We use an enhanced methodology combining specific forms of AI techniques, opinion mining and artificial mathematical intelligence (AMI), with public data on the spread of the coronavirus SARS-CoV-2 and the incidence of COVID-19 disease in Colombia during the first three months since the first reported positive case. The results obtained, together with conceptual tools coming from the global taxonomy of fundamental cognitive mechanisms emerging in AMI and with suitable contextual information from Colombian public health and mainstream social media, allowed us to stating specific preventive guidelines for a better restructuring of initial safe and stable life conditions in Colombia, and in an extended manner in similar Latin American Countries. More specifically, we describe three major guidelines: 1) regular creative visualization and effective planning, 2) the continuous use of constructive linguistic frameworks, and 3) frequent and moderate use of kinesthetic routines. They should be understood as effective tools from a cognitive and behavioural perspective, rather than from a biological one. Even more, the first two guidelines should be acknowledged in integral cooperation with the third one regarding the global effect of COVID-19 in human beings as a whole, this includes the mind and body.

This paper is concerned with multi-modal data fusion (MMDF) under unexpected modality failures in nonlinear non-Gaussian dynamic processes. An efficient framework to tackle this problem is proposed. In particular, a notion termed modality "usefulness", which takes a value of 1 or 0, is used for indicating whether the observation of this modality is useful or not. For $n$ modalities involved, $2^n$ combinations of their "usefulness" values exist. Each combination defines one hypothetical model of the true data generative process. Then the problem of concern is formalized as a task of nonlinear non-Gaussian state filtering under model uncertainty, which is addressed by a dynamic model averaging based particle filter algorithm. Experimental results show that the proposed solution outperforms remarkably state-of-the-art methods. Code and data are available at https://github.com/robinlau1981/fusion.

Mixed linear regression (MLR) model is among the most exemplary statistical tools for modeling non-linear distributions using a mixture of linear models. When the additive noise in MLR model is Gaussian, Expectation-Maximization (EM) algorithm is a widely-used algorithm for maximum likelihood estimation of MLR parameters. However, when noise is non-Gaussian, the steps of EM algorithm may not have closed-form update rules, which makes EM algorithm impractical. In this work, we study the maximum likelihood estimation of the parameters of MLR model when the additive noise has non-Gaussian distribution. In particular, we consider the case that noise has Laplacian distribution and we first show that unlike the the Gaussian case, the resulting sub-problems of EM algorithm in this case does not have closed-form update rule, thus preventing us from using EM in this case. To overcome this issue, we propose a new algorithm based on combining the alternating direction method of multipliers (ADMM) with EM algorithm idea. Our numerical experiments show that our method outperforms the EM algorithm in statistical accuracy and computational time in non-Gaussian noise case.

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function - the matrix analog of the perspective function-and characterize explicitly the convex hull of epigraphs of convex quadratic, matrix exponential, and matrix power functions under low-rank constraints. Further, we exploit these characterizations to develop strong relaxations for a variety of low-rank problems including reduced rank regression, non-negative matrix factorization, and factor analysis. We establish that these relaxations can be modeled via semidefinite and matrix power cone constraints, and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization, and leads to new relaxations for a broad class of problems.

We introduce kernel thinning, a simple algorithm for generating better-than-Monte-Carlo approximations to distributions $\mathbb{P}$ on $\mathbb{R}^d$. Given $n$ input points, a suitable reproducing kernel $\mathbf{k}$, and $\mathcal{O}(n^2)$ time, kernel thinning returns $\sqrt{n}$ points with comparable integration error for every function in the associated reproducing kernel Hilbert space. With high probability, the maximum discrepancy in integration error is $\mathcal{O}_d(n^{-\frac{1}{2}}\sqrt{\log n})$ for compactly supported $\mathbb{P}$ and $\mathcal{O}_d(n^{-\frac{1}{2}} \sqrt{(\log n)^{d+1}\log\log n})$ for sub-exponential $\mathbb{P}$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n^{-\frac14})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\mathbb{R}^d$ and a wide range of common kernels. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat\'ern, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning.

Metric based comparison operations such as finding maximum, nearest and farthest neighbor are fundamental to studying various clustering techniques such as $k$-center clustering and agglomerative hierarchical clustering. These techniques crucially rely on accurate estimation of pairwise distance between records. However, computing exact features of the records, and their pairwise distances is often challenging, and sometimes not possible. We circumvent this challenge by leveraging weak supervision in the form of a comparison oracle that compares the relative distance between the queried points such as `Is point u closer to v or w closer to x?'. However, it is possible that some queries are easier to answer than others using a comparison oracle. We capture this by introducing two different noise models called adversarial and probabilistic noise. In this paper, we study various problems that include finding maximum, nearest/farthest neighbor search under these noise models. Building upon the techniques we develop for these comparison operations, we give robust algorithms for k-center clustering and agglomerative hierarchical clustering. We prove that our algorithms achieve good approximation guarantees with a high probability and analyze their query complexity. We evaluate the effectiveness and efficiency of our techniques empirically on various real-world datasets.

Classification problems with a large number of labels arise in language modelling [Mikolov et al., 2013, Levy and Goldberg, 2014], recommender systems [Covington et al., 2016, Xu et al., 2016], and information retrieval [Agrawal et al., 2013, Prabhu and Varma, 2014]. Such large-output problems pose a core challenge: losses such as the softmax cross-entropy can be prohibitive to optimise, as they depend on the entire set of labels. Several works have thus devised negative sampling schemes for efficiently and effectively approximating such losses [Bengio and Senecal, 2008, Blanc and Rendle, 2018, Ruiz et al., 2018, Bamler and Mandt, 2020]. Broadly, negative sampling techniques sample a subset of "negative" labels, which are used to contrast against the observed "positive" labels. One further applies a suitable weighting on these "negatives", which ostensibly corrects the sampling bias introduced by the dependence on a random subset of labels. Intuitively, such bias assesses how closely a scheme approximates the unsampled loss on the full label set. This bias is well understood for sampled softmax schemes (see, e.g., Bengio and Senecal [2008]); surprisingly, however, far less is understood about other popular schemes, e.g., within-batch and uniform sampling (cf.

A primary motivation for the development and implementation of structural health monitoring systems, is the prospect of gaining the ability to make informed decisions regarding the operation and maintenance of structures and infrastructure. Unfortunately, descriptive labels for measured data corresponding to health-state information for the structure of interest are seldom available prior to the implementation of a monitoring system. This issue limits the applicability of the traditional supervised and unsupervised approaches to machine learning in the development of statistical classifiers for decision-supporting SHM systems. The current paper presents a risk-based formulation of active learning, in which the querying of class-label information is guided by the expected value of said information for each incipient data point. When applied to structural health monitoring, the querying of class labels can be mapped onto the inspection of a structure of interest in order to determine its health state. In the current paper, the risk-based active learning process is explained and visualised via a representative numerical example and subsequently applied to the Z24 Bridge benchmark. The results of the case studies indicate that a decision-maker's performance can be improved via the risk-based active learning of a statistical classifier, such that the decision process itself is taken into account.

We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior performance of MsIGN over previous approaches in posterior approximation and multiple mode capture. On the natural image synthesis task, MsIGN achieves superior performance in bits-per-dimension over baseline models and yields great interpret-ability of its neurons in intermediate layers.

$\ell_1$ regularization has been used for logistic regression to circumvent the overfitting and use the estimated sparse coefficient for feature selection. However, the challenge of such a regularization is that the $\ell_1$ norm is not differentiable, making the standard algorithms for convex optimization not applicable to this problem. This paper presents a simple projection neural network for $\ell_1$-regularized logistics regression. In contrast to many available solvers in the literature, the proposed neural network does not require any extra auxiliary variable nor any smooth approximation, and its complexity is almost identical to that of the gradient descent for logistic regression without $\ell_1$ regularization, thanks to the projection operator. We also investigate the convergence of the proposed neural network by using the Lyapunov theory and show that it converges to a solution of the problem with any arbitrary initial value. The proposed neural solution significantly outperforms state-of-the-art methods with respect to the execution time and is competitive in terms of accuracy and AUROC.