This paper studies the problem of estimating the covariance of a collection of vectors using only highly compressed measurements of each vector. An estimator based on back-projections of these compressive samples is proposed and analyzed. A distribution-free analysis shows that by observing just a single linear measurement of each vector, one can consistently estimate the covariance matrix, in both infinity and spectral norm, and this analysis leads to precise rates of convergence in both norms. Through information-theoretic techniques, lower bounds showing that this estimator is minimax-optimal for both infinity and spectral norm estimation problems are established. These results are also specialized to give matching upper and lower bounds for estimating the population covariance of a collection of Gaussian vectors, again in the compressive measurement model. The analysis conducted in this paper shows that the effective sample complexity for this problem is scaled by a factor of m2/d2, where m is the compression dimension and d is the ambient dimension. Applications to subspace learning (principal components analysis) and learning over distributed sensor networks are also discussed.

We consider the problem of clustering data points in high dimensions, i.e., when the number of data points may be much smaller than the number of dimensions. Specifically, we consider a Gaussian mixture model (GMM) with two non-spherical Gaussian components, where the clusters are distinguished by only a few relevant dimensions. The method we propose is a combination of a recent approach for learning parameters of a Gaussian mixture model and sparse linear discriminant analysis (LDA). In addition to cluster assignments, the method returns an estimate of the set of features relevant for clustering. Our results indicate that the sample complexity of clustering depends on the sparsity of the relevant feature set, while only scaling logarithmically with the ambient dimension. Further, we require much milder assumptions than existing work on clustering in high dimensions. In particular, we do not require spherical clusters nor necessitate mean separation along relevant dimensions.

While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends on the number of relevant dimensions and mean separation, and can be achieved by a simple computationally efficient procedure. Our results provide the first step of a theoretical basis for recent methods that combine feature selection and clustering.

Finding a person in a public place, such as in a library, conference hotel, or shopping mall, can be difficult. The difficulty arises from not knowing where the person may be at that time; even if known, navigating through an unfamiliar place may be frustrating. Maps and floor plans help in some occasions, but such maps may not be always handy. In a small scale poll, 80% of users responded that the ideal solution would be "to have an escort walk me to the desired person".

A growing number of mobile computing applications are centered around the user's location. The notion of location is broad, ranging from physical coordinates (latitude/longitude) to logical labels (like Starbucks, McDonalds). While extensive research has been performed in physical localization, there have been few attempts in recognizing logical locations. This paper argues that the increasing number of sensors on mobile phones presents new opportunities for logical localization. We postulate that ambient sound, light, and color in a place convey a photo-acoustic signature that can be sensed by the phone's camera and microphone. In-built accelerometers in some phones may also be useful in inferring broad classes of user-motion, often dictated by the nature of the place. By combining these optical, acoustic, and motion attributes, it may be feasible to construct an identifiable fingerprint for logical localization. Hence, users in adjacent stores can be separated logically, even when their physical positions are extremely close. We propose SurroundSense, a mobile phone based system that explores logical localization via ambience fingerprinting. Evaluation results from 51 different stores show that SurroundSense can achieve an average accuracy of 87% when all sensing modalities are employed. We believe this is an encouraging result, opening new possibilities in indoor localization.