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

Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications

In computer vision, many problems such as image segmentation, pixel labelling, and scene parsing can be formulated as binary quadratic programs (BQPs). For submodular problems, cuts based methods can be employed to efficiently solve large-scale problems. However, general nonsubmodular problems are significantly more challenging to solve. Finding a solution when the problem is of large size to be of practical interest, however, typically requires relaxation. Two standard relaxation methods are widely used for solving general BQPs--spectral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high, especially for large scale problems. In this work, we present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, compared with conventional SDP methods, the new SDP formulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the dual problem. Both of them are significantly more efficiently than standard interior-point methods. In practice, the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems. Our experiments on a few computer vision applications including clustering, image segmentation, co-segmentation and registration show the potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern Analysis and Machine Intelligenc

Regularization and Kernelization of the Maximin Correlation Approach

Robust classification becomes challenging when each class consists of multiple subclasses. Examples include multi-font optical character recognition and automated protein function prediction. In correlation-based nearest-neighbor classification, the maximin correlation approach (MCA) provides the worst-case optimal solution by minimizing the maximum misclassification risk through an iterative procedure. Despite the optimality, the original MCA has drawbacks that have limited its wide applicability in practice. That is, the MCA tends to be sensitive to outliers, cannot effectively handle nonlinearities in datasets, and suffers from having high computational complexity. To address these limitations, we propose an improved solution, named regularized maximin correlation approach (R-MCA). We first reformulate MCA as a quadratically constrained linear programming (QCLP) problem, incorporate regularization by introducing slack variables in the primal problem of the QCLP, and derive the corresponding Lagrangian dual. The dual formulation enables us to apply the kernel trick to R-MCA so that it can better handle nonlinearities. Our experimental results demonstrate that the regularization and kernelization make the proposed R-MCA more robust and accurate for various classification tasks than the original MCA. Furthermore, when the data size or dimensionality grows, R-MCA runs substantially faster by solving either the primal or dual (whichever has a smaller variable dimension) of the QCLP.Comment: Submitted to IEEE Acces

Parameter Selection and Pre-Conditioning for a Graph Form Solver

In a recent paper, Parikh and Boyd describe a method for solving a convex optimization problem, where each iteration involves evaluating a proximal operator and projection onto a subspace. In this paper we address the critical practical issues of how to select the proximal parameter in each iteration, and how to scale the original problem variables, so as the achieve reliable practical performance. The resulting method has been implemented as an open-source software package called POGS (Proximal Graph Solver), that targets multi-core and GPU-based systems, and has been tested on a wide variety of practical problems. Numerical results show that POGS can solve very large problems (with, say, more than a billion coefficients in the data), to modest accuracy in a few tens of seconds. As just one example, a radiation treatment planning problem with around 100 million coefficients in the data can be solved in a few seconds, as compared to around one hour with an interior-point method.Comment: 28 pages, 1 figure, 1 open source implementatio

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