In press, SIAM J. Matrix Anal. Appl.Abstract. In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

Abstract. The techniques and analysis presented in this paper provide new methods to solve optimization problems posed on Riemannian manifolds. A new point of view is offered for the solution of constrained optimization problems. Some classical optimization techniques on Euclidean space are generalized to Riemannian manifolds. Several algorithms are presented and their convergence properties are analyzed employing the Riemannian structure of the manifold. Specifically, two apparently new algorithms, which can be thought of as Newton's method and the conjugate gradient method on Riemannian manifolds, are presented and shown to possess, respectively, quadratic and superlinear convergence. Examples of each method on certain Riemannian manifolds are given with the results of numerical experiments. Rayleigh's quotient defined on the sphere is one example. It is shown that Newton's method applied to this function converges cubically, and that the Rayleigh quotient iteration is an efficient approximation of Newton's method. The Riemannian version of the conjugate gradient method applied to this function gives a new algorithm for finding the eigenvectors corresponding to the extreme eigenvalues of a symmetric matrix. Another example arises from extremizing the function tr Θ T QΘN on the special orthogonal group. In a similar example, it is shown that Newton's method applied to the sum of the squares of the off-diagonal entries of a symmetric matrix converges cubically.

Cramér-Rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists. The frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail. The set of subspaces, called the Grassmann manifold, and the set of covariance (positive-definite Hermitian) matrices have no fixed coordinate system associated with them and do not possess a vector space structure, both of which are required for deriving classical Cramér-Rao bounds. Intrinsic versions of the Cramér-Rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators. In the example of covariance matrix estimation, closed-form expressions for both the intrinsic and flat bounds are derived and compared with the root-mean-square error (RMSE) of the sample covariance matrix (SCM) estimator for varying sample support. The accuracy bound on unbiased covariance matrix estimators is shown to be about (10 log 10) 1 2 dB, where is the matrix order. Remarkably, it is shown that from an intrinsic perspective, the SCM is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice. The RMSE of the standard method of estimating subspaces using the singular value decomposition (SVD) is compared with the intrinsic subspace Cramér-Rao bound derived in closed form by varying both the signal-to-noise ratio (SNR) of the unknown-dimensional subspace and the sample support. In the simplest case, the Cramér-Rao bound on subspace estimation accuracy is shown to be about (()) 1 2 1 2 SNR 1 2 rad for-dimensional subspaces. It is seen that the SVD-based method yields accuracies very close to the Cramér-Rao bound, establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace. The analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere, such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation.

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Halophytes are plants that are adapted to grow in saline soils, and have been widely studied for their physiological and molecular characteristics, but little is known about their associated microbiomes. Bacteria were isolated from the rhizosphere and as root endophytes of Salicornia rubra, Sarcocornia utahensis, and Allenrolfea occidentalis , three native Utah halophytes. A total of 41 independent isolates were identified by 16S rRNA gene sequencing analysis. Isolates were tested for maximum salt tolerance, and some were able to grow in the presence of up to 4 M NaCl. Pigmentation, Gram stain characteristics, optimal temperature for growth, and biofilm formation of each isolate aided in species identification. Some variation in the bacterial population was observed in samples collected at different times of the year, while most of the genera were present regardless of the sampling time. Halomonas , Bacillus , and Kushneria species were consistently isolated both from the soil and as endophytes from roots of all three plant species at all collection times. Non-culturable bacterial species were analyzed by Illumina DNA sequencing. The most commonly identified bacteria were from several phyla commonly found in soil or extreme environments: Acidobacteria, Actinobacteria, Bacteroidetes, Chloroflexi, and Gamma- and Delta-Proteobacteria. Isolates were tested for the ability to stimulate growth of alfalfa under saline conditions. This screening led to the identification of one Halomonas and one Bacillus isolate that, when used to inoculate young alfalfa seedlings, stimulate plant growth in the presence of 1% NaCl, a level that significantly inhibits growth of uninoculated plants. The same bacteria used in the inoculation were recovered from surface sterilized alfalfa roots, indicating the ability of the inoculum to become established as an endophyte. The results with these isolates have exciting promise for enhancing the growth of inoculated alfalfa in salty soil.

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Abstract-An important objective for analyzing realworld graphs is to achieve scalable performance on large, streaming graphs. A challenging and relevant example is the graph partition problem. As a combinatorial problem, graph partition is NP-hard, but existing relaxation methods provide reasonable approximate solutions that can be scaled for large graphs. Competitive benchmarks and challenges have proven to be an effective means to advance state-of-the-art performance and foster community collaboration. This paper describes a graph partition challenge with a baseline partition algorithm of sub-quadratic complexity. The algorithm employs rigorous Bayesian inferential methods based on a statistical model that captures characteristics of the real-world graphs. This strong foundation enables the algorithm to address limitations of well-known graph partition approaches such as modularity maximization. This paper describes various aspects of the challenge including: (1) the data sets and streaming graph generator, (2) the baseline partition algorithm with pseudocode, (3) an argument for the correctness of parallelizing the Bayesian inference, (4) different parallel computation strategies such as node-based parallelism and matrix-based parallelism, (5) evaluation metrics for partition correctness and computational requirements, (6) preliminary timing of a Python-based demonstration code and the open source C++ code, and (7) considerations for partitioning the graph in streaming fashion. Data sets and source code for the algorithm as well as metrics, with detailed documentation are available at GraphChallenge.org.