Optimal Estimation and Rank Detection for Sparse Spiked Covariance Matrices

This paper considers sparse spiked covariance matrix models in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We also establish the minimax rate under the spectral norm for estimating the principal subspace, the primary object of interest in principal component analysis. In addition, the optimal rate for the rank detection boundary is obtained. This result also resolves the gap in a recent paper by Berthet and Rigollet [1] where the special case of rank one is considered

Sparse PCA: Optimal rates and adaptive estimation

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish the optimal rates of convergence for estimating the principal subspace which are sharp with respect to all the parameters, thus providing a complete characterization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy and an application of Fano's lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible. We then introduce an adaptive procedure for estimating the principal subspace which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construction is a reduction scheme which reduces the sparse PCA problem to a high-dimensional multivariate regression problem. This method is potentially also useful for other related problems.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1178 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Register automata with linear arithmetic

We propose a novel automata model over the alphabet of rational numbers, which we call register automata over the rationals (RA-Q). It reads a sequence of rational numbers and outputs another rational number. RA-Q is an extension of the well-known register automata (RA) over infinite alphabets, which are finite automata equipped with a finite number of registers/variables for storing values. Like in the standard RA, the RA-Q model allows both equality and ordering tests between values. It, moreover, allows to perform linear arithmetic between certain variables. The model is quite expressive: in addition to the standard RA, it also generalizes other well-known models such as affine programs and arithmetic circuits. The main feature of RA-Q is that despite the use of linear arithmetic, the so-called invariant problem---a generalization of the standard non-emptiness problem---is decidable. We also investigate other natural decision problems, namely, commutativity, equivalence, and reachability. For deterministic RA-Q, commutativity and equivalence are polynomial-time inter-reducible with the invariant problem

Development of first generation aerospace NiMH cells

Gates Aerospace Batteries in conjunction with Gates Energy Products (GEP) has been developing NiMH technology for aerospace use since 1990. GEP undertook the development of NiMH technology for commercial cell applications in 1987. This program focused on wound cell technology for replacement of current NiCd technology. As an off shoot of this program small, wound cells were used to evaluate initial design options for aerospace prismatic cell designs. Early in 1991, the first aerospace prismatic cell designs were built in a 6 Ah cell configuration. These cells were used to initially characterize performance in prismatic configurations and begin early life cycle testing. Soon after the 6 Ah cells were on test, several 22 Ah cells were built to test other options. The results of testing of these cells were used to identify potential problem areas for long lived cells and develop solutions to those problems. Following these two cell builds, a set of 7 Ah cells was built to evaluate improvements to the technology. To date results from these tests are very promising. Cycle lives in excess of 2,200 LEO cycles at 50 percent DoD were achieved with cells continuing on test. Results from these cell tests are discussed and data presented to demonstrate feasibility of this technology for aerospace programs

The development of nickel-metal hydride technology for use in aerospace applications

The nickel metal hydride technology for battery application is relatively immature even though this technology was made widely known by Philips' scientists as long ago as 1970. Recently, because of the international environmental regulatory pressures being placed on cadmium in the workplace and in disposal practices, battery companies have initiated extensive development programs to make this technology a viable commercial operation. These hydrides do not pose a toxilogical threat as does cadmium. Also, they provide a higher energy density and specific energy when compared to the other nickel based battery technologies. For these reasons, the nickel metal hydride electrochemisty is being evaluated as the next power source for varied applications such as laptop computers, cellular telephones, electric vehicles, and satellites. A parallel development effort is under way to look at aerospace applications for nickel metal hydride cells. This effort is focused on life testing of small wound cells of the commercial type to validate design options and development of prismatic design cells for aerospace applications