An Efficient Dual Approach to Distance Metric Learning

Distance metric learning is of fundamental interest in machine learning because the distance metric employed can significantly affect the performance of many learning methods. Quadratic Mahalanobis metric learning is a popular approach to the problem, but typically requires solving a semidefinite programming (SDP) problem, which is computationally expensive. Standard interior-point SDP solvers typically have a complexity of $O(D^{6.5})$ (with $D$ the dimension of input data), and can thus only practically solve problems exhibiting less than a few thousand variables. Since the number of variables is $D (D+1) / 2$, this implies a limit upon the size of problem that can practically be solved of around a few hundred dimensions. The complexity of the popular quadratic Mahalanobis metric learning approach thus limits the size of problem to which metric learning can be applied. Here we propose a significantly more efficient approach to the metric learning problem based on the Lagrange dual formulation of the problem. The proposed formulation is much simpler to implement, and therefore allows much larger Mahalanobis metric learning problems to be solved. The time complexity of the proposed method is $O (D ^ 3)$, which is significantly lower than that of the SDP approach. Experiments on a variety of datasets demonstrate that the proposed method achieves an accuracy comparable to the state-of-the-art, but is applicable to significantly larger problems. We also show that the proposed method can be applied to solve more general Frobenius-norm regularized SDP problems approximately

Statistical inference with anchored Bayesian mixture of regressions models: A case study analysis of allometric data

We present a case study in which we use a mixture of regressions model to improve on an ill-fitting simple linear regression model relating log brain mass to log body mass for 100 placental mammalian species. The slope of this regression model is of particular scientific interest because it corresponds to a constant that governs a hypothesized allometric power law relating brain mass to body mass. A specific line of investigation is to determine whether the regression parameters vary across subgroups of related species. We model these data using an anchored Bayesian mixture of regressions model, which modifies the standard Bayesian Gaussian mixture by pre-assigning small subsets of observations to given mixture components with probability one. These observations (called anchor points) break the relabeling invariance typical of exchangeable model specifications (the so-called label-switching problem). A careful choice of which observations to pre-classify to which mixture components is key to the specification of a well-fitting anchor model. In the article we compare three strategies for the selection of anchor points. The first assumes that the underlying mixture of regressions model holds and assigns anchor points to different components to maximize the information about their labeling. The second makes no assumption about the relationship between x and y and instead identifies anchor points using a bivariate Gaussian mixture model. The third strategy begins with the assumption that there is only one mixture regression component and identifies anchor points that are representative of a clustering structure based on case-deletion importance sampling weights. We compare the performance of the three strategies on the allometric data set and use auxiliary taxonomic information about the species to evaluate the model-based classifications estimated from these models

Worst-Case Linear Discriminant Analysis as Scalable Semidefinite Feasibility Problems

In this paper, we propose an efficient semidefinite programming (SDP) approach to worst-case linear discriminant analysis (WLDA). Compared with the traditional LDA, WLDA considers the dimensionality reduction problem from the worst-case viewpoint, which is in general more robust for classification. However, the original problem of WLDA is non-convex and difficult to optimize. In this paper, we reformulate the optimization problem of WLDA into a sequence of semidefinite feasibility problems. To efficiently solve the semidefinite feasibility problems, we design a new scalable optimization method with quasi-Newton methods and eigen-decomposition being the core components. The proposed method is orders of magnitude faster than standard interior-point based SDP solvers. Experiments on a variety of classification problems demonstrate that our approach achieves better performance than standard LDA. Our method is also much faster and more scalable than standard interior-point SDP solvers based WLDA. The computational complexity for an SDP with $m$ constraints and matrices of size $d$ by $d$ is roughly reduced from $\mathcal{O}(m^3+md^3+m^2d^2)$ to $\mathcal{O}(d^3)$ ($m>d$ in our case).Comment: 14 page

Distance Metric Learning for Conditional Anomaly Detection

International audienceAnomaly detection methods can be very useful in identifying unusual or interesting patterns in data. A recently proposed conditional anomaly detection framework extends anomaly detection to the problem of identifying anomalous patterns on a subset of attributes in the data. The anomaly always depends (is conditioned) on the value of remaining attributes. The work presented in this paper focuses on instance-based methods for detecting conditional anomalies. The methods depend heavily on the distance metric that lets us identify examples in the dataset that are most critical for detecting the anomaly. To optimize the performance of the anomaly detection methods we explore and study metric learning methods. We evaluate the quality of our methods on the Pneumonia PORT dataset by detecting unusual admission decisions for patients with the community-acquired pneumonia. The results of our metric learning methods show an improved detection performance over standard distance metrics, which is very promising for building automated anomaly detection systems for variety of intelligent monitoring applications