Optimal approximate matrix product in terms of stable rank

We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows. Here $\tilde{r}$ is the maximum stable rank, i.e. squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [MZ11, KVZ14], and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future. Our main theorem, via connections with spectral error matrix multiplication shown in prior work, implies quantitative improvements for approximate least squares regression and low rank approximation. Our main result has also already been applied to improve dimensionality reduction guarantees for $k$-means clustering [CEMMP14], and implies new results for nonparametric regression [YPW15]. We also separately point out that the proof of the "BSS" deterministic row-sampling result of [BSS12] can be modified to show that for any matrices $A, B$ of stable rank at most $\tilde{r}$, one can achieve the spectral norm guarantee for approximate matrix multiplication of $A^T B$ by deterministically sampling $O(\tilde{r}/\varepsilon^2)$ rows that can be found in polynomial time. The original result of [BSS12] was for rank instead of stable rank. Our observation leads to a stronger version of a main theorem of [KMST10].Comment: v3: minor edits; v2: fixed one step in proof of Theorem 9 which was wrong by a constant factor (see the new Lemma 5 and its use; final theorem unaffected

Exoskeleton master controller with force-reflecting telepresence

A thorough understanding of the requirements for successful master-slave robotic systems is becoming increasingly desirable. Such systems can aid in the accomplishment of tasks that are hazardous or inaccessible to humans. Although a history of use has proven master-slave systems to be viable, system requirements and the impact of specifications on the human factors side of system performance are not well known. In support of the next phase of teleoperation research being conducted at the Armstrong Research Laboratory, a force-reflecting, seven degree of freedom exoskeleton for master-slave teleoperation has been concepted, and is presently being developed. The exoskeleton has a unique kinematic structure that complements the structure of the human arm. It provides a natural means for teleoperating a dexterous, possibly redundant manipulator. It allows ease of use without operator fatigue and faithfully follows human arm and wrist motions. Reflected forces and moments are remotely transmitted to the operator hand grip using a cable transmission scheme. This paper presents the exoskeleton concept and development results to date. Conceptual design, hardware, algorithms, computer architecture, and software are covered

Free Energies of Isolated 5- and 7-fold Disclinations in Hexatic Membranes

We examine the shapes and energies of 5- and 7-fold disclinations in low-temperature hexatic membranes. These defects buckle at different values of the ratio of the bending rigidity, $\kappa$, to the hexatic stiffness constant, $K_A$, suggesting {\em two} distinct Kosterlitz-Thouless defect proliferation temperatures. Seven-fold disclinations are studied in detail numerically for arbitrary $\kappa/K_A$. We argue that thermal fluctuations always drive $\kappa/K_A$ into an ``unbuckled'' regime at long wavelengths, so that disclinations should, in fact, proliferate at the {\em same} critical temperature. We show analytically that both types of defects have power law shapes with continuously variable exponents in the ``unbuckled'' regime. Thermal fluctuations then lock in specific power laws at long wavelengths, which we calculate for 5- and 7-fold defects at low temperatures.Comment: LaTeX format. 17 pages. To appear in Phys. Rev.

Localization transitions in non-Hermitian quantum mechanics

We study the localization transitions which arise in both one and two dimensions when quantum mechanical particles described by a random Schr\"odinger equation are subjected to a constant imaginary vector potential. A path-integral formulation relates the transition to flux lines depinned from columnar defects by a transverse magnetic field in superconductors. The theory predicts that the transverse Meissner effect is accompanied by stretched exponential relaxation of the field into the bulk and a diverging penetration depth at the transition.Comment: 4 pages (latex) with 3 figures (epsf) embedded in the text using the style file epsf.st

The Emergence of Commercial Scale Offshore Wind: Progress Made and Challenges Ahead

This Article examines the offshore wind development process from leasing and permitting to electric power supply and interconnection. Willing developers may divide the process into three discrete, but not necessarily sequential, endeavors. First, the developer must secure a viable purchaser or market for the output. “Offshore wind energy” is a more complex commercial product than one might envision—it includes the actual electric energy produced, the electric generating capacity that is available to serve load, and both the environmental and clean energy attributes of wind energy. The environmental and clean energy attributes may have an economic and regulatory value separate from, or in addition to, the value of the electric energy itself. These separate complexities give rise to several questions: What are the available markets for actual offshore wind energy? How does a developer find a buyer (off-taker) for the offshore wind electric output? How are the markets for the actual energy and the environmental attributes, normally embodied in a “renewable energy certificate” (REC), combined or otherwise related? How much control can individual states exercise over the decisions of an individual utility or other purchasers of offshore wind energy and RECs (or each of them separately)? If the average cost to the developer of electric energy generation from offshore wind per kilowatt-hour (kWh) is substantially higher than the average cost of energy in the onshore markets, what features of state regulation or policy facilitate the sale? Second, the developer must secure, or acquire by sale or assignment, appropriate offshore sites for development of the physical resource. Most available offshore wind resources are located in the OCS and will be under federal control for leasing. Developers must secure OCS leases either through successful bids in the initial offering or through a later acquisition or assignment from winning bidders. Offshore wind development requires large areas within which to erect the number of turbines needed, as well as a gathering system of cables and substations, to collect and deliver the output of all the turbines via transmission lines to interconnections with the existing mainland grid. The developer also must obtain rights-of-way to lay cable for its gathering and transmission facilities—on the OCS and across state submerged lands and coastal areas. In the alternative, a new offshore wind transmission system may be built by a third party to connect with multiple wind farms and deliver energy to an onshore point of interconnection. These leasing and project configuration scenarios present many questions. If the offshore wind developer and the transmission facility developer are separate entities, how much coordination is required? What is the appropriate scope of environmental impact studies needed in connection with the OCS leasing process? What are the mechanics for acquiring the necessary property rights and leases between winning bidders and other interested developers? Third, the offshore wind developer, alone or with a third-party transmission developer, must be concerned about the interconnection of the offshore cable to the onshore transmission grid. Most onshore transmission and distribution grids were planned, constructed and operated on the assumption that electricity consumers on the coast are the end of the delivery line. While transmission grids are somewhat more robust at these isolated coastal locations—particularly when large nuclear and fossil generation exists at water’s edge—these more robust coastal grid systems are limited and may be neither geographically nor electrically proximate to offshore wind generation locations. With advances in turbine technology and the overall economics of offshore wind farm development most proposed commercial-scale projects are likely to have generation capacity in the hundreds of megawatts (MWs). Typically, interconnection of offshore wind and related transmission delivery facilities require not only reconfiguration and enlargement of the receiving onshore transmission grid to accept the input of such electric capacity at water’s edge, but also delivery to load centers that may be located a substantial distance inland. Owners of the onshore grid may not be the same as the utility purchaser or other off-taker of the offshore electric energy. The complexities of onshore interconnection raise vexing questions, such as: (i) how to reconfigure and enlarge the grid to interconnect with offshore generation, accept the energy output, and deliver to load centers; and (ii) who should bear the costs of that reconfiguration and enlargement. This Article is intended to provide a helpful roadmap or guidance for major issues in three principal areas—securing a viable purchaser, siting the offshore development farm, and onshore interconnection of the offshore cable. To date, most offshore wind development efforts in the United States occur off the Northeast and Mid-Atlantic coast. This Article highlights the emerging federal-state dynamic in the development of offshore wind generation and illuminates several key uncertainties developers face today