Dimensionality Reduction for k-Means Clustering and Low Rank Approximation

We show how to approximate a data matrix $\mathbf{A}$ with a much smaller sketch $\mathbf{\tilde A}$ that can be used to solve a general class of constrained k-rank approximation problems to within $(1+\epsilon)$ error. Importantly, this class of problems includes $k$-means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just $O(k)$ dimensions, our methods generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For $k$-means dimensionality reduction, we provide $(1+\epsilon)$ relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only `cover' a good subspace for \bv{A}, but can be used directly to compute this subspace. Finally, for $k$-means clustering, we show how to achieve a $(9+\epsilon)$ approximation by Johnson-Lindenstrauss projecting data points to just $O(\log k/\epsilon^2)$ dimensions. This gives the first result that leverages the specific structure of $k$-means to achieve dimension independent of input size and sublinear in $k$

The alchemy of ideas

This article presents an assessment of the power of ideas and their role in initiating change and progress. The enormous potential cascade effect is illustrated by examining the movement of Modernism in the arts. Next, the immense scope and capabilities of the modern scientific endeavor—with robotic space exploration at the scale of 10⁹ meters at one extreme and the wonders of nanoscience at the scale of 10⁻⁹ m at the other—are examined. The attitudes and philosophies of neurological surgery are related to those involved in the Modernist movement and placed on the defined scale of contemporary scientific activity

Glassy phases and driven response of the phase-field-crystal model with random pinning

We study the structural correlations and the nonlinear response to a driving force of a two-dimensional phase-field-crystal model with random pinning. The model provides an effective continuous description of lattice systems in the presence of disordered external pinning centers, allowing for both elastic and plastic deformations. We find that the phase-field crystal with disorder assumes an amorphous glassy ground state, with only short-ranged positional and orientational correlations even in the limit of weak disorder. Under increasing driving force, the pinned amorphous-glass phase evolves into a moving plastic-flow phase and then finally a moving smectic phase. The transverse response of the moving smectic phase shows a vanishing transverse critical force for increasing system sizes

Grain boundary motion in layered phases

We study the motion of a grain boundary that separates two sets of mutually perpendicular rolls in Rayleigh-B\'enard convection above onset. The problem is treated either analytically from the corresponding amplitude equations, or numerically by solving the Swift-Hohenberg equation. We find that if the rolls are curved by a slow transversal modulation, a net translation of the boundary follows. We show analytically that although this motion is a nonlinear effect, it occurs in a time scale much shorter than that of the linear relaxation of the curved rolls. The total distance traveled by the boundary scales as $\epsilon^{-1/2}$, where $\epsilon$ is the reduced Rayleigh number. We obtain analytical expressions for the relaxation rate of the modulation and for the time dependent traveling velocity of the boundary, and especially their dependence on wavenumber. The results agree well with direct numerical solutions of the Swift-Hohenberg equation. We finally discuss the implications of our results on the coarsening rate of an ensemble of differently oriented domains in which grain boundary motion through curved rolls is the dominant coarsening mechanism.Comment: 16 pages, 5 figure

Generational research: between historical and sociological imaginations

This paper reflects on Julia Brannen’s contribution to the development of theory and methods for intergenerational research. The discussion is contextualised within a contemporary ‘turn to time’ within sociology, involving tensions and synergies between sociological and historical imagination. These questions are informed by a juxtaposition of Brannen’s four-generation study of family change and social historian Angela Davis’s exploration women and the family in England between 1945 and 2000. These two studies give rise to complementary findings, yet have distinctive orientations towards the status and treatment of sources, the role of geography in research design and limits of generalisatio

Sharp interface limits of phase-field models

The use of continuum phase-field models to describe the motion of well-defined interfaces is discussed for a class of phenomena, that includes order/disorder transitions, spinodal decomposition and Ostwald ripening, dendritic growth, and the solidification of eutectic alloys. The projection operator method is used to extract the ``sharp interface limit'' from phase field models which have interfaces that are diffuse on a length scale $\xi$. In particular,phase-field equations are mapped onto sharp interface equations in the limits $\xi \kappa \ll 1$ and $\xi v/D \ll 1$, where $\kappa$ and $v$ are respectively the interface curvature and velocity and $D$ is the diffusion constant in the bulk. The calculations provide one general set of sharp interface equations that incorporate the Gibbs-Thomson condition, the Allen-Cahn equation and the Kardar-Parisi-Zhang equation.Comment: 17 pages, 9 figure

Microscopic theory of network glasses

A molecular theory of the glass transition of network forming liquids is developed using a combination of self-consistent phonon and liquid state approaches. Both the dynamical transition and the entropy crisis characteristic of random first order transitions are mapped out as a function of the degree of bonding and the density. Using a scaling relation for a soft-core model to crudely translate the densities into temperatures, the theory predicts that the ratio of the dynamical transition temperature to the laboratory transition temperature rises as the degree of bonding increases, while the Kauzmann temperature falls relative to the laboratory transition. These results indicate why highly coordinated liquids should be "strong" while van der Waals liquids without coordination are "fragile".Comment: slightly revised version that has been accepted for publication in Phys. Rev. Let

Temperature dependence of the diffuse scattering fine structure in equiatomic CuAu

The temperature dependence of the diffuse scattering fine structure from disordered equiatomic CuAu was studied using {\it in situ} x-ray scattering. In contrast to Cu$_3$Au the diffuse peak splitting in CuAu was found to be relatively insensitive to temperature. Consequently, no evidence for a divergence of the antiphase length-scale at the transition temperature was found. At all temperatures studied the peak splitting is smaller than the value corresponding to the CuAuII modulated phase. An extended Ginzburg-Landau approach is used to explain the temperature dependence of the diffuse peak profiles in the ordering and modulation directions. The estimated mean-field instability point is considerably lower than is the case for Cu$_3$Au.Comment: 4 pages, 5 figure

Optical interconnect with densely integrated plasmonic modulator and germanium photodetector arrays

We demonstrate the first chip-to-chip interconnect utilizing a densely integrated plasmonic Mach-Zehnder modulator array operating at 3 x 10 Gbit/s. A multicore fiber provides a compact optical interface, while the receiver consists of germanium photodetectors

Flame propagation in random media

We introduce a phase-field model to describe the dynamics of a self-sustaining propagating combustion front within a medium of randomly distributed reactants. Numerical simulations of this model show that a flame front exists for reactant concentration $c > c^* > 0$, while its vanishing at $c^*$ is consistent with mean-field percolation theory. For $c > c^*$, we find that the interface associated with the diffuse combustion zone exhibits kinetic roughening characteristic of the Kardar-Parisi-Zhang equation.Comment: 4, LR541