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Technology of data collection and information transmission is based on various mathematical models of encoding. The words "Geometry of information" refer to such models, whereas the words "Moufang patterns" refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.

Many statistical settings call for estimating a population parameter, most typically the population mean, based on a sample of matrices. The most natural estimate of the population mean is the arithmetic mean, but there are many other matrix means that may behave differently, especially in high dimensions. Here we consider the matrix harmonic mean as an alternative to the arithmetic matrix mean. We show that in certain high-dimensional regimes, the harmonic mean yields an improvement over the arithmetic mean in estimation error as measured by the operator norm. Counter-intuitively, studying the asymptotic behavior of these two matrix means in a spiked covariance estimation problem, we find that this improvement in operator norm error does not imply better recovery of the leading eigenvector. We also show that a Rao-Blackwellized version of the harmonic mean is equivalent to a linear shrinkage estimator studied previously in the high-dimensional covariance estimation literature, while applying a similar Rao-Blackwellization to regularized sample covariance matrices yields a novel nonlinear shrinkage estimator. Simulations complement the theoretical results, illustrating the conditions under which the harmonic matrix mean yields an empirically better estimate.

Distribution function is essential in statistical inference, and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space, but without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric between random objects and a fixed location in metric spaces. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces that lie the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop homogeneity test and mutual independence test for non-Euclidean random objects, and present comprehensive empirical evidence to support the performance of our proposed methods.

Motivated by the problem of exploring discrete but very complex state spaces in Bayesian models, we propose a novel Markov Chain Monte Carlo search algorithm: the taxicab sampler. We describe the construction of this sampler and discuss how its interpretation and usage differs from that of standard Metropolis-Hastings as well as the closely-related Hamming ball sampler. The proposed taxicab sampling algorithm is then shown to demonstrate substantial improvement in computation time relative to a na\"ive Metropolis-Hastings search in a motivating Bayesian regression tree count model, in which we leverage the discrete state space assumption to construct a novel likelihood function that allows for flexibly describing different mean-variance relationships while preserving parameter interpretability compared to existing likelihood functions for count data.

We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.

In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In particular, the random diffusivity coefficient is described by a finite-dimensional random variable or a truncated combination of random variables with the Karhunen-Lo\`{e}ve decomposition, then a probabilistic collocation method (PCM) with sparse grids is employed to sample the stochastic process. On each sample, the deterministic nonlocal diffusion problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random coefficients space as the number of collocation points grows. Finally, to validate the applicability of this approach we consider a randomly heterogeneous nonlocal problem with a given spatial correlation structure, demonstrating that the proposed PCM approach achieves substantial speed-up compared to conventional Monte Carlo simulations.

We study the behavior of Thompson sampling from the perspective of weak convergence. In the regime where the gaps between arm means scale as $1/\sqrt{n}$ with the time horizon $n$, we show that the dynamics of Thompson sampling evolve according to discrete versions of SDEs and random ODEs. As $n \to \infty$, we show that the dynamics converge weakly to solutions of the corresponding SDEs and random ODEs. (Recently, Wager and Xu (arXiv:2101.09855) independently proposed this regime and developed similar SDE and random ODE approximations for Thompson sampling in the multi-armed bandit setting.) Our weak convergence theory, which covers both multi-armed and linear bandit settings, is developed from first principles using the Continuous Mapping Theorem and can be directly adapted to analyze other sampling-based bandit algorithms, for example, algorithms using the bootstrap for exploration. We also establish an invariance principle for multi-armed bandits with gaps scaling as $1/\sqrt{n}$ -- for Thompson sampling and related algorithms involving posterior approximation or the bootstrap, the weak diffusion limits are in general the same regardless of the specifics of the reward distributions or the choice of prior. In particular, as suggested by the classical Bernstein-von Mises normal approximation for posterior distributions, the weak diffusion limits generally coincide with the limit for normally-distributed rewards and priors.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

We propose an Active Learning approach to image segmentation that exploits geometric priors to streamline the annotation process. We demonstrate this for both background-foreground and multi-class segmentation tasks in 2D images and 3D image volumes. Our approach combines geometric smoothness priors in the image space with more traditional uncertainty measures to estimate which pixels or voxels are most in need of annotation. For multi-class settings, we additionally introduce two novel criteria for uncertainty. In the 3D case, we use the resulting uncertainty measure to show the annotator voxels lying on the same planar patch, which makes batch annotation much easier than if they were randomly distributed in the volume. The planar patch is found using a branch-and-bound algorithm that finds a patch with the most informative instances. We evaluate our approach on Electron Microscopy and Magnetic Resonance image volumes, as well as on regular images of horses and faces. We demonstrate a substantial performance increase over state-of-the-art approaches.