This paper presents a polynomial algorithm for learning mixtures of logconcave distributions in ℝn in the presence of malicious noise. That is, each sample is corrupted with some small probability, being replaced by a point about which we can make no assumptions. A key element of the algorithm is Robust Principle Components Analysis (PCA), which is less susceptible to corruption by noisy points. While noise may cause standard PCA to collapse well-separated mixture components so that they are indistinguishable, Robust PCA preserves the distance between some of the components, making a partition possible. It then recurses on each half of the mixture until every component is isolated. The success of this algorithm requires only a O*(log n) factor increase in the required separation between components of the mixture compared to the noiseless case.

We present an extension of Principal Component Analysis (PCA) and a new algorithm for clustering points in $\R^n$ based on it. The key property of the algorithm is that it is affine-invariant. When the input is a sample from a mixture of two arbitrary Gaussians, the algorithm correctly classifies the sample assuming only that the two components are separable by a hyperplane, i.e., there exists a halfspace that contains most of one Gaussian and almost none of the other in probability mass. This is nearly the best possible, improving known results substantially. For k≫2 components, the algorithm requires only that there be some (k-1)-dimensional subspace in which the ``overlap'' in every direction is small. Our main tools are isotropic transformation, spectral projection and a simple reweighting technique. We call this combination isotropic PCA.

A cascade face detector uses a sequence of node classifiers to distinguish faces from non-faces. This paper presents a new approach to design node classifiers in the cascade detector. Previous methods used machine learning algorithms that simultaneously select features and form ensemble classifiers. We argue that if these two parts are decoupled, we have the freedom to design a classifier that explicitly addresses the difficulties caused by the asymmetric learning goal. There are three contributions in this paper. The first is a categorization of asymmetries in the learning goal, and why they make face detection hard. The second is the Forward Feature Selection (FFS) algorithm and a fast pre- omputing strategy for AdaBoost. FFS and the fast AdaBoost can reduce the training time by approximately 100 and 50 times, in comparison to a naive implementation of the AdaBoost feature selection method. The last contribution is Linear Asymmetric Classifier (LAC), a classifier that explicitly handles the asymmetric learning goal as a well-defined constrained optimization problem. We demonstrated experimentally that LAC results in improved ensemble classifier performance.