Low-distortion Subspace Embeddings in Input-sparsity Time and Applications to Robust Linear Regression

Low-distortion embeddings are critical building blocks for developing random sampling and random projection algorithms for linear algebra problems. We show that, given a matrix $A \in \R^{n \times d}$ with $n \gg d$ and a $p \in [1, 2)$, with a constant probability, we can construct a low-distortion embedding matrix \Pi \in \R^{O(\poly(d)) \times n} that embeds \A_p, the $\ell_p$ subspace spanned by $A$'s columns, into (\R^{O(\poly(d))}, \| \cdot \|_p); the distortion of our embeddings is only O(\poly(d)), and we can compute $\Pi A$ in O(\nnz(A)) time, i.e., input-sparsity time. Our result generalizes the input-sparsity time $\ell_2$ subspace embedding by Clarkson and Woodruff [STOC'13]; and for completeness, we present a simpler and improved analysis of their construction for $\ell_2$. These input-sparsity time $\ell_p$ embeddings are optimal, up to constants, in terms of their running time; and the improved running time propagates to applications such as $(1\pm \epsilon)$-distortion $\ell_p$ subspace embedding and relative-error $\ell_p$ regression. For $\ell_2$, we show that a $(1+\epsilon)$-approximate solution to the $\ell_2$ regression problem specified by the matrix $A$ and a vector $b \in \R^n$ can be computed in O(\nnz(A) + d^3 \log(d/\epsilon) /\epsilon^2) time; and for $\ell_p$, via a subspace-preserving sampling procedure, we show that a $(1\pm \epsilon)$-distortion embedding of \A_p into \R^{O(\poly(d))} can be computed in O(\nnz(A) \cdot \log n) time, and we also show that a $(1+\epsilon)$-approximate solution to the $\ell_p$ regression problem $\min_{x \in \R^d} \|A x - b\|_p$ can be computed in O(\nnz(A) \cdot \log n + \poly(d) \log(1/\epsilon)/\epsilon^2) time. Moreover, we can improve the embedding dimension or equivalently the sample size to $O(d^{3+p/2} \log(1/\epsilon) / \epsilon^2)$ without increasing the complexity.Comment: 22 page

Maximum thickness of amorphous NiZr interlayers formed by a solid-state reaction technique

Formation of the equilibrium intermetallic compound NiZr in sputter deposited Ni/Zr diffusion couples is suppressed by the formation of a metastable amorphous NiZr alloy until a critical thickness of the amorphous NiZr interlayer is reached. The temperature dependence of this critical thickness is studied experimentally. A phenomenological model based on the premise of interfacial heterogeneous nucleation is proposed to understand the evolution of Ni/Zr diffusion couples

Automatic Classification of Text Databases through Query Probing

Many text databases on the web are "hidden" behind search interfaces, and their documents are only accessible through querying. Search engines typically ignore the contents of such search-only databases. Recently, Yahoo-like directories have started to manually organize these databases into categories that users can browse to find these valuable resources. We propose a novel strategy to automate the classification of search-only text databases. Our technique starts by training a rule-based document classifier, and then uses the classifier's rules to generate probing queries. The queries are sent to the text databases, which are then classified based on the number of matches that they produce for each query. We report some initial exploratory experiments that show that our approach is promising to automatically characterize the contents of text databases accessible on the web.Comment: 7 pages, 1 figur

Solid-state interdiffusion reactions in Ni/Ti and Ni/Zr multilayered thin films

We have performed a comparative transmission electron microscopy study of solid-state interdiffusion reactions in multilayered Ni/Zr and Ni/Ti thin films. The Ni-Zr reaction product was amorphous while the Ni-Ti reaction product was a simple intermetallic compound. Because thermodynamic and chemical properties of these two alloy systems are similar, we suggest kinetic origins for this difference in reaction product

Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules

For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an \mr{Spin}(2, 2n+2) dynamical symmetry which extends the manifest \mr{Spin}(2n+1) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight \mr{Spin}(2, 2n+2)-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight \mr{Spin}(2, 2n+2)-module.Comment: 27 pages, Refs. update

Simultaneous planar growth of amorphous and crystalline Ni silicides

We report a solid-state interdiffusion reaction induced by rapid thermal annealing and vacuum furnace annealing in evaporated Ni/Si bilayers. Upon heat treatment of a Ni film overlaid on a film of amorphous Si evaporated from a graphite crucible, amorphous and crystalline silicide layers grow uniformly side by side as revealed by cross-sectional transmission electron microscopy and backscattering spectrometry. This phenomenon contrasts with the silicide formation behavior previously observed in the Ni-Si system, and constitutes an interesting counterpart of the solid-state interdiffusion-induced amorphization in Ni/Zr thin-film diffusion couples. Carbon impurity contained in the amorphous Si film stabilizes the amorphous phase. Kinetic and thermodynamic factors that account for the experimental findings are discussed

Assessment of the Lunar Surface Layer and in Situ Materials to Sustain Construction-related Applications

Present and future technologies to facilitate lunar composition and resource assessment with applications to lunar surface construction are presented. We are particularly interested in the construction activity associated with lunar-based astronomy. We address, as an example, the use of ground-probing radar to help assess subsurface conditions at sites for observatories and other facilities