The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a similarity space and concepts are represented by convex regions in this space. After pointing out a problem with the convexity requirement, we propose a formalization of conceptual spaces based on fuzzy star-shaped sets. Our formalization uses a parametric definition of concepts and extends the original framework by adding means to represent correlations between different domains in a geometric way. Moreover, we define various operations for our formalization, both for creating new concepts from old ones and for measuring relations between concepts. We present an illustrative toy-example and sketch a research project on concept formation that is based on both our formalization and its implementation.

Hierarchical probabilistic models are able to use a large number of parameters to create a model with a high representation power. However, it is well known that increasing the number of parameters also increases the complexity of the model which leads to a bias-variance trade-off. Although it is a classical problem, the bias-variance trade-off between hidden layers and higher-order interactions have not been well studied. In our study, we propose an efficient inference algorithm for the log-linear formulation of the higher-order Boltzmann machine using a combination of Gibbs sampling and annealed importance sampling. We then perform a bias-variance decomposition to study the differences in hidden layers and higher-order interactions. Our results have shown that using hidden layers and higher-order interactions have a comparable error with a similar order of magnitude and using higher-order interactions produce less variance for smaller sample size.

Modeling unknown systems from data is a precursor of system optimization and sequential decision making. In this paper, we focus on learning a Markov model from a single trajectory of states. Suppose that the transition model has a small rank despite of a large state space, meaning that the system admits a low-dimensional latent structure. We show that one can estimate the full transition model accurately using a trajectory of length that is proportional to the total number of states. We propose two maximum likelihood estimation methods: a convex approach with nuclear-norm regularization and a nonconvex approach with rank constraint. We show that both estimators enjoy optimal statistical rates in terms of the Kullback-Leiber divergence and the $\ell_2$ error. For computing the nonconvex estimator, we develop a novel DC (difference of convex function) programming algorithm that starts with the convex M-estimator and then successively refines the solution till convergence. Empirical experiments demonstrate consistent superiority of the nonconvex estimator over the convex one.

We build upon probabilistic models for Boolean Matrix and Boolean Tensor factorisation that have recently been shown to solve these problems with unprecedented accuracy and to enable posterior inference to scale to Billions of observation. Here, we lift the restriction of a pre-specified number of latent dimensions by introducing an Indian Buffet Process prior over factor matrices. Not only does the full factor-conditional take a computationally convenient form due to the logical dependencies in the model, but also the posterior over the number of non-zero latent dimensions is remarkably simple. It amounts to counting the number false and true negative predictions, whereas positive predictions can be ignored. This constitutes a very transparent example of sampling-based posterior inference with an IBP prior and, importantly, lets us maintain extremely efficient inference. We discuss applications to simulated data, as well as to a real world data matrix with 6 Million entries.

One of the most surprising and exciting discoveries in supervising learning was the benefit of overparametrization (i.e. training a very large model) to improving the optimization landscape of a problem, with minimal effect on statistical performance (i.e. generalization). In contrast, unsupervised settings have been under-explored, despite the fact that it has been observed that overparameterization can be helpful as early as Dasgupta & Schulman (2007). In this paper, we perform an exhaustive study of different aspects of overparameterization in unsupervised learning via synthetic and semi-synthetic experiments. We discuss benefits to different metrics of success (held-out log-likelihood, recovering the parameters of the ground-truth model), sensitivity to variations of the training algorithm, and behavior as the amount of overparameterization increases. We find that, when learning using methods such as variational inference, larger models can significantly increase the number of ground truth latent variables recovered.

We present a consensus Monte Carlo algorithm that scales existing Bayesian nonparametric models for clustering and feature allocation to big data. The algorithm is valid for any prior on random subsets such as partitions and latent feature allocation, under essentially any sampling model. Motivated by three case studies, we focus on clustering induced by a Dirichlet process mixture sampling model, inference under an Indian buffet process prior with a binomial sampling model, and with a categorical sampling model. We assess the proposed algorithm with simulation studies and show results for inference with three datasets: an MNIST image dataset, a dataset of pancreatic cancer mutations, and a large set of electronic health records (EHR). Supplementary materials for this article are available online.

Numerous lung diseases, such as idiopathic pulmonary fibrosis (IPF), exhibit dilation of the airways. Accurate measurement of dilatation enables assessment of the progression of disease. Unfortunately the combination of image noise and airway bifurcations causes high variability in the profiles of cross-sectional areas, rendering the identification of affected regions very difficult. Here we introduce a noise-robust method for automatically detecting the location of progressive airway dilatation given two profiles of the same airway acquired at different time points. We propose a probabilistic model of abrupt relative variations between profiles and perform inference via Reversible Jump Markov Chain Monte Carlo sampling. We demonstrate the efficacy of the proposed method on two datasets; (i) images of healthy airways with simulated dilatation; (ii) pairs of real images of IPF-affected airways acquired at 1 year intervals. Our model is able to detect the starting location of airway dilatation with an accuracy of 2.5mm on simulated data. The experiments on the IPF dataset display reasonable agreement with radiologists. We can compute a relative change in airway volume that may be useful for quantifying IPF disease progression.

We describe a limitation in the expressiveness of the predictive uncertainty estimate given by mean-field variational inference (MFVI), a popular approximate inference method for Bayesian neural networks. In particular, MFVI fails to give calibrated uncertainty estimates in between separated regions of observations. This can lead to catastrophically overconfident predictions when testing on out-of-distribution data. Avoiding such overconfidence is critical for active learning, Bayesian optimisation and out-of-distribution robustness. We instead find that a classical technique, the linearised Laplace approximation, can handle 'in-between' uncertainty much better for small network architectures.

To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the objects are typically embedded in a Euclidean space while satisfying as many triplet comparisons as possible. In this paper, we introduce empirical uncertainty estimates for standard embedding algorithms when few noisy triplets are available, using a bootstrap and a Bayesian approach. In particular, simulations show that these estimates are well calibrated and can serve to select embedding parameters or to quantify uncertainty in scientific applications.

Integer-valued and count data are ubiquitous in many fields, including epidemiology (Osthus et al., 2018; Kowal, 2019), ecology (Dorazio et al., 2005), and insurance (Bening and Korolev, 2012), among others (Cameron and Trivedi, 2013). Count data also serve as an indicator of demand, such as the demand for medical services (Deb and Trivedi, 1997), emergency medical services (Matteson et al., 2011), and call center access (Shen and Huang, 2008). In these applications and many others, integer-valued data are frequently observed jointly with predictors, over time intervals, or across spatial locations. Integer-valued data also exhibit a variety of distributional features, including zero-inflation, skewness, over-or underdispersion, and in some cases may be bounded or censored. Flexible and interpretable models for integervalued processes are therefore highly useful in practice. The most widely-used models for count data build upon the Poisson distribution. However, the limitations of the Poisson distribution are well-known: the distribution is not sufficiently flexible in practice and cannot account for zero-inflation or over-and underdispersion. A common strategy is to generalize the Poisson model by introducing additional parameters.