Label switching is a phenomenon arising in mixture model posterior inference that prevents one from meaningfully assessing posterior statistics using standard Monte Carlo procedures. This issue arises due to invariance of the posterior under actions of a group; for example, permuting the ordering of mixture components has no effect on the likelihood. We propose a resolution to label switching that leverages machinery from optimal transport. Our algorithm efficiently computes posterior statistics in the quotient space of the symmetry group. We give conditions under which there is a meaningful solution to label switching and demonstrate advantages over alternative approaches on simulated and real data.

Objectives This study sought to develop models for predicting mortality and heart failure (HF) hospitalization for outpatients with HF with preserved ejection fraction (HFpEF) in the TOPCAT (Treatment of Preserved Cardiac Function Heart Failure with an Aldosterone Antagonist) trial. Background Although risk assessment models are available for patients with HF with reduced ejection fraction, few have assessed the risks of death and hospitalization in patients with HFpEF. Methods The following 5 methods: logistic regression with a forward selection of variables; logistic regression with a lasso regularization for variable selection; random forest (RF); gradient descent boosting; and support vector machine, were used to train models for assessing risks of mortality and HF hospitalization through 3 years of follow-up and were validated using 5-fold cross-validation. Model discrimination and calibration were estimated using receiver-operating characteristic curves and Brier scores, respectively. The top prediction variables were assessed by using the best performing models, using the incremental improvement of each variable in 5-fold cross-validation. Results The RF was the best performing model with a mean C-statistic of 0.72 (95% confidence interval [CI]: 0.69 to 0.75) for predicting mortality (Brier score: 0.17), and 0.76 (95% CI: 0.71 to 0.81) for HF hospitalization (Brier score: 0.19). Blood urea nitrogen levels, body mass index, and Kansas City Cardiomyopathy Questionnaire (KCCQ) subscale scores were strongly associated with mortality, whereas hemoglobin level, blood urea nitrogen, time since previous HF hospitalization, and KCCQ scores were the most significant predictors of HF hospitalization. Conclusions These models predict the risks of mortality and HF hospitalization in patients with HFpEF and emphasize the importance of health status data in determining prognosis.

For details about this plot and the meaning of all the variables check out part one and the notebook. Now, let's move on to implementing Bayesian Linear Regression in Python. Let's briefly recap Frequentist and Bayesian linear regression. Where the response, y, is generated from the model parameters, β, times the input matrix, X, plus error due to random sampling noise or latent variables. In the ordinary least squares (OLS) method, the model parameters, β, are calculated by finding the parameters which minimize the sum of squared errors on the training data.

Conductor galloping is the high-amplitude, low-frequency oscillation of overhead power lines due to wind. Such movements may lead to severe damages to transmission lines, and hence pose significant risks to the power system operation. In this paper, we target to design a prediction framework for conductor galloping. The difficulty comes from imbalanced dataset as galloping happens rarely. By examining the impacts of data balance and data volume on the prediction performance, we propose to employ proper sample adjustment methods to achieve better performance. Numerical study suggests that using only three features, together with over sampling, the SVM based prediction framework achieves an F_1-score of 98.9%.

Tens of millions of women suffer from infertility worldwide each year. In vitro fertilization (IVF) is the best choice for many such patients. However, IVF is expensive, time-consuming, and both physically and emotionally demanding. The first question that a patient usually asks before the IVF is how likely she will conceive, given her basic medical examination information. This paper proposes three approaches to predict the cumulative pregnancy rate after multiple oocyte pickup cycles. Experiments on 11,190 patients showed that first clustering the patients into different groups and then building a support vector machine model for each group can achieve the best overall performance. Our model could be a quick and economic approach for reliably estimating the cumulative pregnancy rate for a patient, given only her basic medical examination information, well before starting the actual IVF procedure. The predictions can help the patient make optimal decisions on whether to use her own oocyte or donor oocyte, how many oocyte pickup cycles she may need, whether to use embryo frozen, etc. They will also reduce the patient's cost and time to pregnancy, and improve her quality of life.

This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data. Theoretical error bounds are provided when the tangent spaces of the manifold satisfy certain incoherence conditions. We also provide a near optimal choice of the tuning parameters for the proposed optimization formulation with the help of a new curvature estimation method. The efficacy of our method is demonstrated on both synthetic and real datasets.

In this paper, we develop a novel procedure for low-rank tensor regression, namely \underline{I}mportance \underline{S}ketching \underline{L}ow-rank \underline{E}stimation for \underline{T}ensors (ISLET). The central idea behind ISLET is \emph{importance sketching}, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical studies that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods whilst having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$ orders of magnitude faster than baseline methods.

Since this phenomenon was first observed, researchers have attempted to develop methods which produce models that are robust to adversarial perturbations under specific attack models (Wong and Kolter (2018); Sinha et al. (2018); Raghunathan et al. (2018); Mirman et al. (2018); Madry et al. (2018); Zhang et al. (2019)). As machine learning proliferates into society, including security-critical settings like health care (Esteva et al. (2017)) or autonomous vehicles (Codevilla et al. (2018)), it is crucial to develop methods that allow us to understand the vulnerability of our models and design appropriate countermeasures. Additionally there is a growing literature on the theory of adversarial examples. Many of these results attempt to understand adversarial examples by constructing examples of learning problems for which it is difficult to construct a classifier that is robust to adversarial perturbations. This difficultly may arise due to sample complexity (Schmidt et al. (2018)), computational constraints (Bubeck et al. (2019); Degwekar et al. (2019)), or the high-dimensional geometry of the initial feature space (Shafahi et al. (2019); Khoury and Hadfield-Menell (2018)).

Many studies have illustrated the import ance to accurately and precisely measure the attributes of an urban transport system. Due to the rise of Big Data and Internet of Things, there are numerous machine learning methods to measur e attributes of the transport system . Chow ( 1) provides an overview of these techniques including several appl ications like Allahviranloo and Recker ( 2) for activity pattern prediction; Cai et al. ( 3) for short - term traffic forecasting; Luque - Baena et al. ( 4) for vehicle detection; Lv et al. ( 5) for t raffic flow prediction; and Ma et al. ( 6) for network congestion prediction. However, generic machine learning techniques are not specifically designed to exploit the unique structure of urban transport networks. As a result, in recent years a theory of inverse problems (see 7) have emerged to capture network structure, dubbed " inverse transportation problems " by Xu et al. ( 8).

This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order $d$ (i.e., a $d$-fold multiway array) in $\mathbb{R}^{n_1 \times n_2 \times \cdots \times n_d}$. This work focuses on the task of estimating the regression tensor from $m$ realizations of the response variable and the predictors where $m\ll n = \prod \nolimits_{i} n_i$. Despite the ill-posedness of this estimation problem, it can still be solved if the parameter tensor belongs to the space of sparse, low Tucker-rank tensors. Accordingly, the estimation procedure is posed as a non-convex optimization program over the space of sparse, low Tucker-rank tensors, and a tensor variant of projected gradient descent is proposed to solve the resulting non-convex problem. In addition, mathematical guarantees are provided that establish the proposed method converges to the correct solution under the right set of conditions. Further, an upper bound on sample complexity of tensor parameter estimation for the model under consideration is characterized for the special case when the individual (scalar) predictors independently draw values from a sub-Gaussian distribution. The sample complexity bound is shown to have a polylogarithmic dependence on $\bar{n} = \max \big\{n_i: i\in \{1,2,\ldots,d \} \big\}$ and, orderwise, it matches the bound one can obtain from a heuristic parameter counting argument. Finally, numerical experiments demonstrate the efficacy of the proposed tensor model and estimation method on a synthetic dataset and a neuroimaging dataset pertaining to attention deficit hyperactivity disorder. Specifically, the proposed method exhibits better sample complexities on both synthetic and real datasets, demonstrating the usefulness of the model and the method in settings where $n \gg m$.