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

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.

Thermodynamics and the Global Optimization of Lennard-Jones clusters

Theoretical design of global optimization algorithms can profitably utilize recent statistical mechanical treatments of potential energy surfaces (PES's). Here we analyze the basin-hopping algorithm to explain its success in locating the global minima of Lennard-Jones (LJ) clusters, even those such as \LJ{38} for which the PES has a multiple-funnel topography, where trapping in local minima with different morphologies is expected. We find that a key factor in overcoming trapping is the transformation applied to the PES which broadens the thermodynamic transitions. The global minimum then has a significant probability of occupation at temperatures where the free energy barriers between funnels are surmountable.Comment: 13 pages, 13 figures, revte

The double-funnel energy landscape of the 38-atom Lennard-Jones cluster

The 38-atom Lennard-Jones cluster has a paradigmatic double-funnel energy landscape. One funnel ends in the global minimum, a face-centred-cubic (fcc) truncated octahedron. At the bottom of the other funnel is the second lowest energy minimum which is an incomplete Mackay icosahedron. We characterize the energy landscape in two ways. Firstly, from a large sample of minima and transition states we construct a disconnectivity tree showing which minima are connected below certain energy thresholds. Secondly we compute the free energy as a function of a bond-order parameter. The free energy profile has two minima, one which corresponds to the fcc funnel and the other which at low temperature corresponds to the icosahedral funnel and at higher temperatures to the liquid-like state. These two approaches show that the greater width of the icosahedral funnel, and the greater structural similarity between the icosahedral structures and those associated with the liquid-like state, are the cause of the smaller free energy barrier for entering the icosahedral funnel from the liquid-like state and therefore of the cluster's preferential entry into this funnel on relaxation down the energy landscape. Furthermore, the large free energy barrier between the fcc and icosahedral funnels, which is energetic in origin, causes the cluster to be trapped in one of the funnels at low temperature. These results explain in detail the link between the double-funnel energy landscape and the difficulty of global optimization for this cluster.Comment: 12 pages, 11 figures, revte

Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams

In the communication problem $\mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y \in\{0,1\}^n$ with the promise that $x\neq y$. The last player to receive a message must output an index $i$ such that $x_i\neq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $\Theta(\min\{n,\log(1/\delta)\log^2(\frac n{\log(1/\delta)})\})$ for failure probability $\delta$. Our lower bound holds even if promised $\mathop{support}(y)\subset \mathop{support}(x)$. As a corollary, we obtain optimal lower bounds for $\ell_p$-sampling in strict turnstile streams for $0\le p < 2$, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised $x\in\{0,1\}^n$ at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm $\mathcal A$ solving sampling problems in turnstile streams in low memory can be used to encode subsets of $[n]$ of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to $\mathcal A$ throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoder's interactions with $\mathcal A$, which is loosely motivated by techniques in differential privacy. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with $\mathcal A$ adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from $\mathcal A$ in the reduction.Comment: merge of arXiv:1703.08139 and of work of Kapralov, Woodruff, and Yahyazade

Neutrino Oscillations as a Probe of Dark Energy

We consider a class of theories in which neutrino masses depend significantly on environment, as a result of interactions with the dark sector. Such theories of mass varying neutrinos (MaVaNs) were recently introduced to explain the origin of the cosmological dark energy density and why its magnitude is apparently coincidental with that of neutrino mass splittings. In this Letter we argue that in such theories neutrinos can exhibit different masses in matter and in vacuum, dramatically affecting neutrino oscillations. Both long and short baseline experiments are essential to test for these interactions. As an example of modifications to the standard picture, we consider simple models which may simultaneously account for the LSND anomaly, KamLAND, K2K and studies of solar and atmospheric neutrinos, while providing motivation to continue to search for neutrino oscillations in short baseline experiments such as BooNE.Comment: 5 pages, 1 figure, refs added, additional data considered, minor change in conclusions about LSN

Anomalous coupling between topological defects and curvature

We investigate a counterintuitive geometric interaction between defects and curvature in thin layers of superfluids, superconductors and liquid crystals deposited on curved surfaces. Each defect feels a geometric potential whose functional form is determined only by the shape of the surface, but whose sign and strength depend on the transformation properties of the order parameter. For superfluids and superconductors, the strength of this interaction is proportional to the square of the charge and causes all defects to be repelled (attracted) by regions of positive (negative) Gaussian curvature. For liquid crystals in the one elastic constant approximation, charges between 0 and $4\pi$ are attracted by regions of positive curvature while all other charges are repelled.Comment: 5 pages, 4 figures, minor changes, accepted for publication in Phys. Rev. Let

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