A new index for internal evaluation of clustering is introduced. The index is defined as a mixture of two sub-indices. The first sub-index $ I_a $ is called the Ambiguous Index; the second sub-index $ I_s $ is called the Similarity Index. Calculation of the two sub-indices is based on density estimation to each cluster of a partition of the data. An experiment is conducted to test the performance of the new index, and compared with six other internal clustering evaluation indices -- Calinski-Harabasz index, Silhouette coefficient, Davies-Bouldin index, CDbw, DBCV, and VIASCKDE, on a set of 145 datasets. The result shows the new index significantly improves other internal clustering evaluation indices.

Probabilistic predictions from neural networks which account for predictive uncertainty during classification is crucial in many real-world and high-impact decision making settings. However, in practice most datasets are trained on non-probabilistic neural networks which by default do not capture this inherent uncertainty. This well-known problem has led to the development of post-hoc calibration procedures, such as Platt scaling (logistic), isotonic and beta calibration, which transforms the scores into well calibrated empirical probabilities. A plausible alternative to the calibration approach is to use Bayesian neural networks, which directly models a predictive distribution. Although they have been applied to images and text datasets, they have seen limited adoption in the tabular and small data regime. In this paper, we demonstrate that Bayesian neural networks yields competitive performance when compared to calibrated neural networks and conduct experiments across a wide array of datasets.

Search-and-rescue (SaR) in unknown environments requires precise, optimal, and fast decisions. Robots are promising candidates for autonomously performing SaR tasks in unknown environments. While humans use their heuristics to effectively deal with uncertainties, optimisation of multiple objectives in the presence of physical and control constraints is a mathematical challenge that requires machine computations. Thus having both human-inspired and mathematical control capabilities is desired for SaR robots. Moreover, coordinating the decisions of robots with little computation cost in large-scale SaR missions is an open challenge. Finally, in real-life data perceived by SaR robots may be prone to uncertainties. We introduce a hierarchical multi-agent control architecture that exploits non-homogeneous and imperfect perception capabilities of SaR robots, as well as the computational efficiency and robustness to failure of decentralised control methods and global performance improvement of centralised control methods. The integrated structure of the proposed control framework allows to combine human-inspired and mathematical decision making methods in a coordinated and computationally efficient way. The results of various computer-based simulations show that while the area coverage of the proposed approach is comparable to existing heuristic methods that are particularly developed for coverage-oriented SaR, the efficiency of the introduced approach in locating the trapped victims is significantly higher. Furthermore, with comparable computation times, the proposed control approach successfully avoids potential conflicts that exist in non-cooperative methods. These results confirm that the proposed multi-agent control system is capable of combining coverage-oriented and target-oriented SaR in a balanced and coordinated way.

We address object tracking by radar and the robustness of the current state-of-the-art methods to process outliers. The standard tracking algorithms extract detections from radar image space to use it in the filtering stage. Filtering is performed by a Kalman filter, which assumes Gaussian distributed noise. However, this assumption does not account for large modeling errors and results in poor tracking performance during abrupt motions. We take the Gaussian Sum Filter (single-object variant of the Multi Hypothesis Tracker) as our baseline and propose a modification by modelling process noise with a distribution that has heavier tails than a Gaussian. Variational Bayes provides a fast, computationally cheap inference algorithm. Our simulations show that - in the presence of process outliers - the robust tracker outperforms the Gaussian Sum filter when tracking single objects.

Over the course of the last several years, a significant amount of scholarly attention has been drawn to the issue of feature selection. At a high level, feature selection can be considered as a branch of reducing data dimensionality of which the two primary methods are feature learning and feature selection. The problem of feature learning involves the creation of new features from the original data. In contrast, the feature selection problem does not change the original representation of the data variables, so the physical meaning of each variable is preserved. To be more specific, the feature selection problem can be subdivided into two scenarios: supervised and unsupervised. Since we do not have target variables, selecting unsupervised features is more challenging. Typically, the unsupervised feature selection relies on matrix decomposition (Cheng et al., 2005; Liberty et al., 2007; Martinsson et al., 2011; Lu, 2022a), filter (Dash et al., 2002), and embeddings (Dy & Brodley, 2004; Hou et al., 2011). On the other hand, matrix decomposition algorithms such as QR decomposition, and singular value decomposition have been used extensively over the years to reveal hidden structures of data matrices in scientific and engineering areas such as collaborative filtering (Marlin, 2003; Lim & Teh, 2007; Mnih & Salakhutdinov, 2007; Lu, 2022c;a), recommendation systems (Lu, 2022c), clustering and classification (Li et al., 2009; Wang et al., 2013).

Contrast pattern mining (CPM) is an important and popular subfield of data mining. Traditional sequential patterns cannot describe the contrast information between different classes of data, while contrast patterns involving the concept of contrast can describe the significant differences between datasets under different contrast conditions. Based on the number of papers published in this field, we find that researchers' interest in CPM is still active. Since CPM has many research questions and research methods. It is difficult for new researchers in the field to understand the general situation of the field in a short period of time. Therefore, the purpose of this article is to provide an up-to-date comprehensive and structured overview of the research direction of contrast pattern mining. First, we present an in-depth understanding of CPM, including basic concepts, types, mining strategies, and metrics for assessing discriminative ability. Then we classify CPM methods according to their characteristics into boundary-based algorithms, tree-based algorithms, evolutionary fuzzy system-based algorithms, decision tree-based algorithms, and other algorithms. In addition, we list the classical algorithms of these methods and discuss their advantages and disadvantages. Advanced topics in CPM are presented. Finally, we conclude our survey with a discussion of the challenges and opportunities in this field.

We rigorously prove that deep Gaussian process priors can outperform Gaussian process priors if the target function has a compositional structure. To this end, we study information-theoretic lower bounds for posterior contraction rates for Gaussian process regression in a continuous regression model. We show that if the true function is a generalized additive function, then the posterior based on any mean-zero Gaussian process can only recover the truth at a rate that is strictly slower than the minimax rate by a factor that is polynomially suboptimal in the sample size $n$.

Normalizing flows model complex probability distributions using maps obtained by composing invertible layers. Special linear layers such as masked and 1x1 convolutions play a key role in existing architectures because they increase expressive power while having tractable Jacobians and inverses. We propose a new family of invertible linear layers based on butterfly layers, which are known to theoretically capture complex linear structures including permutations and periodicity, yet can be inverted efficiently. This representational power is a key advantage of our approach, as such structures are common in many real-world datasets. Based on our invertible butterfly layers, we construct a new class of normalizing flow models called ButterflyFlow. Empirically, we demonstrate that ButterflyFlows not only achieve strong density estimation results on natural images such as MNIST, CIFAR-10, and ImageNet 32x32, but also obtain significantly better log-likelihoods on structured datasets such as galaxy images and MIMIC-III patient cohorts -- all while being more efficient in terms of memory and computation than relevant baselines.

Importance sampling (IS) is a powerful Monte Carlo (MC) methodology for approximating integrals, for instance in the context of Bayesian inference. In IS, the samples are simulated from the so-called proposal distribution, and the choice of this proposal is key for achieving a high performance. In adaptive IS (AIS) methods, a set of proposals is iteratively improved. AIS is a relevant and timely methodology although many limitations remain yet to be overcome, e.g., the curse of dimensionality in high-dimensional and multi-modal problems. Moreover, the Hamiltonian Monte Carlo (HMC) algorithm has become increasingly popular in machine learning and statistics. HMC has several appealing features such as its exploratory behavior, especially in high-dimensional targets, when other methods suffer. In this paper, we introduce the novel Hamiltonian adaptive importance sampling (HAIS) method. HAIS implements a two-step adaptive process with parallel HMC chains that cooperate at each iteration. The proposed HAIS efficiently adapts a population of proposals, extracting the advantages of HMC. HAIS can be understood as a particular instance of the generic layered AIS family with an additional resampling step. HAIS achieves a significant performance improvement in high-dimensional problems w.r.t. state-of-the-art algorithms. We discuss the statistical properties of HAIS and show its high performance in two challenging examples.

Surrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties. For models that show a strongly non-linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response. Through applying a rational approximation instead, the approximation error can be efficiently reduced for models whose non-linearity is accurately described through a rational function. Specifically, our aim is to approximate complex-valued models. A common approach to obtain the coefficients in the surrogate is to minimize the sample-based error between model and surrogate in the least-square sense. In order to obtain an accurate representation of the original model and to avoid overfitting, the sample set has be two to three times the number of polynomial terms in the expansion. For models that require a high polynomial degree or are high-dimensional in terms of their input parameters, this number often exceeds the affordable computational cost. To overcome this issue, we apply a sparse Bayesian learning approach to the rational approximation. Through a specific prior distribution structure, sparsity is induced in the coefficients of the surrogate model. The denominator polynomial coefficients as well as the hyperparameters of the problem are determined through a type-II-maximum likelihood approach. We apply a quasi-Newton gradient-descent algorithm in order to find the optimal denominator coefficients and derive the required gradients through application of $\mathbb{CR}$-calculus.