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In this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system trough a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method. We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called ``coarse-fine'' strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer. The estimates obtained here are rigourous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretiazation and the approximations induced by finite-precision arithmetic. We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time. We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package.

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer a great modeling flexibility, and often, also mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the interval of support for some parameter values, the number of their parameters is usually much larger than for canonical distributions, and the interval of support must be finite. In particular, polynomial distributions are defined here assuming three forms of polynomial function. The transformation of polynomial distributions and fitting a histogram to a polynomial distribution are considered. The key properties of polynomial distributions are derived in closed-form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. Finally, the problems of estimating parameters of polynomial distributions and generating polynomially distributed samples are also studied.

This paper proposes a novel multivariate definition of statistical dependence using a functional methodology inspired by Alfred R\'enyi. We define a new symmetric and self-adjoint cross density kernel through a recursive bidirectional statistical mapping between conditional densities of continuous random processes, which estimates their statistical dependence. Therefore, the kernel eigenspectrum is proposed as a new multivariate statistical dependence measure, and the formulation requires fewer assumptions about the data generation model than current methods. The measure can also be estimated from realizations. The proposed functional maximum correlation algorithm (FMCA) is applied to a learning architecture with two multivariate neural networks. The FMCA optimal solution is an equilibrium point that estimates the eigenspectrum of the cross density kernel. Preliminary results with synthetic data and medium size image datasets corroborate the theory. Four different strategies of applying the cross density kernel are thoroughly discussed and implemented to show the versatility and stability of the methodology, and it transcends supervised learning. When two random processes are high-dimensional real-world images and white uniform noise, respectively, the algorithm learns a factorial code i.e., the occurrence of a code guarantees that a certain input in the training set was present, which is quite important for feature learning.

A recently developed measure-theoretic framework solves a stochastic inverse problem (SIP) for models where uncertainties in model output data are predominantly due to aleatoric (i.e., irreducible) uncertainties in model inputs (i.e., parameters). The subsequent inferential target is a distribution on parameters. Another type of inverse problem is to quantify uncertainties in estimates of "true" parameter values under the assumption that such uncertainties should be reduced as more data are incorporated into the problem, i.e., the uncertainty is considered epistemic. A major contribution of this work is the formulation and solution of such a parameter identification problem (PIP) within the measure-theoretic framework developed for the SIP. The approach is novel in that it utilizes a solution to a stochastic forward problem (SFP) to update an initial density only in the parameter directions informed by the model output data. In other words, this method performs "selective regularization" only in the parameter directions not informed by data. The solution is defined by a maximal updated density (MUD) point where the updated density defines the measure-theoretic solution to the PIP. Another significant contribution of this work is the full theory of existence and uniqueness of MUD points for linear maps with Gaussian distributions. Data-constructed Quantity of Interest (QoI) maps are also presented and analyzed for solving the PIP within this measure-theoretic framework as a means of reducing uncertainties in the MUD estimate. We conclude with a demonstration of the general applicability of the method on two problems involving either spatial or temporal data for estimating uncertain model parameters.

Mathematical models are essential for understanding and making predictions about systems arising in nature and engineering. Yet, mathematical models are a simplification of true phenomena, thus making predictions subject to uncertainty. Hence, the ability to quantify uncertainties is essential to any modelling framework, enabling the user to assess the importance of certain parameters on quantities of interest and have control over the quality of the model output by providing a rigorous understanding of uncertainty. Peridynamic models are a particular class of mathematical models that have proven to be remarkably accurate and robust for a large class of material failure problems. However, the high computational expense of peridynamic models remains a major limitation, hindering outer-loop applications that require a large number of simulations, for example, uncertainty quantification. This contribution provides a framework to make such computations feasible. By employing a Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The results show a speed-up factor of 16x over a standard Monte Carlo estimator, enabling the forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thus improving the confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.

The main purpose of this paper is to introduce the random tensor with normal distribution, which promotes the matrix normal distribution to a higher order case. Some basic knowledge on tensors are introduced before we focus on the random tensors whose entries follow normal distribution. The random tensor with standard normal distribution(SND) is introduced as an extension of random normal matrices. As a random multi-array deduced from an affine transformation on a SND tensor, the general normal random tensor is initialised in the paper. We then investigate some equivalent definitions of a normal tensor and present the description of the density function, characteristic function, moments, and some other functions related to a random matrix. A general form of an even-order multi-variance tensor is also introduced to tackle a random tensor. Finally some equivalent definitions for the tensor normal distribution are described.

Results on the spectral behavior of random matrices as the dimension increases are applied to the problem of detecting the number of sources impinging on an array of sensors. A common strategy to solve this problem is to estimate the multiplicity of the smallest eigenvalue of the spatial covariance matrix $R$ of the sensed data from the sample covariance matrix $\widehat{R}$. Existing approaches, such as that based on information theoretic criteria, rely on the closeness of the noise eigenvalues of $\widehat R$ to each other and, therefore, the sample size has to be quite large when the number of sources is large in order to obtain a good estimate. The analysis presented in this report focuses on the splitting of the spectrum of $\widehat{R}$ into noise and signal eigenvalues. It is shown that, when the number of sensors is large, the number of signals can be estimated with a sample size considerably less than that required by previous approaches. The practical significance of the main result is that detection can be achieved with a number of samples comparable to the number of sensors in large dimensional array processing.

We present a new method of modelling numerical systems where there are two distinct output solution classes, for example tipping points or bifurcations. Gaussian process emulation is a useful tool in understanding these complex systems and provides estimates of uncertainty, but we aim to include systems where there are discontinuities between the two output solutions. Due to continuity assumptions, we consider current methods of classification to split our input space into two output regions. Classification and logistic regression methods currently rely on drawing from an independent Bernoulli distribution, which neglects any information known in the neighbouring area. We build on this by including correlation between our input points. Gaussian processes are still a vital element, but used in latent space to model the two regions. Using the input values and an associated output class label, the latent variable is estimated using MCMC sampling and a unique likelihood. A threshold (usually at zero) defines the boundary. We apply our method to a motivating example provided by the hormones associated with the reproductive system in mammals, where the two solutions are associated with high and low rates of reproduction.

We consider the problem of estimating the optimal transport map between a (fixed) source distribution $P$ and an unknown target distribution $Q$, based on samples from $Q$. The estimation of such optimal transport maps has become increasingly relevant in modern statistical applications, such as generative modeling. At present, estimation rates are only known in a few settings (e.g. when $P$ and $Q$ have densities bounded above and below and when the transport map lies in a H\"older class), which are often not reflected in practice. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure $P$ satisfies a Poincar\'e inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for bounded densities and H\"older transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when $P$ is the normal distribution and the transport map is given by an infinite-width shallow neural network.

Deep neural networks (DNNs) have achieved unprecedented success in the field of artificial intelligence (AI), including computer vision, natural language processing and speech recognition. However, their superior performance comes at the considerable cost of computational complexity, which greatly hinders their applications in many resource-constrained devices, such as mobile phones and Internet of Things (IoT) devices. Therefore, methods and techniques that are able to lift the efficiency bottleneck while preserving the high accuracy of DNNs are in great demand in order to enable numerous edge AI applications. This paper provides an overview of efficient deep learning methods, systems and applications. We start from introducing popular model compression methods, including pruning, factorization, quantization as well as compact model design. To reduce the large design cost of these manual solutions, we discuss the AutoML framework for each of them, such as neural architecture search (NAS) and automated pruning and quantization. We then cover efficient on-device training to enable user customization based on the local data on mobile devices. Apart from general acceleration techniques, we also showcase several task-specific accelerations for point cloud, video and natural language processing by exploiting their spatial sparsity and temporal/token redundancy. Finally, to support all these algorithmic advancements, we introduce the efficient deep learning system design from both software and hardware perspectives.