We present a new method for high-dimensional linear regression when a scale parameter of the additive errors is unknown. The proposed estimator is based on a penalized Huber $M$-estimator, for which theoretical results on estimation error have recently been proposed in high-dimensional statistics literature. However, the variance of the error term in the linear model is intricately connected to the optimal parameter used to define the shape of the Huber loss. Our main idea is to use an adaptive technique, based on Lepski's method, to overcome the difficulties in solving a joint nonconvex optimization problem with respect to the location and scale parameters.

We propose a sparse and low-rank tensor regression model to relate a univariate outcome to a feature tensor, in which each unit-rank tensor from the CP decomposition of the coefficient tensor is assumed to be sparse. This structure is both parsimonious and highly interpretable, as it implies that the outcome is related to the features through a few distinct pathways, each of which may only involve subsets of feature dimensions. We take a divide-and-conquer strategy to simplify the task into a set of sparse unit-rank tensor regression problems. To make the computation efficient and scalable, for the unit-rank tensor regression, we propose a stagewise estimation procedure to efficiently trace out its entire solution path. We show that as the step size goes to zero, the stagewise solution paths converge exactly to those of the corresponding regularized regression. The superior performance of our approach is demonstrated on various real-world and synthetic examples.

Such information includes: the database in modern hospital systems, usually known as Electronic Health Records (EHR), which store the patients' diagnosis, medication, laboratory test results, medical image data, etc.; information on various health behaviors tracked and stored by wearable devices, ubiquitous sensors and mobile applications, such as the smoking status, alcoholism history, exercise level, sleeping conditions, etc.; information collected by census or various surveys regarding sociodemographic factors of the target cohort; and information on people's mental health inferred from their social media activities or social networks such as Twitter, Facebook, etc. These health-related data come from heterogeneous sources, describe assorted aspects of the individual's health conditions. Such data is rich in structure and information which has great research potentials for revealing unknown medical knowledge about genomic epidemiology, disease developments and correlations, drug discoveries, medical diagnosis, mental illness prevention, health behavior adaption, etc. In real-world problems, the number of features relating to a certain health condition could grow exponentially with the development of new information techniques for collecting and measuring data. To reveal the causal influence between various factors and a certain disease or to discover the correlations among diseases from data at such a tremendous scale, requires the assistance of advanced information technology such as data mining, machine learning, text mining, etc. Machine learning technology not only provides a way for learning qualitative relationships among features and patients, but also the quantitative parameters regarding the strength of such correlations.

A machine learning model may exhibit discrimination when used to make decisions involving people. One potential cause for such outcomes is that the model uses a statistical proxy for a protected demographic attribute. In this paper we formulate a definition of proxy use for the setting of linear regression and present algorithms for detecting proxies. Our definition follows recent work on proxies in classification models, and characterizes a model's constituent behavior that: 1) correlates closely with a protected random variable, and 2) is causally influential in the overall behavior of the model. We show that proxies in linear regression models can be efficiently identified by solving a second-order cone program, and further extend this result to account for situations where the use of a certain input variable is justified as a "business necessity". Finally, we present empirical results on two law enforcement datasets that exhibit varying degrees of racial disparity in prediction outcomes, demonstrating that proxies shed useful light on the causes of discriminatory behavior in models.

We present a method that "meta" classifies whether segments (objects) predicted by a semantic segmentation neural network intersect with the ground truth. To this end, we employ measures of dispersion for predicted pixel-wise class probability distributions, like classification entropy, that yield heat maps of the input scene's size. We aggregate these dispersion measures segment-wise and derive metrics that are well-correlated with the segment-wise $\mathit{IoU}$ of prediction and ground truth. In our tests, we use two publicly available DeepLabv3+ networks (pre-trained on the Cityscapes data set) and analyze the predictive power of different metrics and different sets of metrics. To this end, we compute logistic LASSO regression fits for the task of classifying $\mathit{IoU}=0$ vs. $\mathit{IoU} > 0$ per segment and obtain classification rates of up to $81.91\%$ and AUROC values of up to $87.71\%$ without the incorporation of advanced techniques like Monte-Carlo dropout. We complement these tests with linear regression fits to predict the segment-wise $\mathit{IoU}$ and obtain prediction standard deviations of down to $0.130$ as well as $R^2$ values of up to $81.48\%$. We show that these results clearly outperform single-metric baseline approaches.

Due to its high computational speed and accuracy compared to ab-initio quantum chemistry and forcefield modeling, the prediction of molecular properties using machine learning has received great attention in the fields of materials design and drug discovery. A main ingredient required for machine learning is a training dataset consisting of molecular features\textemdash for example fingerprint bits, chemical descriptors, etc. that adequately characterize the corresponding molecules. However, choosing features for any application is highly non-trivial. No "universal" method for feature selection exists. In this work, we propose a data fusion framework that uses Independent Vector Analysis to exploit underlying complementary information contained in different molecular featurization methods, bringing us a step closer to automated feature generation. Our approach takes an arbitrary number of individual feature vectors and automatically generates a single, compact (low dimensional) set of molecular features that can be used to enhance the prediction performance of regression models. At the same time our methodology retains the possibility of interpreting the generated features to discover relationships between molecular structures and properties. We demonstrate this on the QM7b dataset for the prediction of several properties such as atomization energy, polarizability, frontier orbital eigenvalues, ionization potential, electron affinity, and excitation energies. In addition, we show how our method helps improve the prediction of experimental binding affinities for a set of human BACE-1 inhibitors. To encourage more widespread use of IVA we have developed the PyIVA Python package, an open source code which is available for download on Github.

Modern sensors generate large amounts of timestamped measurement data. These data sets are critical in a wide range of applications including traffic flow prediction, transportation management, GPS navigation, and city planning. Machine learning-based prediction algorithms typically adjust their parameters automatically based on the data, but also require users to set additional parameters, known as hyperparameters. For example, in a kernel-based regression model, the (ordinary) parameters are the regression weights, whereas the hyperparameters include the kernel scales and regularization constants. These hyperparameters have a strong influence on the prediction accuracy. Often, their values are set based on past experience or through time-consuming grid searches. In applications where the characteristics of the data change, such as unusual traffic pattern due to upcoming concert events, these hyperparameters have to be adjusted dynamically in order to maintain prediction quality. In this paper, we use the term hyperparameter learning, hyperparameter optimization, and hyperparameter selection/tuning interchangeably, referring to the process of configuring the model specification before model fitting.

Sparse regression such as Lasso has achieved great success in dealing with high dimensional data for several decades. However, there are few methods applicable to missing data, which often occurs in high dimensional data. Recently, CoCoLasso was proposed to deal with high dimensional missing data, but it still suffers from highly missing data. In this paper, we propose a novel Lasso-type regression technique for Highly Missing data, called `HMLasso'. We use the mean imputed covariance matrix, which is notorious in general due to its estimation bias for missing data. However, we effectively incorporate it into Lasso, by using a useful connection with the pairwise covariance matrix. The resulting optimization problem can be seen as a weighted modification of CoCoLasso with the missing ratios, and is quite effective for highly missing data. To the best of our knowledge, this is the first method that can efficiently deal with both high dimensional and highly missing data. We show that the proposed method is beneficial with regards to non-asymptotic properties of the covariance matrix. Numerical experiments show that the proposed method is highly advantageous in terms of estimation error and generalization error.

Exploiting low-rank structure of the user-item rating matrix has been the crux of many recommendation engines. However, existing recommendation engines force raters with heterogeneous behavior profiles to map their intrinsic rating scales to a common rating scale (e.g. 1-5). This non-linear transformation of the rating scale shatters the low-rank structure of the rating matrix, therefore resulting in a poor fit and consequentially, poor recommendations. In this paper, we propose Clustered Monotone Transforms for Rating Factorization (CMTRF), a novel approach to perform regression up to unknown monotonic transforms over unknown population segments. Essentially, for recommendation systems, the technique searches for monotonic transformations of the rating scales resulting in a better fit. This is combined with an underlying matrix factorization regression model that couples the user-wise ratings to exploit shared low dimensional structure. The rating scale transformations can be generated for each user, for a cluster of users, or for all the users at once, forming the basis of three simple and efficient algorithms proposed in this paper, all of which alternate between transformation of the rating scales and matrix factorization regression. Despite the non-convexity, CMTRF is theoretically shown to recover a unique solution under mild conditions. Experimental results on two synthetic and seven real-world datasets show that CMTRF outperforms other state-of-the-art baselines.

Predicting the response of cancer cells to drugs is an important problem in pharmacogenomics. Recent efforts in generation of large scale datasets profiling gene expression and drug sensitivity in cell lines have provided a unique opportunity to study this problem. However, one major challenge is the small number of samples (cell lines) compared to the number of features (genes) even in these large datasets. We propose a collaborative filtering (CF) like algorithm for modeling gene-drug relationship to identify patients most likely to benefit from a treatment. Due to the correlation of gene expressions in different cell lines, the gene expression matrix is approximately low-rank, which suggests that drug responses could be estimated from a reduced dimension latent space of the gene expression. Towards this end, we propose a joint low-rank matrix factorization and latent linear regression approach. Experiments with data from the Genomics of Drug Sensitivity in Cancer database are included to show that the proposed method can predict drug-gene associations better than the state-of-the-art methods.