or random) of groups of observables of interest. The relationships between the measured variables are often complex, in many cases displaying inter-dependencies among each other. For example, the spreading of Covid-19 in Indonesia was dependent on weather conditions [11]; the Sustainable Development Goals have extensive interlinkages [12]; there are strong interconnections between foreign exchange and cryptocurrencies [13]; and the brain displays multiple spatial and temporal scales of functional connectivity [14]. Driven by technological advances (e.g., imaging techniques in the brain sciences [15], or the increased connectivity of personal devices via the Internet of Things [16]), there is a rapid expansion in the collection and storage of multivariate time series data sets, which underlines the need for mathematical tools to analyze the interdependencies within complex high-dimensional time series data.

Self-Supervised Learning (SSL) methods such as VICReg, Barlow Twins or W-MSE avoid collapse of their joint embedding architectures by constraining or regularizing the covariance matrix of their projector's output. This study highlights important properties of such strategy, which we coin Variance-Covariance regularization (VCReg). More precisely, we show that VCReg enforces pairwise independence between the features of the learned representation. This result emerges by bridging VCReg applied on the projector's output to kernel independence criteria applied on the projector's input. This provides the first theoretical motivations and explanations of VCReg. We empirically validate our findings where (i) we observe that SSL methods employing VCReg learn visual representations with greater pairwise independence than other methods, (i) we put in evidence which projector's characteristics favor pairwise independence, and show it to emerge independently from learning the projector, (ii) we use these findings to obtain nontrivial performance gains for VICReg, (iii) we demonstrate that the scope of VCReg goes beyond SSL by using it to solve Independent Component Analysis. We hope that our findings will support the adoption of VCReg in SSL and beyond.

A secure key generation depends on three principles: channel The renewed interest in physical layer security (PLS) technologies reciprocity between Alice and Bob, spatial decorrelation for sixth-generation (6G) systems stems from the and temporal variations [7]. Spatial decorrelation is particularly emergence of massive-scale Internet of things (IoT) networks, important because a passive eavesdropper (Eve) present which have an extensive range of non-functional (security) close to the legitimate users can generate the duplicate keys constraints as well as computational, power and energy limitations, by exploiting the shared spatial correlation. Based on Jakes' delay and latency constraints, etc. [1], [2]. One of model, the channel will be uncorrelated when a third party is the most popular physical layer security (PLS) techniques is located half-wavelength away [5]. Under this assumption, to facilitate reconciliation, the authors for the transmitter (Alice) and the receiver (Bob) to extract of [8] carry out a theoretical study on pre-processing a key from the wireless channel realisations exploiting the algorithms such as principal component analysis (PCA) to common randomness of the wireless channels during the establish a high-agreement uncorrelated secret key by retaining channel coherence time [3], [4].

We introduce Information Condensing Active Learning (ICAL), a batch mode model agnostic Active Learning (AL) method targeted at Deep Bayesian Active Learning that focuses on acquiring labels for points which have as much information as possible about the still unacquired points. ICAL uses the Hilbert Schmidt Independence Criterion (HSIC) to measure the strength of the dependency between a candidate batch of points and the unlabeled set. We develop key optimizations that allow us to scale our method to large unlabeled sets. We show significant improvements in terms of model accuracy and negative log likelihood (NLL) on several image datasets compared to state of the art batch mode AL methods for deep learning.

We present a framework for testing independence between two random vectors that is scalable to massive data. Taking a "divide-and-conquer" approach, we break down the nonparametric multivariate test of independence into simple univariate independence tests on a collection of $2\times 2$ contingency tables, constructed by sequentially discretizing the original sample space at a cascade of scales from coarse to fine. This transforms a complex nonparametric testing problem---that traditionally requires quadratic computational complexity with respect to the sample size---into a multiple testing problem that can be addressed with a computational complexity that scales almost linearly with the sample size. We further consider the scenario when the dimensionality of the two random vectors also grows large, in which case the curse of dimensionality arises in the proposed framework through an explosion in the number of univariate tests to be completed. To overcome this difficulty, we propose a data-adaptive version of our method that completes a fraction of the univariate tests, judged to be more likely to contain evidence for dependency based on exploiting the spatial characteristics of the dependency structure in the data. We provide an inference recipe based on multiple testing adjustment that guarantees the inferential validity in terms of properly controlling the family-wise error rate. We demonstrate the tremendous computational advantage of the algorithm in comparison to existing approaches while achieving desirable statistical power through an extensive simulation study. In addition, we illustrate how our method can be used for learning the nature of the underlying dependency in addition to hypothesis testing. We demonstrate the use of our method through analyzing a data set from flow cytometry.

We investigate the problem of testing whether $d$ random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the $d$-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the $d$ variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small $d$, it performs well in practice. Finally, we apply the test to a problem in causal discovery.