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For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.

Many problems in computational science and engineering can be described in terms of approximating a smooth function of $d$ variables, defined over an unknown domain of interest $\Omega\subset \mathbb{R}^d$, from sample data. Here both the curse of dimensionality ($d\gg 1$) and the lack of domain knowledge with $\Omega$ potentially irregular and/or disconnected are confounding factors for sampling-based methods. Na\"{i}ve approaches often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upwards of 20\% wasted samples in some problems. In surrogate model construction in computational uncertainty quantification (UQ), the high cost of computing samples needs a more efficient sampling procedure. In the last years, methods for computing such approximations from sample data have been studied in the case of irregular domains. The advantages of computing sampling measures depending on an approximation space $P$ of $\dim(P)=N$ have been shown. In particular, such methods confer advantages such as stability and well-conditioning, with $\mathcal{O}(N\log(N))$ as sample complexity. The recently-proposed adaptive sampling for general domains (ASGD) strategy is one method to construct these sampling measures. The main contribution of this paper is to improve ASGD by adaptively updating the sampling measures over unknown domains. We achieve this by first introducing a general domain adaptivity strategy (GDAS), which approximates the function and domain of interest from sample points. Second, we propose adaptive sampling for unknown domains (ASUD), which generates sampling measures over a domain that may not be known in advance. Then, we derive least squares techniques for polynomial approximation on unknown domains. Numerical results show that the ASUD approach can reduce the computational cost by as 50\% when compared with uniform sampling.

We propose a new algorithm for k-means clustering in a distributed setting, where the data is distributed across many machines, and a coordinator communicates with these machines to calculate the output clustering. Our algorithm guarantees a cost approximation factor and a number of communication rounds that depend only on the computational capacity of the coordinator. Moreover, the algorithm includes a built-in stopping mechanism, which allows it to use fewer communication rounds whenever possible. We show both theoretically and empirically that in many natural cases, indeed 1-4 rounds suffice. In comparison with the popular k-means|| algorithm, our approach allows exploiting a larger coordinator capacity to obtain a smaller number of rounds. Our experiments show that the k-means cost obtained by the proposed algorithm is usually better than the cost obtained by k-means||, even when the latter is allowed a larger number of rounds. Moreover, the machine running time in our approach is considerably smaller than that of k-means||. Code for running the algorithm and experiments is available at //github.com/selotape/distributed_k_means.

A central goal in designing clinical trials is to find the test that maximizes power (or equivalently minimizes required sample size) for finding a true research hypothesis subject to the constraint of type I error. When there is more than one test, such as in clinical trials with multiple endpoints, the issues of optimal design and optimal policies become more complex. In this paper we address the question of how such optimal tests should be defined and how they can be found. We review different notions of power and how they relate to study goals, and also consider the requirements of type I error control and the nature of the policies. This leads us to formulate the optimal policy problem as an explicit optimization problem with objective and constraints which describe its specific desiderata. We describe a complete solution for deriving optimal policies for two hypotheses, which have desired monotonicity properties, and are computationally simple. For some of the optimization formulations this yields optimal policies that are identical to existing policies, such as Hommel's procedure or the procedure of Bittman et al. (2009), while for others it yields completely novel and more powerful policies than existing ones. We demonstrate the nature of our novel policies and their improved power extensively in simulation and on the APEX study (Cohen et al., 2016).

Phase-type (PH) distributions are a popular tool for the analysis of univariate risks in numerous actuarial applications. Their multivariate counterparts (MPH$^\ast$), however, have not seen such a proliferation, due to lack of explicit formulas and complicated estimation procedures. A simple construction of multivariate phase-type distributions -- mPH -- is proposed for the parametric description of multivariate risks, leading to models of considerable probabilistic flexibility and statistical tractability. The main idea is to start different Markov processes at the same state, and allow them to evolve independently thereafter, leading to dependent absorption times. By dimension augmentation arguments, this construction can be cast into the umbrella of MPH$^\ast$ class, but enjoys explicit formulas which the general specification lacks, including common measures of dependence. Moreover, it is shown that the class is still rich enough to be dense on the set of multivariate risks supported on the positive orthant, and it is the smallest known sub-class to have this property. In particular, the latter result provides a new short proof of the denseness of the MPH$^\ast$ class. In practice this means that the mPH class allows for modeling of bivariate risks with any given correlation or copula. We derive an EM algorithm for its statistical estimation, and illustrate it on bivariate insurance data. Extensions to more general settings are outlined.

We investigate the quantize and binning scheme, known as the Shimokawa-Han-Amari (SHA) scheme, for the distributed hypothesis testing. We develop tools to evaluate the critical rate attainable by the SHA scheme. For a product of binary symmetric double sources, we present a sequential scheme that improves upon the SHA scheme.

Optimizing multiple, non-preferential objectives for mixed-variable, expensive black-box problems is important in many areas of engineering and science. The expensive, noisy black-box nature of these problems makes them ideal candidates for Bayesian optimization (BO). Mixed-variable and multi-objective problems, however, are a challenge due to the BO's underlying smooth Gaussian process surrogate model. Current multi-objective BO algorithms cannot deal with mixed-variable problems. We present MixMOBO, the first mixed variable multi-objective Bayesian optimization framework for such problems. Using a genetic algorithm to sample the surrogate surface, optimal Pareto-fronts for multi-objective, mixed-variable design spaces can be found efficiently while ensuring diverse solutions. The method is sufficiently flexible to incorporate many different kernels and acquisition functions, including those that were developed for mixed-variable or multi-objective problems by other authors. We also present HedgeMO, a modified Hedge strategy that uses a portfolio of acquisition functions in multi-objective problems. We present a new acquisition function SMC. We show that MixMOBO performs well against other mixed-variable algorithms on synthetic problems. We apply MixMOBO to the real-world design of an architected material and show that our optimal design, which was experimentally fabricated and validated, has a normalized strain energy density $10^4$ times greater than existing structures.

The low-rank matrix approximation problem is ubiquitous in computational mathematics. Traditionally, this problem is solved in spectral or Frobenius norms, where the accuracy of the approximation is related to the rate of decrease of the singular values of the matrix. However, recent results indicate that this requirement is not necessary for other norms. In this paper, we propose a method for solving the low-rank approximation problem in the Chebyshev norm, which is capable of efficiently constructing accurate approximations for matrices, whose singular values do not decrease or decrease slowly.

This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycenteric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.

One of the main features of interest in analysing the light curves of stars is the underlying periodic behaviour. The corresponding observations are a complex type of time series with unequally spaced time points and are sometimes accompanied by varying measures of accuracy. The main tools for analysing these type of data rely on the periodogram-like functions, constructed with a desired feature so that the peaks indicate the presence of a potential period. In this paper, we explore a particular periodogram for the irregularly observed time series data, similar to Thieler et. al. (2013). We identify the potential periods at the appropriate peaks and more importantly with a quantifiable uncertainty. Our approach is shown to easily generalise to non-parametric methods including a weighted Gaussian process regression periodogram. We also extend this approach to correlated background noise. The proposed method for period detection relies on a test based on quadratic forms with normally distributed components. We implement the saddlepoint approximation, as a faster and more accurate alternative to the simulation-based methods that are currently used. The power analysis of the testing methodology is reported together with applications using light curves from the Hunting Outbursting Young Stars citizen science project.