Fine-grained visual categorization (FGVC) is to categorize objects into subordinate classes instead of basic classes. One major challenge in FGVC is the co-occurrence of two issues: 1) many subordinate classes are highly correlated and are difficult to distinguish, and 2) there exists the large intra-class variation (e.g., due to object pose). This paper proposes to explicitly address the above two issues via distance metric learning (DML). DML addresses the first issue by learning an embedding so that data points from the same class will be pulled together while those from different classes should be pushed apart from each other; and it addresses the second issue by allowing the flexibility that only a portion of the neighbors (not all data points) from the same class need to be pulled together. However, feature representation of an image is often high dimensional, and DML is known to have difficulty in dealing with high dimensional feature vectors since it would require $\mathcal{O}(d^2)$ for storage and $\mathcal{O}(d^3)$ for optimization. To this end, we proposed a multi-stage metric learning framework that divides the large-scale high dimensional learning problem to a series of simple subproblems, achieving $\mathcal{O}(d)$ computational complexity. The empirical study with FVGC benchmark datasets verifies that our method is both effective and efficient compared to the state-of-the-art FGVC approaches.

The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gaussian graphical model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry of a precision matrix under a sparseness condition relative to the sample size. When the precision matrix is not sufficiently sparse, or equivalently the sample size is not sufficiently large, a lower bound is established to show that it is no longer possible to achieve the parametric rate in the estimation of each entry. This lower bound result, which provides an answer to the delicate sample size question, is established with a novel construction of a subset of sparse precision matrices in an application of Le Cam's lemma. Moreover, the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved, under a minimal sample requirement. The proposed estimator is applied to test the presence of an edge in the Gaussian graphical model or to recover the support of the entire model, to obtain adaptive rate-optimal estimation of the entire precision matrix as measured by the matrix $\ell_q$ operator norm and to make inference in latent variables in the graphical model. All of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum. This significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix, irrepresentability condition on the Hessian tensor operator of the covariance matrix or the $\ell_1$ constraint on the precision matrix. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the popular GLasso algorithm.

We present a new, fully generative model of optical telescope image sets, along with a variational procedure for inference. Each pixel intensity is treated as a Poisson random variable, with a rate parameter dependent on latent properties of stars and galaxies. Key latent properties are themselves random, with scientific prior distributions constructed from large ancillary data sets. We check our approach on synthetic images. We also run it on images from a major sky survey, where it exceeds the performance of the current state-of-the-art method for locating celestial bodies and measuring their colors.

This paper is concerned with the problems of interaction screening and nonlinear classification in a high-dimensional setting. We propose a two-step procedure, IIS-SQDA, where in the first step an innovated interaction screening (IIS) approach based on transforming the original $p$-dimensional feature vector is proposed, and in the second step a sparse quadratic discriminant analysis (SQDA) is proposed for further selecting important interactions and main effects and simultaneously conducting classification. Our IIS approach screens important interactions by examining only $p$ features instead of all two-way interactions of order $O(p^2)$. Our theory shows that the proposed method enjoys sure screening property in interaction selection in the high-dimensional setting of $p$ growing exponentially with the sample size. In the selection and classification step, we establish a sparse inequality on the estimated coefficient vector for QDA and prove that the classification error of our procedure can be upper-bounded by the oracle classification error plus some smaller order term. Extensive simulation studies and real data analysis show that our proposal compares favorably with existing methods in interaction selection and high-dimensional classification.

Data analysis and machine learning have become an integrative part of the modern scientific methodology, offering automated procedures for the prediction of a phenomenon based on past observations, unraveling underlying patterns in data and providing insights about the problem. Yet, caution should avoid using machine learning as a black-box tool, but rather consider it as a methodology, with a rational thought process that is entirely dependent on the problem under study. In particular, the use of algorithms should ideally require a reasonable understanding of their mechanisms, properties and limitations, in order to better apprehend and interpret their results. Accordingly, the goal of this thesis is to provide an in-depth analysis of random forests, consistently calling into question each and every part of the algorithm, in order to shed new light on its learning capabilities, inner workings and interpretability. The first part of this work studies the induction of decision trees and the construction of ensembles of randomized trees, motivating their design and purpose whenever possible. Our contributions follow with an original complexity analysis of random forests, showing their good computational performance and scalability, along with an in-depth discussion of their implementation details, as contributed within Scikit-Learn. In the second part of this work, we analyse and discuss the interpretability of random forests in the eyes of variable importance measures. The core of our contributions rests in the theoretical characterization of the Mean Decrease of Impurity variable importance measure, from which we prove and derive some of its properties in the case of multiway totally randomized trees and in asymptotic conditions. In consequence of this work, our analysis demonstrates that variable importances [...].

Community detection, which aims to cluster $N$ nodes in a given graph into $r$ distinct groups based on the observed undirected edges, is an important problem in network data analysis. In this paper, the popular stochastic block model (SBM) is extended to the generalized stochastic block model (GSBM) that allows for adversarial outlier nodes, which are connected with the other nodes in the graph in an arbitrary way. Under this model, we introduce a procedure using convex optimization followed by $k$-means algorithm with $k=r$. Both theoretical and numerical properties of the method are analyzed. A theoretical guarantee is given for the procedure to accurately detect the communities with small misclassification rate under the setting where the number of clusters can grow with $N$. This theoretical result admits to the best-known result in the literature of computationally feasible community detection in SBM without outliers. Numerical results show that our method is both computationally fast and robust to different kinds of outliers, while some popular computationally fast community detection algorithms, such as spectral clustering applied to adjacency matrices or graph Laplacians, may fail to retrieve the major clusters due to a small portion of outliers. We apply a slight modification of our method to a political blogs data set, showing that our method is competent in practice and comparable to existing computationally feasible methods in the literature. To the best of the authors' knowledge, our result is the first in the literature in terms of clustering communities with fast growing numbers under the GSBM where a portion of arbitrary outlier nodes exist.

We introduce Bayesian optimization, a technique developed for optimizing time-consuming engineering simulations and for fitting machine learning models on large datasets. Bayesian optimization guides the choice of experiments during materials design and discovery to find good material designs in as few experiments as possible. We focus on the case when materials designs are parameterized by a low-dimensional vector. Bayesian optimization is built on a statistical technique called Gaussian process regression, which allows predicting the performance of a new design based on previously tested designs. After providing a detailed introduction to Gaussian process regression, we introduce two Bayesian optimization methods: expected improvement, for design problems with noise-free evaluations; and the knowledge-gradient method, which generalizes expected improvement and may be used in design problems with noisy evaluations. Both methods are derived using a value-of-information analysis, and enjoy one-step Bayes-optimality.

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

Consider the problem of sampling sequentially from a finite number of $N \geq 2$ populations, specified by random variables $X^i_k$, $ i = 1,\ldots , N,$ and $k = 1, 2, \ldots$; where $X^i_k$ denotes the outcome from population $i$ the $k^{th}$ time it is sampled. It is assumed that for each fixed $i$, $\{ X^i_k \}_{k \geq 1}$ is a sequence of i.i.d. normal random variables, with unknown mean $\mu_i$ and unknown variance $\sigma_i^2$. The objective is to have a policy $\pi$ for deciding from which of the $N$ populations to sample form at any time $n=1,2,\ldots$ so as to maximize the expected sum of outcomes of $n$ samples or equivalently to minimize the regret due to lack on information of the parameters $\mu_i$ and $\sigma_i^2$. In this paper, we present a simple inflated sample mean (ISM) index policy that is asymptotically optimal in the sense of Theorem 4 below. This resolves a standing open problem from Burnetas and Katehakis (1996). Additionally, finite horizon regret bounds are given.

Bayesian hierarchical clustering (BHC) is an agglomerative clustering method, where a probabilistic model is defined and its marginal likelihoods are evaluated to decide which clusters to merge. While BHC provides a few advantages over traditional distance-based agglomerative clustering algorithms, successive evaluation of marginal likelihoods and careful hyperparameter tuning are cumbersome and limit the scalability. In this paper we relax BHC into a non-probabilistic formulation, exploring small-variance asymptotics in conjugate-exponential models. We develop a novel clustering algorithm, referred to as relaxed BHC (RBHC), from the asymptotic limit of the BHC model that exhibits the scalability of distance-based agglomerative clustering algorithms as well as the flexibility of Bayesian nonparametric models. We also investigate the reducibility of the dissimilarity measure emerged from the asymptotic limit of the BHC model, allowing us to use scalable algorithms such as the nearest neighbor chain algorithm. Numerical experiments on both synthetic and real-world datasets demonstrate the validity and high performance of our method.