We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall be used to predict the class label of a future data function. Unlike the popular functional discriminant analysis approaches which rely on Gaussian assumption, the proposed FDNN approach applies to general non-Gaussian multi-dimensional functional data. Moreover, when the log density ratio possesses a locally connected functional modular structure, we show that FDNN achieves minimax optimality. The superiority of our approach is demonstrated through both simulated and real-world datasets.

Hierarchical multi-label classification (HMC) has drawn increasing attention in the past few decades. It is applicable when hierarchical relationships among classes are available and need to be incorporated along with the multi-label classification whereby each object is assigned to one or more classes. There are two key challenges in HMC: i) optimizing the classification accuracy, and meanwhile ii) ensuring the given class hierarchy. To address these challenges, in this article, we introduce a new statistic called the multidimensional local precision rate (mLPR) for each object in each class. We show that classification decisions made by simply sorting objects across classes in descending order of their true mLPRs can, in theory, ensure the class hierarchy and lead to the maximization of CATCH, an objective function we introduce that is related to the area under a hit curve. This approach is the first of its kind that handles both challenges in one objective function without additional constraints, thanks to the desirable statistical properties of CATCH and mLPR. In practice, however, true mLPRs are not available. In response, we introduce HierRank, a new algorithm that maximizes an empirical version of CATCH using estimated mLPRs while respecting the hierarchy. The performance of this approach was evaluated on a synthetic data set and two real data sets; ours was found to be superior to several comparison methods on evaluation criteria based on metrics such as precision, recall, and $F_1$ score.

This paper uses Gaussian mixture model instead of linear Gaussian model to fit the distribution of every node in Bayesian network. We will explain why and how we use Gaussian mixture models in Bayesian network. Meanwhile we propose a new method, called double iteration algorithm, to optimize the mixture model, the double iteration algorithm combines the expectation maximization algorithm and gradient descent algorithm, and it performs perfectly on the Bayesian network with mixture models. In experiments we test the Gaussian mixture model and the optimization algorithm on different graphs which is generated by different structure learning algorithm on real data sets, and give the details of every experiment.

Physics-informed extreme learning machine (PIELM) has recently received significant attention as a rapid version of physics-informed neural network (PINN) for solving partial differential equations (PDEs). The key characteristic is to fix the input layer weights with random values and use Moore-Penrose generalized inverse for the output layer weights. The framework is effective, but it easily suffers from overfitting noisy data and lacks uncertainty quantification for the solution under noise scenarios.To this end, we develop the Bayesian physics-informed extreme learning machine (BPIELM) to solve both forward and inverse linear PDE problems with noisy data in a unified framework. In our framework, a prior probability distribution is introduced in the output layer for extreme learning machine with physic laws and the Bayesian method is used to estimate the posterior of parameters. Besides, for inverse PDE problems, problem parameters considered as new output layer weights are unified in a framework with forward PDE problems. Finally, we demonstrate BPIELM considering both forward problems, including Poisson, advection, and diffusion equations, as well as inverse problems, where unknown problem parameters are estimated. The results show that, compared with PIELM, BPIELM quantifies uncertainty arising from noisy data and provides more accurate predictions. In addition, BPIELM is considerably cheaper than PINN in terms of the computational cost.

A new Bayesian modelling framework is introduced for piece-wise homogeneous variable-memory Markov chains, along with a collection of effective algorithmic tools for change-point detection and segmentation of discrete time series. Building on the recently introduced Bayesian Context Trees (BCT) framework, the distributions of different segments in a discrete time series are described as variable-memory Markov chains. Inference for the presence and location of change-points is then performed via Markov chain Monte Carlo sampling. The key observation that facilitates effective sampling is that, using one of the BCT algorithms, the prior predictive likelihood of the data can be computed exactly, integrating out all the models and parameters in each segment. This makes it possible to sample directly from the posterior distribution of the number and location of the change-points, leading to accurate estimates and providing a natural quantitative measure of uncertainty in the results. Estimates of the actual model in each segment can also be obtained, at essentially no additional computational cost. Results on both simulated and real-world data indicate that the proposed methodology performs better than or as well as state-of-the-art techniques.

Applied category theory has recently developed libraries for computing with morphisms in interesting categories, while machine learning has developed ways of learning programs in interesting languages. Taking the analogy between categories and languages seriously, this paper defines a probabilistic generative model of morphisms in free monoidal categories over domain-specific generating objects and morphisms. The paper shows how acyclic directed wiring diagrams can model specifications for morphisms, which the model can use to generate morphisms. Amortized variational inference in the generative model then enables learning of parameters (by maximum likelihood) and inference of latent variables (by Bayesian inversion). A concrete experiment shows that the free category prior achieves competitive reconstruction performance on the Omniglot dataset.

This paper studies the cooperative learning of two generative flow models, in which the two models are iteratively updated based on the jointly synthesized examples. The first flow model is a normalizing flow that transforms an initial simple density into a target density by applying a sequence of invertible transformations. The second flow model is a Langevin flow that runs finite steps of gradient-based MCMC toward an energy-based model. We start from proposing a generative framework that trains an energy-based model with a normalizing flow as an amortized sampler to initialize the MCMC chains of the energy-based model. In each learning iteration, we generate synthesized examples by using a normalizing flow initialization followed by a short-run Langevin flow revision toward the current energy-based model. Then we treat the synthesized examples as fair samples from the energy-based model and update the model parameters with the maximum likelihood learning gradient, while the normalizing flow directly learns from the synthesized examples by maximizing the tractable likelihood. Under the short-run non-mixing MCMC scenario, the estimation of the energy-based model is shown to follow the perturbation of maximum likelihood, and the short-run Langevin flow and the normalizing flow form a two-flow generator that we call CoopFlow. We provide an understating of the CoopFlow algorithm by information geometry and show that it is a valid generator as it converges to a moment matching estimator. We demonstrate that the trained CoopFlow is capable of synthesizing realistic images, reconstructing images, and interpolating between images. Normalizing flows (Dinh et al., 2015; 2017; Kingma & Dhariwal, 2018) are a family of generative models that construct a complex distribution by transforming a simple probability density, such as Gaussian distribution, through a sequence of invertible and differentiable mappings. Due to the tractability of the exact log-likelihood and the efficiency of the inference and synthesis, normalizing flows have gained popularity in density estimation (Kingma & Dhariwal, 2018; Ho et al., 2019; Yang et al., 2019; Prenger et al., 2019; Kumar et al., 2020) and variational inference (Rezende & Mohamed, 2015; Kingma et al., 2016).

We introduce a conditional compression problem and propose a fast framework for tackling it. The problem is how to quickly compress a pretrained large neural network into optimal smaller networks given target contexts, e.g. a context involving only a subset of classes or a context where only limited compute resource is available. To solve this, we propose an efficient Bayesian framework to compress a given large network into much smaller size tailored to meet each contextual requirement. We employ a hypernetwork to parameterize the posterior distribution of weights given conditional inputs and minimize a variational objective of this Bayesian neural network. To further reduce the network sizes, we propose a new input-output group sparsity factorization of weights to encourage more sparseness in the generated weights. Our methods can quickly generate compressed networks with significantly smaller sizes than baseline methods.

This is an interesting paper that distils structure learning in Bayesian networks (BNs) and kernel methods in a quest to produce more flexible distributional assumptions. Conditional (linear) Gaussian Bayesian networks (CGBNs) have been well explored in the literature for some time, to the point that they now appear in many recent textbooks [1-3]. The authors address one of the key limitations of CGBNS, that they can only capture linear dependencies between the continuous variables they contain, and remove it by replacing (mixtures of) linear regression models with more general kernel densities. Dependencies between discrete variables were already flexible, because the conditional probability tables that parametrise them essentially act as a saturated model[4]. It is not obvious that more flexibility will produce better models for whatever task we have in mind: it can also lead to overfitting, instability and hyperparameter tuning problems.

Autonomous systems that operate in the open world often use incomplete models of their environment. Model incompleteness is inevitable due to the practical limitations in precise model specification and data collection about open-world environments. Due to the limited fidelity of the model, agent actions may produce negative side effects (NSEs) when deployed. Negative side effects are undesirable, unmodeled effects of agent actions on the environment. NSEs are inherently challenging to identify at design time and may affect the reliability, usability and safety of the system. We present two complementary approaches to mitigate the NSE via: (1) learning from feedback, and (2) environment shaping. The solution approaches target settings with different assumptions and agent responsibilities. In learning from feedback, the agent learns a penalty function associated with a NSE. We investigate the efficiency of different feedback mechanisms, including human feedback and autonomous exploration. The problem is formulated as a multi-objective Markov decision process such that optimizing the agent’s assigned task is prioritized over mitigating NSE. A slack parameter denotes the maximum allowed deviation from the optimal expected reward for the agent’s task in order to mitigate NSE. In environment shaping, we examine how a human can assist an agent, beyond providing feedback, and utilize their broader scope of knowledge to mitigate the impacts of NSE. We formulate the problem as a human-agent collaboration with decoupled objectives. The agent optimizes its assigned task and may produce NSE during its operation. The human assists the agent by performing modest reconfigurations of the environment so as to mitigate the impacts of NSE, without affecting the agent’s ability to complete its assigned task. We present an algorithm for shaping and analyze its properties. Empirical evaluations demonstrate the trade-offs in the performance of different approaches in mitigating NSE in different settings.