We present a Gaussian Process - Latent Class Choice Model (GP-LCCM) to integrate a non-parametric class of probabilistic machine learning within discrete choice models (DCMs). Gaussian Processes (GPs) are kernel-based algorithms that incorporate expert knowledge by assuming priors over latent functions rather than priors over parameters, which makes them more flexible in addressing nonlinear problems. By integrating a Gaussian Process within a LCCM structure, we aim at improving discrete representations of unobserved heterogeneity. The proposed model would assign individuals probabilistically to behaviorally homogeneous clusters (latent classes) using GPs and simultaneously estimate class-specific choice models by relying on random utility models. Furthermore, we derive and implement an Expectation-Maximization (EM) algorithm to jointly estimate/infer the hyperparameters of the GP kernel function and the class-specific choice parameters by relying on a Laplace approximation and gradient-based numerical optimization methods, respectively. The model is tested on two different mode choice applications and compared against different LCCM benchmarks. Results show that GP-LCCM allows for a more complex and flexible representation of heterogeneity and improves both in-sample fit and out-of-sample predictive power. Moreover, behavioral and economic interpretability is maintained at the class-specific choice model level while local interpretation of the latent classes can still be achieved, although the non-parametric characteristic of GPs lessens the transparency of the model.

Machine learning (ML) has become a vital part in many aspects of our daily life. However, building well performing machine learning applications requires highly specialized data scientists and domain experts. Automated machine learning (AutoML) aims to reduce the demand for data scientists by enabling domain experts to build machine learning applications automatically without extensive knowledge of statistics and machine learning. This paper is a combination of a survey on current AutoML methods and a benchmark of popular AutoML frameworks on real data sets. Driven by the selected frameworks for evaluation, we summarize and review important AutoML techniques and methods concerning every step in building an ML pipeline. The selected AutoML frameworks are evaluated on 137 data sets from established AutoML benchmark suites.

Inferring dynamics from time series is an important objective in data analysis. In particular, it is challenging to infer stochastic dynamics given incomplete data. We propose an expectation maximization (EM) algorithm that iterates between alternating two steps: E-step restores missing data points, while M-step infers an underlying network model of restored data. Using synthetic data generated by a kinetic Ising model, we confirm that the algorithm works for restoring missing data points as well as inferring the underlying model. At the initial iteration of the EM algorithm, the model inference shows better model-data consistency with observed data points than with missing data points. As we keep iterating, however, missing data points show better model-data consistency. We find that demanding equal consistency of observed and missing data points provides an effective stopping criterion for the iteration to prevent overshooting the most accurate model inference. Armed with this EM algorithm with this stopping criterion, we infer missing data points and an underlying network from a time-series data of real neuronal activities. Our method recovers collective properties of neuronal activities, such as time correlations and firing statistics, which have previously never been optimized to fit.

This paper provides elementary analyses of the regret and generalization of minimum-norm interpolating classifiers (MNIC). The MNIC is the function of smallest Reproducing Kernel Hilbert Space norm that perfectly interpolates a label pattern on a finite data set. We derive a mistake bound for MNIC and a regularized variant that holds for all data sets. This bound follows from elementary properties of matrix inverses. Under the assumption that the data is independently and identically distributed, the mistake bound implies that MNIC generalizes at a rate proportional to the norm of the interpolating solution and inversely proportional to the number of data points. This rate matches similar rates derived for margin classifiers and perceptrons. We derive several plausible generative models where the norm of the interpolating classifier is bounded or grows at a rate sublinear in $n$. We also show that as long as the population class conditional distributions are sufficiently separable in total variation, then MNIC generalizes with a fast rate.

To measure the similarity of documents, the Wasserstein distance is a powerful tool, but it requires a high computational cost. Recently, for fast computation of the Wasserstein distance, methods for approximating the Wasserstein distance using a tree metric have been proposed. These tree-based methods allow fast comparisons of a large number of documents; however, they are unsupervised and do not learn task-specific distances. In this work, we propose the Supervised Tree-Wasserstein (STW) distance, a fast, supervised metric learning method based on the tree metric. Specifically, we rewrite the Wasserstein distance on the tree metric by the parent-child relationships of a tree, and formulate it as a continuous optimization problem using a contrastive loss. Experimentally, we show that the STW distance can be computed fast, and improves the accuracy of document classification tasks. Furthermore, the STW distance is formulated by matrix multiplications, runs on a GPU, and is suitable for batch processing. Therefore, we show that the STW distance is extremely efficient when comparing a large number of documents.

Social media platforms, such as Twitter, provide a totally new perspective in dealing with the traffic problems and is anticipated to complement the traditional methods. The geo-tagged tweets can provide the Twitter users' location information and is being applied in traveler behavior analysis. This paper explores the full potentials of Twitter in deriving travel behavior information and conducts a case study in Manhattan Area. A systematic method is proposed to extract displacement information from Twitter locations. Our study shows that Twitter has a unique demographics which combine not only local residents but also the tourists or passengers. For individual user, Twitter can uncover his/her travel behavior features including the time-of-day and location distributions on both weekdays and weekends. For all Twitter users, the aggregated travel behavior results also show that the time-of-day travel patterns in Manhattan Island resemble that of the traffic flow; the identification of OD pattern is also promising by comparing with the results of travel survey.

Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM) and kernel mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or vector-valued RKHS (vv RKHS), analysis with RKHM enables us to capture and extract structural properties in multivariate data, functional data and other structured data. We show a branch of theories for RKHM to apply to data analysis, including the representer theorem, and the injectivity and universality of the proposed KME. We also show RKHM generalizes RKHS and vv RKHS. Then, we provide concrete procedures for employing RKHM and the proposed KME to data analysis.

Hypergraphs are a natural modeling paradigm for a wide range of complex relational systems with multibody interactions. A standard analysis task is to identify clusters of closely related or densely interconnected nodes. While many probabilistic generative models for graph clustering have been proposed, there are relatively few such models for hypergraphs. We propose a Poisson degree-corrected hypergraph stochastic blockmodel (DCHSBM), an expressive generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum-likelihood inference in the DCHSBM naturally leads to a clustering objective that generalizes the popular modularity objective for graphs. We derive a general Louvain-type algorithm for this objective, as well as a a faster, specialized "All-Or-Nothing" (AON) variant in which edges are expected to lie fully within clusters. This special case encompasses a recent proposal for modularity in hypergraphs, while also incorporating flexible resolution and edge-size parameters. We show that hypergraph Louvain is highly scalable, including as an example an experiment on a synthetic hypergraph of one million nodes. We also demonstrate through synthetic experiments that the detectability regimes for hypergraph community detection differ from methods based on dyadic graph projections. In particular, there are regimes in which hypergraph methods can recover planted partitions even though graph based methods necessarily fail due to information-theoretic limits. We use our model to analyze different patterns of higher-order structure in school contact networks, U.S. congressional bill cosponsorship, U.S. congressional committees, product categories in co-purchasing behavior, and hotel locations from web browsing sessions, that it is able to recover ground truth clusters in empirical data sets exhibiting the corresponding higher-order structure.

Stochastic gradient MCMC methods, such as stochastic gradient Langevin dynamics (SGLD), enable large-scale posterior inference by leveraging noisy but cheap gradient estimates. However, when federated data are non-IID, the variance of distributed gradient estimates is amplified compared to its centralized version, and delayed communication rounds lead chains to diverge from the target posterior. In this work, we introduce the concept of conducive gradients, zero-mean stochastic gradients that serve as a mechanism for sharing probabilistic information between data shards. We propose a novel stochastic gradient estimator that incorporates the conducive gradients, and we show that it improves convergence on federated data when compared to distributed SGLD (DSGLD). We evaluate, conducive gradient DSGLD (CG-DSGLD) on metric learning and deep MLPs tasks. Experiments show that it outperforms standard DSGLD for non-IID federated data.

A key challenge in scaling Gaussian Process (GP) regression to massive datasets is that exact inference requires computation with a dense n x n kernel matrix, where n is the number of data points. Significant work focuses on approximating the kernel matrix via interpolation using a smaller set of m inducing points. Structured kernel interpolation (SKI) is among the most scalable methods: by placing inducing points on a dense grid and using structured matrix algebra, SKI achieves per-iteration time of O(n + m log m) for approximate inference. This linear scaling in n enables inference for very large data sets; however the cost is per-iteration, which remains a limitation for extremely large n. We show that the SKI per-iteration time can be reduced to O(m log m) after a single O(n) time precomputation step by reframing SKI as solving a natural Bayesian linear regression problem with a fixed set of m compact basis functions. With per-iteration complexity independent of the dataset size n for a fixed grid, our method scales to truly massive data sets. We demonstrate speedups in practice for a wide range of m and n and apply the method to GP inference on a three-dimensional weather radar dataset with over 100 million points.