We introduce a kernel method for manifold alignment (KEMA) and domain adaptation that can match an arbitrary number of data sources without needing corresponding pairs, just few labeled examples in all domains. KEMA has interesting properties: 1) it generalizes other manifold alignment methods, 2) it can align manifolds of very different complexities, performing a sort of manifold unfolding plus alignment, 3) it can define a domain-specific metric to cope with multimodal specificities, 4) it can align data spaces of different dimensionality, 5) it is robust to strong nonlinear feature deformations, and 6) it is closed-form invertible which allows transfer across-domains and data synthesis. We also present a reduced-rank version for computational efficiency and discuss the generalization performance of KEMA under Rademacher principles of stability. KEMA exhibits very good performance over competing methods in synthetic examples, visual object recognition and recognition of facial expressions tasks.

Preference learning (PL) is a core area of machine learning that handles datasets with ordinal relations. As the number of generated data of ordinal nature is increasing, the importance and role of the PL field becomes central within machine learning research and practice. This paper introduces an open source, scalable, efficient and accessible preference learning toolbox that supports the key phases of the data training process incorporating various popular data preprocessing, feature selection and preference learning methods.

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.

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.

Joint peak detection is a central problem when comparing samples in genomic data analysis, but current algorithms for this task are unsupervised and limited to at most 2 sample types. We propose PeakSegJoint, a new constrained maximum likelihood segmentation model for any number of sample types. To select the number of peaks in the segmentation, we propose a supervised penalty learning model. To infer the parameters of these two models, we propose to use a discrete optimization heuristic for the segmentation, and convex optimization for the penalty learning. In comparisons with state-of-the-art peak detection algorithms, PeakSegJoint achieves similar accuracy, faster speeds, and a more interpretable model with overlapping peaks that occur in exactly the same positions across all samples.

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.

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 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.

Multi-task feature learning aims to identity the shared features among tasks to improve generalization. It has been shown that by minimizing non-convex learning models, a better solution than the convex alternatives can be obtained. Therefore, a non-convex model based on the capped-$\ell_{1},\ell_{1}$ regularization was proposed in \cite{Gong2013}, and a corresponding efficient multi-stage multi-task feature learning algorithm (MSMTFL) was presented. However, this algorithm harnesses a prescribed fixed threshold in the definition of the capped-$\ell_{1},\ell_{1}$ regularization and the lack of adaptivity might result in suboptimal performance. In this paper we propose to employ an adaptive threshold in the capped-$\ell_{1},\ell_{1}$ regularized formulation, where the corresponding variant of MSMTFL will incorporate an additional component to adaptively determine the threshold value. This variant is expected to achieve a better feature selection performance over the original MSMTFL algorithm. In particular, the embedded adaptive threshold component comes from our previously proposed iterative support detection (ISD) method \cite{Wang2010}. Empirical studies on both synthetic and real-world data sets demonstrate the effectiveness of this new variant over the original MSMTFL.

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.