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

Efficient Semidefinite Spectral Clustering via Lagrange Duality

We propose an efficient approach to semidefinite spectral clustering (SSC), which addresses the Frobenius normalization with the positive semidefinite (p.s.d.) constraint for spectral clustering. Compared with the original Frobenius norm approximation based algorithm, the proposed algorithm can more accurately find the closest doubly stochastic approximation to the affinity matrix by considering the p.s.d. constraint. In this paper, SSC is formulated as a semidefinite programming (SDP) problem. In order to solve the high computational complexity of SDP, we present a dual algorithm based on the Lagrange dual formalization. Two versions of the proposed algorithm are proffered: one with less memory usage and the other with faster convergence rate. The proposed algorithm has much lower time complexity than that of the standard interior-point based SDP solvers. Experimental results on both UCI data sets and real-world image data sets demonstrate that 1) compared with the state-of-the-art spectral clustering methods, the proposed algorithm achieves better clustering performance; and 2) our algorithm is much more efficient and can solve larger-scale SSC problems than those standard interior-point SDP solvers.Comment: 13 page

Synergy-Based Hand Pose Sensing: Optimal Glove Design

In this paper we study the problem of improving human hand pose sensing device performance by exploiting the knowledge on how humans most frequently use their hands in grasping tasks. In a companion paper we studied the problem of maximizing the reconstruction accuracy of the hand pose from partial and noisy data provided by any given pose sensing device (a sensorized "glove") taking into account statistical a priori information. In this paper we consider the dual problem of how to design pose sensing devices, i.e. how and where to place sensors on a glove, to get maximum information about the actual hand posture. We study the continuous case, whereas individual sensing elements in the glove measure a linear combination of joint angles, the discrete case, whereas each measure corresponds to a single joint angle, and the most general hybrid case, whereas both continuous and discrete sensing elements are available. The objective is to provide, for given a priori information and fixed number of measurements, the optimal design minimizing in average the reconstruction error. Solutions relying on the geometrical synergy definition as well as gradient flow-based techniques are provided. Simulations of reconstruction performance show the effectiveness of the proposed optimal design.Comment: Submitted to International Journal of Robotics Research 201