The Dirichlet Process Mixture Model (DPMM) is a Bayesian non-parametric approach widely used for density estimation and clustering. In this manuscript, we study the choice of prior for the variance or precision matrix when Gaussian kernels are adopted. Typically, in the relevant literature, the assessment of mixture models is done by considering observations in a space of only a handful of dimensions. Instead, we are concerned with more realistic problems of higher dimensionality, in a space of up to 20 dimensions. We observe that the choice of prior is increasingly important as the dimensionality of the problem increases. After identifying certain undesirable properties of standard priors in problems of higher dimensionality, we review and implement possible alternative priors. The most promising priors are identified, as well as other factors that affect the convergence of MCMC samplers. Our results show that the choice of prior is critical for deriving reliable posterior inferences. This manuscript offers a thorough overview and comparative investigation into possible priors, with detailed guidelines for their implementation. Although our work focuses on the use of the DPMM in clustering, it is also applicable to density estimation.
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In this paper we present an algebraic dimension-oblivious two-level domain decomposition solver for discretizations of elliptic partial differential equations. The proposed parallel solver is based on a space-filling curve partitioning approach that is applicable to any discretization, i.e. it directly operates on the assembled matrix equations. Moreover, it allows for the effective use of arbitrary processor numbers independent of the dimension of the underlying partial differential equation while maintaining optimal convergence behavior. This is the core property required to attain a sparse grid based combination method with extreme scalability which can utilize exascale parallel systems efficiently. Moreover, this approach provides a basis for the development of a fault-tolerant solver for the numerical treatment of high-dimensional problems. To achieve the required data redundancy we are therefore concerned with large overlaps of our domain decomposition which we construct via space-filling curves. In this paper, we propose our space-filling curve based domain decomposition solver and present its convergence properties and scaling behavior. The results of numerical experiments clearly show that our approach provides optimal convergence and scaling behavior in arbitrary dimension utilizing arbitrary processor numbers.
In the realm of unsupervised learning, Bayesian nonparametric mixture models, exemplified by the Dirichlet Process Mixture Model (DPMM), provide a principled approach for adapting the complexity of the model to the data. Such models are particularly useful in clustering tasks where the number of clusters is unknown. Despite their potential and mathematical elegance, however, DPMMs have yet to become a mainstream tool widely adopted by practitioners. This is arguably due to a misconception that these models scale poorly as well as the lack of high-performance (and user-friendly) software tools that can handle large datasets efficiently. In this paper we bridge this practical gap by proposing a new, easy-to-use, statistical software package for scalable DPMM inference. More concretely, we provide efficient and easily-modifiable implementations for high-performance distributed sampling-based inference in DPMMs where the user is free to choose between either a multiple-machine, multiple-core, CPU implementation (written in Julia) and a multiple-stream GPU implementation (written in CUDA/C++). Both the CPU and GPU implementations come with a common (and optional) python wrapper, providing the user with a single point of entry with the same interface. On the algorithmic side, our implementations leverage a leading DPMM sampler from (Chang and Fisher III, 2013). While Chang and Fisher III's implementation (written in MATLAB/C++) used only CPU and was designed for a single multi-core machine, the packages we proposed here distribute the computations efficiently across either multiple multi-core machines or across mutiple GPU streams. This leads to speedups, alleviates memory and storage limitations, and lets us fit DPMMs to significantly larger datasets and of higher dimensionality than was possible previously by either (Chang and Fisher III, 2013) or other DPMM methods.
Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.
Linear mixed models (LMMs) are instrumental for regression analysis with structured dependence, such as grouped, clustered, or multilevel data. However, selection among the covariates--while accounting for this structured dependence--remains a challenge. We introduce a Bayesian decision analysis for subset selection with LMMs. Using a Mahalanobis loss function that incorporates the structured dependence, we derive optimal linear coefficients for (i) any given subset of variables and (ii) all subsets of variables that satisfy a cardinality constraint. Crucially, these estimates inherit shrinkage or regularization and uncertainty quantification from the underlying Bayesian model, and apply for any well-specified Bayesian LMM. More broadly, our decision analysis strategy deemphasizes the role of a single "best" subset, which is often unstable and limited in its information content, and instead favors a collection of near-optimal subsets. This collection is summarized by key member subsets and variable-specific importance metrics. Customized subset search and out-of-sample approximation algorithms are provided for more scalable computing. These tools are applied to simulated data and a longitudinal physical activity dataset, and demonstrate excellent prediction, estimation, and selection ability.
A High-dimensional and sparse (HiDS) matrix is frequently encountered in a big data-related application like an e-commerce system or a social network services system. To perform highly accurate representation learning on it is of great significance owing to the great desire of extracting latent knowledge and patterns from it. Latent factor analysis (LFA), which represents an HiDS matrix by learning the low-rank embeddings based on its observed entries only, is one of the most effective and efficient approaches to this issue. However, most existing LFA-based models perform such embeddings on a HiDS matrix directly without exploiting its hidden graph structures, thereby resulting in accuracy loss. To address this issue, this paper proposes a graph-incorporated latent factor analysis (GLFA) model. It adopts two-fold ideas: 1) a graph is constructed for identifying the hidden high-order interaction (HOI) among nodes described by an HiDS matrix, and 2) a recurrent LFA structure is carefully designed with the incorporation of HOI, thereby improving the representa-tion learning ability of a resultant model. Experimental results on three real-world datasets demonstrate that GLFA outperforms six state-of-the-art models in predicting the missing data of an HiDS matrix, which evidently supports its strong representation learning ability to HiDS data.
Materialized model query aims to find the most appropriate materialized model as the initial model for model reuse. It is the precondition of model reuse, and has recently attracted much attention. Nonetheless, the existing methods suffer from low privacy protection, limited range of applications, and inefficiency since they do not construct a suitable metric to measure the target-related knowledge of materialized models. To address this, we present MMQ, a privacy-protected, general, efficient, and effective materialized model query framework. It uses a Gaussian mixture-based metric called separation degree to rank materialized models. For each materialized model, MMQ first vectorizes the samples in the target dataset into probability vectors by directly applying this model, then utilizes Gaussian distribution to fit for each class of probability vectors, and finally uses separation degree on the Gaussian distributions to measure the target-related knowledge of the materialized model. Moreover, we propose an improved MMQ (I-MMQ), which significantly reduces the query time while retaining the query performance of MMQ. Extensive experiments on a range of practical model reuse workloads demonstrate the effectiveness and efficiency of MMQ.
We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and because of its meshless nature, it can also deal with problems posed in high-dimensional domains. We prove the $\Gamma$-convergence of the neural network approximation towards the solution of the continuous problem, and extend the convergence proof to some well-known related methods. Finally, we present several numerical examples illustrating the performance of our discretization.
Click-through rate (CTR) prediction plays a critical role in recommender systems and online advertising. The data used in these applications are multi-field categorical data, where each feature belongs to one field. Field information is proved to be important and there are several works considering fields in their models. In this paper, we proposed a novel approach to model the field information effectively and efficiently. The proposed approach is a direct improvement of FwFM, and is named as Field-matrixed Factorization Machines (FmFM, or $FM^2$). We also proposed a new explanation of FM and FwFM within the FmFM framework, and compared it with the FFM. Besides pruning the cross terms, our model supports field-specific variable dimensions of embedding vectors, which acts as soft pruning. We also proposed an efficient way to minimize the dimension while keeping the model performance. The FmFM model can also be optimized further by caching the intermediate vectors, and it only takes thousands of floating-point operations (FLOPs) to make a prediction. Our experiment results show that it can out-perform the FFM, which is more complex. The FmFM model's performance is also comparable to DNN models which require much more FLOPs in runtime.
Machine learning plays a role in many deployed decision systems, often in ways that are difficult or impossible to understand by human stakeholders. Explaining, in a human-understandable way, the relationship between the input and output of machine learning models is essential to the development of trustworthy machine-learning-based systems. A burgeoning body of research seeks to define the goals and methods of explainability in machine learning. In this paper, we seek to review and categorize research on counterfactual explanations, a specific class of explanation that provides a link between what could have happened had input to a model been changed in a particular way. Modern approaches to counterfactual explainability in machine learning draw connections to the established legal doctrine in many countries, making them appealing to fielded systems in high-impact areas such as finance and healthcare. Thus, we design a rubric with desirable properties of counterfactual explanation algorithms and comprehensively evaluate all currently-proposed algorithms against that rubric. Our rubric provides easy comparison and comprehension of the advantages and disadvantages of different approaches and serves as an introduction to major research themes in this field. We also identify gaps and discuss promising research directions in the space of counterfactual explainability.
While existing machine learning models have achieved great success for sentiment classification, they typically do not explicitly capture sentiment-oriented word interaction, which can lead to poor results for fine-grained analysis at the snippet level (a phrase or sentence). Factorization Machine provides a possible approach to learning element-wise interaction for recommender systems, but they are not directly applicable to our task due to the inability to model contexts and word sequences. In this work, we develop two Position-aware Factorization Machines which consider word interaction, context and position information. Such information is jointly encoded in a set of sentiment-oriented word interaction vectors. Compared to traditional word embeddings, SWI vectors explicitly capture sentiment-oriented word interaction and simplify the parameter learning. Experimental results show that while they have comparable performance with state-of-the-art methods for document-level classification, they benefit the snippet/sentence-level sentiment analysis.
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