A general Bayesian framework is introduced for mixture modelling and inference with real-valued time series. At the top level, the state space is partitioned via the choice of a discrete context tree, so that the resulting partition depends on the values of some of the most recent samples. At the bottom level, a different model is associated with each region of the partition. This defines a very rich and flexible class of mixture models, for which we provide algorithms that allow for efficient, exact Bayesian inference. In particular, we show that the maximum a posteriori probability (MAP) model (including the relevant MAP context tree partition) can be precisely identified, along with its exact posterior probability. The utility of this general framework is illustrated in detail when a different autoregressive (AR) model is used in each state-space region, resulting in a mixture-of-AR model class. The performance of the associated algorithmic tools is demonstrated in the problems of model selection and forecasting on both simulated and real-world data, where they are found to provide results as good or better than state-of-the-art methods.

This paper investigates the theoretical and empirical performance of Fisher-Pitman-type permutation tests for assessing the equality of unknown Poisson mixture distributions. Building on nonparametric maximum likelihood estimators (NPMLEs) of the mixing distribution, these tests are theoretically shown to be able to adapt to complicated unspecified structures of count data and also consistent against their corresponding ANOVA-type alternatives; the latter is a result in parallel to classic claims made by Robinson (Robinson, 1973). The studied methods are then applied to a single-cell RNA-seq data obtained from different cell types from brain samples of autism subjects and healthy controls; empirically, they unveil genes that are differentially expressed between autism and control subjects yet are missed using common tests. For justifying their use, rate optimality of NPMLEs is also established in settings similar to nonparametric Gaussian (Wu and Yang, 2020a) and binomial mixtures (Tian et al., 2017; Vinayak et al., 2019).

In practice, by simply training a generator and a discriminator together consisting of multi-layer neural networks with non-linear activation functions, using local search algorithms such as stochastic gradient descent ascent (SGDA), the generator network can be trained efficiently to generate samples from highly-complicated distributions (such as the distribution of images). Despite the great empirical success of GAN, it remains to be one of the least understood models on the theory side of deep learning. Most of existing theories focus on the statistical properties of GANs at the global-optimum [15, 16, 20, 87]. However, on the training side, gradient descent ascent only enjoys efficient convergence to a global optimum when the loss function is convex-concave, or efficient convergence to a critical point in general settings [37, 38, 48, 53, 71, 73, 75, 77, 78]. Due to the extreme non-linearity of the networks in both the generator and the discriminator, it is highly unlikely that the training objective of GANs can be convex-concave. In particular, even if the generator and the discriminator are linear functions over prescribed feature mappings-- such as the neural tangent kernel (NTK) feature mappings [3, 8, 9, 17, 18, 32, 35, 40, 41, 47, 51, 54, 65, 69, 92, 97] -- the training objective can still be non-convex-concave.

We introduce CAFLOW, a new diverse image-to-image translation model that simultaneously leverages the power of auto-regressive modeling and the modeling efficiency of conditional normalizing flows. We transform the conditioning image into a sequence of latent encodings using a multi-scale normalizing flow and repeat the process for the conditioned image. We model the conditional distribution of the latent encodings by modeling the auto-regressive distributions with an efficient multi-scale normalizing flow, where each conditioning factor affects image synthesis at its respective resolution scale. Our proposed framework performs well on a range of image-to-image translation tasks. It outperforms former designs of conditional flows because of its expressive auto-regressive structure.

Explanation techniques are commonly evaluated using human-grounded methods, limiting the possibilities for large-scale evaluations and rapid progress in the development of new techniques. We propose a functionally-grounded evaluation procedure for local model-agnostic explanation techniques. In our approach, we generate ground truth for explanations when the black-box model is Logistic Regression and Gaussian Naive Bayes and compare how similar each explanation is to the extracted ground truth. In our empirical study, explanations of Local Interpretable Model-agnostic Explanations (LIME), SHapley Additive exPlanations (SHAP), and Local Permutation Importance (LPI) are compared in terms of how similar they are to the extracted ground truth. In the case of Logistic Regression, we find that the performance of the explanation techniques is highly dependent on the normalization of the data. In contrast, Local Permutation Importance outperforms the other techniques on Naive Bayes, irrespective of normalization. We hope that this work lays the foundation for further research into functionally-grounded evaluation methods for explanation techniques.

There has been an intense recent activity in embedding of very high dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph based methods and kernel based methods. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. The final part of the survey deals with embedding in $\mathbb{R}^2$, which is visualization. Three methods are presented: $t$-SNE, UMAP and LargeVis based on methods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triple of noisy Ranunculoid curves, and one consisting of networks of increasing complexity and with two types of nodes.

Integrating discrete probability distributions and combinatorial optimization problems into neural networks has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable: it only requires the ability to compute the most probable states; and does not rely on smooth relaxations. The framework encompasses several approaches, such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.

While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction $\textit{is}$ has so far has largely eluded mathematical formalization. In this paper, we aim to set out a mathematical theory of abstraction. We provide a precise characterisation of what an abstraction is and, perhaps more importantly, suggest how abstractions can be learnt directly from data both for static datasets and for dynamical systems. We define an abstraction to be a small set of `summaries' of a system which can be used to answer a set of queries about the system or its behaviour. The difference between the ground truth behaviour of the system on the queries and the behaviour of the system predicted only by the abstraction provides a measure of the `leakiness' of the abstraction which can be used as a loss function to directly learn abstractions from data. Our approach can be considered a generalization of classical statistics where we are not interested in reconstructing `the data' in full, but are instead only concerned with answering a set of arbitrary queries about the data. While highly theoretical, our results have deep implications for statistical inference and machine learning and could be used to develop explicit methods for learning precise kinds of abstractions directly from data.

We present a novel framework for designing multiplierless kernel machines that can be used on resource-constrained platforms like intelligent edge devices. The framework uses a piecewise linear (PWL) approximation based on a margin propagation (MP) technique and uses only addition/subtraction, shift, comparison, and register underflow/overflow operations. We propose a hardware-friendly MP-based inference and online training algorithm that has been optimized for a Field Programmable Gate Array (FPGA) platform. Our FPGA implementation eliminates the need for DSP units and reduces the number of LUTs. By reusing the same hardware for inference and training, we show that the platform can overcome classification errors and local minima artifacts that result from the MP approximation. Using the FPGA platform, we also show that the proposed multiplierless MP-kernel machine demonstrates superior performance in terms of power, performance, and area compared to other comparable implementations.

Incomplete instances with various missing attributes in many real-world scenes have brought challenges to the classification task. There are some missing values imputation methods to fill the missing values with substitute values before classification. However, the separation between imputation and classification may lead to inferior performance since label information are ignored during imputation. Moreover, these imputation methods tend to initialize these missing values with strong prior assumptions, while the unreliability of such initialization is rarely considered. To tackle these problems, a novel semi-supervised conditional normalizing flow (SSCFlow) is proposed in this paper. SSCFlow explicitly utilizes the observed labels to facilitate the imputation and classification simultaneously by employing a semi-supervised algorithm to estimate the conditional probability density of missing values. Moreover, SSCFlow takes the initialized missing values as corrupted initial imputation and iteratively reconstructs their latent representations with an overcomplete denoising autoencoder to approximate the true conditional probability density of missing values. Experiments have been conducted with real-world datasets to demonstrate the robustness and efficiency of the proposed algorithm.