Convex optimizations for distance metric learning and pattern classification

The goal of machine learning is to build automated systems that can classify and recognize com-plex patterns in data. Not surprisingly, the representation of the data plays an important role in deter-mining what types of patterns can be automatically discovered. Many algorithms for machine learning assume that the data are represented as elements in a metric space. For example, in popular algorithms such as nearest-neighbor classification, vector quantization, and kernel density estimation, the metric distances between different examples provide a measure of their dissimilarity [1]. The performance of these algorithms can depend sensitively on the manner in which distances are measured. When data are represented as points in a multidimensional vector space, simple Euclidean dis-tances are often used to measure the dissimilarity between different examples. However, such dis-tances often do not yield reliable judgments; in addition, they cannot highlight the distinctive features that play a role in certain types of classification, but not others. For example, consider two schemes for clustering images of faces: one by age, one by gender. Images can be represented as points in a multidimensional vector space in many ways—for example, by enumerating their pixel values, or by computing color histograms. However the images are represented, different components of these feature vectors are likely to be relevant for clustering by age versus clustering by gender. Naturally