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Information diagram and the I-measure are useful mnemonics where random variables are treated as sets, and entropy and mutual information are treated as a signed measure. Although the I-measure has been successful in machine proofs of entropy inequalities, the theoretical underpinning of the ``random variables as sets'' analogy has been unclear until the recent works on mappings from random variables to sets by Ellerman (recovering order-$2$ Tsallis entropy over general probability space), and Down and Mediano (recovering Shannon entropy over discrete probability space). We generalize these constructions by designing a mapping which recovers the Shannon entropy (and the information density) over general probability space. Moreover, it has an intuitive interpretation based on the arrival time in a Poisson process, allowing us to understand the union, intersection and difference between (sets corresponding to) random variables and events. Cross entropy, KL divergence, and conditional entropy given an event, can be obtained as set intersections. We propose a generalization of the information diagram that also includes events, and demonstrate its usage by a diagrammatic proof of Fano's inequality.

Mixtures of factor analysers (MFA) models represent a popular tool for finding structure in data, particularly high-dimensional data. While in most applications the number of clusters, and especially the number of latent factors within clusters, is mostly fixed in advance, in the recent literature models with automatic inference on both the number of clusters and latent factors have been introduced. The automatic inference is usually done by assigning a nonparametric prior and allowing the number of clusters and factors to potentially go to infinity. The MCMC estimation is performed via an adaptive algorithm, in which the parameters associated with the redundant factors are discarded as the chain moves. While this approach has clear advantages, it also bears some significant drawbacks. Running a separate factor-analytical model for each cluster involves matrices of changing dimensions, which can make the model and programming somewhat cumbersome. In addition, discarding the parameters associated with the redundant factors could lead to a bias in estimating cluster covariance matrices. At last, identification remains problematic for infinite factor models. The current work contributes to the MFA literature by providing for the automatic inference on the number of clusters and the number of cluster-specific factors while keeping both cluster and factor dimensions finite. This allows us to avoid many of the aforementioned drawbacks of the infinite models. For the automatic inference on the cluster structure, we employ the dynamic mixture of finite mixtures (MFM) model. Automatic inference on cluster-specific factors is performed by assigning an exchangeable shrinkage process (ESP) prior to the columns of the factor loading matrices. The performance of the model is demonstrated on several benchmark data sets as well as real data applications.

We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix $A\in\mathbb{R}^{m\times n}$ and a vector $b\in\mathbb{R}^m$, we propose classical algorithms that produce a data structure for the solution $x\in\mathbb{R}^{n}$ of the linear system $Ax=b$ with the ability to sample and query its entries. The resulting $x$ satisfies $\|x-A^{+}b\|\leq\epsilon\|A^{+}b\|$, where $\|\cdot\|$ is the spectral norm and $A^+$ is the Moore-Penrose inverse of $A$. Our algorithm has time complexity $\widetilde{O}(\kappa_F^4/\kappa\epsilon^2)$ in the general case, where $\kappa_{F} =\|A\|_F\|A^+\|$ and $\kappa=\|A\|\|A^+\|$ are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When $A$ is $s$-sparse, our algorithm has complexity $\widetilde{O}(s \kappa\log(1/\epsilon))$, matching the quantum lower bound for solving linear systems in $\kappa$ and $1/\epsilon$ up to poly-logarithmic factors [Harrow and Kothari]. When $A$ is $s$-sparse and symmetric positive-definite, our algorithm has complexity $\widetilde{O}(s\sqrt{\kappa}\log(1/\epsilon))$. Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.

Population-based structural health monitoring (PBSHM) aims to share valuable information among members of a population, such as normal- and damage-condition data, to improve inferences regarding the health states of the members. Even when the population is comprised of nominally-identical structures, benign variations among the members will exist as a result of slight differences in material properties, geometry, boundary conditions, or environmental effects (e.g., temperature changes). These discrepancies can affect modal properties and present as changes in the characteristics of the resonance peaks of the frequency response function (FRF). Many SHM strategies depend on monitoring the dynamic properties of structures, so benign variations can be challenging for the practical implementation of these systems. Another common challenge with vibration-based SHM is data loss, which may result from transmission issues, sensor failure, a sample-rate mismatch between sensors, and other causes. Missing data in the time domain will result in decreased resolution in the frequency domain, which can impair dynamic characterisation. The hierarchical Bayesian approach provides a useful modelling structure for PBSHM, because statistical distributions at the population and individual (or domain) level are learnt simultaneously to bolster statistical strength among the parameters. As a result, variance is reduced among the parameter estimates, particularly when data are limited. In this paper, combined probabilistic FRF models are developed for a small population of nominally-identical helicopter blades under varying temperature conditions, using a hierarchical Bayesian structure. These models address critical challenges in SHM, by accommodating benign variations that present as differences in the underlying dynamics, while also considering (and utilising), the similarities among the blades.

We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. This model arises as a natural special case of submodular function maximization: on query $S \subseteq V$, the oracle returns the total weight of the cut between $S$ and $V \backslash S$. For most constants $c \in (0,1]$, we nail down the query complexity of achieving a $c$-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at $c = 1/2$: we design a deterministic algorithm for global $c$-approximate max-cut in $O(\log n)$ queries for any $c < 1/2$, and show that any randomized algorithm requires $\tilde{\Omega}(n)$ queries to find a $c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any deterministic algorithm requires $\Omega(n^2)$ queries to find an exact max-cut (enough to learn the entire graph), and develop a $\tilde{O}(n)$-query randomized $c$-approximation for any $c < 1$. Our approach provides two technical contributions that may be of independent interest. One is a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). Another is an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions).

We study a fundamental problem in optimization under uncertainty. There are $n$ boxes; each box $i$ contains a hidden reward $x_i$. Rewards are drawn i.i.d. from an unknown distribution $\mathcal{D}$. For each box $i$, we see $y_i$, an unbiased estimate of its reward, which is drawn from a Normal distribution with known standard deviation $\sigma_i$ (and an unknown mean $x_i$). Our task is to select a single box, with the goal of maximizing our reward. This problem captures a wide range of applications, e.g. ad auctions, where the hidden reward is the click-through rate of an ad. Previous work in this model [BKMR12] proves that the naive policy, which selects the box with the largest estimate $y_i$, is suboptimal, and suggests a linear policy, which selects the box $i$ with the largest $y_i - c \cdot \sigma_i$, for some $c > 0$. However, no formal guarantees are given about the performance of either policy (e.g., whether their expected reward is within some factor of the optimal policy's reward). In this work, we prove that both the naive policy and the linear policy are arbitrarily bad compared to the optimal policy, even when $\mathcal{D}$ is well-behaved, e.g. has monotone hazard rate (MHR), and even under a "small tail" condition, which requires that not too many boxes have arbitrarily large noise. On the flip side, we propose a simple threshold policy that gives a constant approximation to the reward of a prophet (who knows the realized values $x_1, \dots, x_n$) under the same "small tail" condition. We prove that when this condition is not satisfied, even an optimal clairvoyant policy (that knows $\mathcal{D}$) cannot get a constant approximation to the prophet, even for MHR distributions, implying that our threshold policy is optimal against the prophet benchmark, up to constants.

Providing a model that achieves a strong predictive performance and at the same time is interpretable by humans is one of the most difficult challenges in machine learning research due to the conflicting nature of these two objectives. To address this challenge, we propose a modification of the Radial Basis Function Neural Network model by equipping its Gaussian kernel with a learnable precision matrix. We show that precious information is contained in the spectrum of the precision matrix that can be extracted once the training of the model is completed. In particular, the eigenvectors explain the directions of maximum sensitivity of the model revealing the active subspace and suggesting potential applications for supervised dimensionality reduction. At the same time, the eigenvectors highlight the relationship in terms of absolute variation between the input and the latent variables, thereby allowing us to extract a ranking of the input variables based on their importance to the prediction task enhancing the model interpretability. We conducted numerical experiments for regression, classification, and feature selection tasks, comparing our model against popular machine learning models and the state-of-the-art deep learning-based embedding feature selection techniques. Our results demonstrate that the proposed model does not only yield an attractive prediction performance with respect to the competitors but also provides meaningful and interpretable results that potentially could assist the decision-making process in real-world applications. A PyTorch implementation of the model is available on GitHub at the following link. //github.com/dannyzx/GRBF-NNs

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

Graph machine learning has been extensively studied in both academic and industry. However, as the literature on graph learning booms with a vast number of emerging methods and techniques, it becomes increasingly difficult to manually design the optimal machine learning algorithm for different graph-related tasks. To tackle the challenge, automated graph machine learning, which aims at discovering the best hyper-parameter and neural architecture configuration for different graph tasks/data without manual design, is gaining an increasing number of attentions from the research community. In this paper, we extensively discuss automated graph machine approaches, covering hyper-parameter optimization (HPO) and neural architecture search (NAS) for graph machine learning. We briefly overview existing libraries designed for either graph machine learning or automated machine learning respectively, and further in depth introduce AutoGL, our dedicated and the world's first open-source library for automated graph machine learning. Last but not least, we share our insights on future research directions for automated graph machine learning. This paper is the first systematic and comprehensive discussion of approaches, libraries as well as directions for automated graph machine learning.

Image segmentation is still an open problem especially when intensities of the interested objects are overlapped due to the presence of intensity inhomogeneity (also known as bias field). To segment images with intensity inhomogeneities, a bias correction embedded level set model is proposed where Inhomogeneities are Estimated by Orthogonal Primary Functions (IEOPF). In the proposed model, the smoothly varying bias is estimated by a linear combination of a given set of orthogonal primary functions. An inhomogeneous intensity clustering energy is then defined and membership functions of the clusters described by the level set function are introduced to rewrite the energy as a data term of the proposed model. Similar to popular level set methods, a regularization term and an arc length term are also included to regularize and smooth the level set function, respectively. The proposed model is then extended to multichannel and multiphase patterns to segment colourful images and images with multiple objects, respectively. It has been extensively tested on both synthetic and real images that are widely used in the literature and public BrainWeb and IBSR datasets. Experimental results and comparison with state-of-the-art methods demonstrate that advantages of the proposed model in terms of bias correction and segmentation accuracy.