We consider a problem of ecological inference, in which individual-level covariates are known, but labeled data is available only at the aggregate level. The intended application is modeling voter preferences in elections. In Rosenman and Viswanathan (2018), we proposed modeling individual voter probabilities via a logistic regression, and posing the problem as a maximum likelihood estimation for the parameter vector beta. The likelihood is a Poisson binomial, the distribution of the sum of independent but not identically distributed Bernoulli variables, though we approximate it with a heteroscedastic Gaussian for computational efficiency. Here, we extend the prior work by proving results about the existence of the MLE and the curvature of this likelihood, which is not log-concave in general. We further demonstrate the utility of our method on a real data example. Using data on voters in Morris County, NJ, we demonstrate that our approach outperforms other ecological inference methods in predicting a related, but known outcome: whether an individual votes.

Online Peer to Peer Lending (P2PL) systems connect lenders and borrowers directly, thereby making it convenient to borrow and lend money without intermediaries such as banks. Many recommendation systems have been developed for lenders to achieve higher interest rates and avoid defaulting loans. However, there has not been much research in developing recommendation systems to help borrowers make wise decisions. On P2PL platforms, borrowers can either apply for bidding loans, where the interest rate is determined by lenders bidding on a loan or traditional loans where the P2PL platform determines the interest rate. Different borrower grades -- determining the credit worthiness of borrowers get different interest rates via these two mechanisms. Hence, it is essential to determine which type of loans borrowers should apply for. In this paper, we build a recommendation system that recommends to any new borrower the type of loan they should apply for. Using our recommendation system, any borrower can achieve lowered interest rates with a higher likelihood of getting funded.

Mode decomposition is a prototypical pattern recognition problem that can be addressed from the (a priori distinct) perspectives of numerical approximation, statistical inference and deep learning. Could its analysis through these combined perspectives be used as a Rosetta stone for deciphering mechanisms at play in deep learning? Motivated by this question we introduce programmable and interpretable regression networks for pattern recognition and address mode decomposition as a prototypical problem. The programming of these networks is achieved by assembling elementary modules decomposing and recomposing kernels and data. These elementary steps are repeated across levels of abstraction and interpreted from the equivalent perspectives of optimal recovery, game theory and Gaussian process regression (GPR). The prototypical mode/kernel decomposition module produces an optimal approximation $(w_1,w_2,\cdots,w_m)$ of an element $(v_1,v_2,\ldots,v_m)$ of a product of Hilbert subspaces of a common Hilbert space from the observation of the sum $v:=v_1+\cdots+v_m$. The prototypical mode/kernel recomposition module performs partial sums of the recovered modes $w_i$ based on the alignment between each recovered mode $w_i$ and the data $v$. We illustrate the proposed framework by programming regression networks approximating the modes $v_i= a_i(t)y_i\big(\theta_i(t)\big)$ of a (possibly noisy) signal $\sum_i v_i$ when the amplitudes $a_i$, instantaneous phases $\theta_i$ and periodic waveforms $y_i$ may all be unknown and show near machine precision recovery under regularity and separation assumptions on the instantaneous amplitudes $a_i$ and frequencies $\dot{\theta}_i$. The structure of some of these networks share intriguing similarities with convolutional neural networks while being interpretable, programmable and amenable to theoretical analysis.

Prevalent efforts have been put in automatically inferring genres of musical items. Yet, the propose solutions often rely on simplifications and fail to address the diversity and subjectivity of music genres. Accounting for these has, though, many benefits for aligning knowledge sources, integrating data and enriching musical items with tags. Here, we choose a new angle for the genre study by seeking to predict what would be the genres of musical items in a target tag system, knowing the genres assigned to them within source tag systems. We call this a translation task and identify three cases: 1) no common annotated corpus between source and target tag systems exists, 2) such a large corpus exists, 3) only few common annotations exist. We propose the related solutions: a knowledge-based translation modeled as taxonomy mapping, a statistical translation modeled with maximum likelihood logistic regression; a hybrid translation modeled with maximum a posteriori logistic regression with priors given by the knowledge-based translation. During evaluation, the solutions fit well the identified cases and the hybrid translation is systematically the most effective w.r.t. multilabel classification metrics. This is a first attempt to unify genre tag systems by leveraging both representation and interpretation diversity.

We propose sparse estimation methods for the generalized linear models, which run Least Angle Regression (LARS) and Least Absolute Shrinkage and Selection Operator (LASSO) in the tangent space of the manifold of the statistical model. Our approach is to roughly approximate the statistical model and to subsequently use exact calculations. LARS was proposed as an efficient algorithm for parameter estimation and variable selection for the normal linear model. The LARS algorithm is described in terms of Euclidean geometry with regarding correlation as metric of the space. Since the LARS algorithm only works in Euclidean space, we transform a manifold of the statistical model into the tangent space at the origin. In the generalized linear regression, this transformation allows us to run the original LARS algorithm for the generalized linear models. The proposed methods are efficient and perform well. Real-data analysis shows that the proposed methods output similar results as that of the $l_1$-penalized maximum likelihood estimation for the generalized linear models. Numerical experiments show that our methods work well and they can be better than the $l_1$-penalization for the generalized linear models in generalization, parameter estimation, and model selection.

A large number and diversity of techniques have been offered in the literature in recent years for solving multi-label classification tasks, including classifier chains where predictions are cascaded to other models as additional features. The idea of extending this chaining methodology to multi-output regression has already been suggested and trialed: regressor chains. However, this has so-far been limited to greedy inference and has provided relatively poor results compared to individual models, and of limited applicability. In this paper we identify and discuss the main limitations, including an analysis of different base models, loss functions, explainability, and other desiderata of real-world applications. To overcome the identified limitations we study and develop methods for regressor chains. In particular we present a sequential Monte Carlo scheme in the framework of a probabilistic regressor chain, and we show it can be effective, flexible and useful in several types of data. We place regressor chains in context in general terms of multi-output learning with continuous outputs, and in doing this shed additional light on classifier chains.

For many important problems the quantity of interest (or output) is an unknown function of the parameter space (or input), which is a random vector with known statistics. Since the dependence of the output on this random vector is unknown, the challenge is to identify its statistics, using the minimum number of function evaluations. This is a problem that can been seen in the context of active learning or optimal experimental design. We employ Bayesian regression to represent the derived model uncertainty due to finite and small number of input-output pairs. In this context we evaluate existing methods for optimal sample selection, such as model error minimization and mutual information maximization. We show that the commonly employed criteria in the literature do not take into account the output values of the existing input-output pairs. To overcome this deficiency we introduce a new criterion that explicitly takes into account the values of the output for the existing samples and adaptively selects inputs from regions or dimensions of the parameter space which have important contribution to the output. The new method allows for application to a large number of input variables, paving the way for optimal experimental design in very high-dimensions.

A tacit assumption in linear regression is that (response, predictor)-pairs correspond to identical observational units. A series of recent works have studied scenarios in which this assumption is violated under terms such as ``Unlabeled Sensing and ``Regression with Unknown Permutation''. In this paper, we study the setup of multiple response variables and a notion of mismatches that generalizes permutations in order to allow for missing matches as well as for one-to-many matches. A two-stage method is proposed under the assumption that most pairs are correctly matched. In the first stage, the regression parameter is estimated by handling mismatches as contaminations, and subsequently the generalized permutation is estimated by a basic variant of matching. The approach is both computationally convenient and equipped with favorable statistical guarantees. Specifically, it is shown that the conditions for permutation recovery become considerably less stringent as the number of responses $m$ per observation increase. Particularly, for $m = \Omega(\log n)$, the required signal-to-noise ratio does no longer depend on the sample size $n$. Numerical results on synthetic and real data are presented to support the main findings of our analysis.

This work proposes the Bregman-Tweedie classification model and analyzes the domain structure of the extended exponential function, an extension of the classic generalized exponential function with additional scaling parameter, and related high-level mathematical structures, such as the Bregman-Tweedie loss function and the Bregman-Tweedie divergence. The base function of this divergence is the convex function of Legendre type induced from the extended exponential function. The Bregman-Tweedie loss function of the proposed classification model is the regular Legendre transformation of the Bregman-Tweedie divergence. This loss function is a polynomial parameterized function between unhinge loss and the logistic loss function. Actually, we have two sub-models of the Bregman-Tweedie classification model; H-Bregman with hinge-like loss function and L-Bregman with logisticlike loss function. Although the proposed classification model is nonconvex and unbounded, empirically, we have observed that the H-Bregman and L-Bregman outperform, in terms of the Friedman ranking, logistic regression and SVM and show reasonable performance in terms of the classification accuracy in the category of the binary linear classification problem. Keywords: Extended exponential function, convex function of Legendre type, Bregman-Tweedie divergence, Bregman-Tweedie classification model, hinge loss, logistic loss.

Risk adjustment has become an increasingly important tool in healthcare. It has been extensively applied to payment adjustment for health plans to reflect the expected cost of providing coverage for members. Risk adjustment models are typically estimated using linear regression, which does not fully exploit the information in claims data. Moreover, the development of such linear regression models requires substantial domain expert knowledge and computational effort for data preprocessing. In this paper, we propose a novel approach for risk adjustment that uses semantic embeddings to represent patient medical histories. Embeddings efficiently represent medical concepts learned from diagnostic, procedure, and prescription codes in patients' medical histories. This approach substantially reduces the need for feature engineering. Our results show that models using embeddings had better performance than a commercial risk adjustment model on the task of prospective risk score prediction.