Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations

Low rank matrix approximation is an important tool in machine learning. Given a data matrix, low rank approximation helps to find factors, patterns and provides concise representations for the data. Research on low rank approximation usually focus on real matrices. However, in many applications data are binary (categorical) rather than continuous. This leads to the problem of low rank approximation of binary matrix. Here we are given a $d \times n$ binary matrix $A$ and a small integer $k$. The goal is to find two binary matrices $U$ and $V$ of sizes $d \times k$ and $k \times n$ respectively, so that the Frobenius norm of $A - U V$ is minimized. There are two models of this problem, depending on the definition of the dot product of binary vectors: The $\mathrm{GF}(2)$ model and the Boolean semiring model. Unlike low rank approximation of real matrix which can be efficiently solved by Singular Value Decomposition, approximation of binary matrix is $NP$-hard even for $k=1$. In this paper, we consider the problem of Column Subset Selection (CSS), in which one low rank matrix must be formed by $k$ columns of the data matrix. We characterize the approximation ratio of CSS for binary matrices. For $GF(2)$ model, we show the approximation ratio of CSS is bounded by $\frac{k}{2}+1+\frac{k}{2(2^k-1)}$ and this bound is asymptotically tight. For Boolean model, it turns out that CSS is no longer sufficient to obtain a bound. We then develop a Generalized CSS (GCSS) procedure in which the columns of one low rank matrix are generated from Boolean formulas operating bitwise on columns of the data matrix. We show the approximation ratio of GCSS is bounded by $2^{k-1}+1$, and the exponential dependency on $k$ is inherent.Comment: 38 page

Daily partition of urinary nitrogen and nitrogen balance during treatment of protein-depleted infants

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A statistical-mechanical explanation of dark matter halo properties

Cosmological N-body simulations have revealed many empirical relationships of dark matter halos, yet the physical origin of these halo properties still remains unclear. On the other hand, the attempts to establish the statistical mechanics for self-gravitating systems have encountered many formal difficulties, and little progress has been made for about fifty years. The aim of this work is to strengthen the validity of the statistical-mechanical approach we have proposed previously to explain the dark matter halo properties. By introducing an effective pressure instead of the radial pressure to construct the specific entropy, we use the entropy principle and proceed in a similar way as previously to obtain an entropy stationary equation. An equation of state for equilibrated dark halos is derived from this entropy stationary equation, by which the dark halo density profiles with finite mass can be obtained. We also derive the anisotropy parameter and pseudo-phase-space density profile. All these predictions agree well with numerical simulations in the outer regions of dark halos. Our work provides further support to the idea that statistical mechanics for self-gravitating systems is a viable tool for investigation.Comment: 5 pages, 4 figures, Accepted by A&

Modelling the WMAP large-angle anomalies as an effect of a local density inhomogeneity

We investigate large-angle scale temperature anisotropy in the Cosmic Microwave Background (CMB) with the Wilkinson Microwave Anisotropy Probe (WMAP) data and model the large-angle anomalies as the effect of the CMB quadrupole anisotropies caused by the local density inhomogeneities. The quadrupole caused by the local density inhomogeneities is different from the special relativity kinematic quadrupole. If the observer inhabits a strong inhomogeneous region, the local quadrupole should not be neglected. We calculate such local quadrupole under the assumption that there is a huge density fluctuation field in direction $(284^{\circ},74^{\circ})$, where the density fluctuation is $10^{-3}$, and its center is $\sim 112h^{-1} \rm {Mpc}$ away from us. After removing such mock signals from WMAP data, the power in quadrupole, $C_2$, increases from the range $(200\sim260\mu \rm{K^2})$ to $\sim1000\mu \rm{K^2}$. The quantity S, which is used to estimate the alignment between the quadrupole and the octopole, decreases from $(0.7\sim0.74)$ to $(0.31\sim0.37)$, while the model predict that $C_2=1071.5\mu \rm{K^2}$, $S=0.412$. So our local density inhomogeneity model can, in part, explain the WMAP low-$\ell$ anomalies.Comment: 7 pages, 5 figures, accepted by RA

Observational Constraints on First-Star Nucleosynthesis. I. Evidence for Multiple Progenitors of CEMP-no Stars

We investigate anew the distribution of absolute carbon abundance, $A$(C) $= \log\,\epsilon$(C), for carbon-enhanced metal-poor (CEMP) stars in the halo of the Milky Way, based on high-resolution spectroscopic data for a total sample of 305 CEMP stars. The sample includes 147 CEMP-$s$ (and CEMP-r/s) stars, 127 CEMP-no stars, and 31 CEMP stars that are unclassified, based on the currently employed [Ba/Fe] criterion. We confirm previous claims that the distribution of $A$(C) for CEMP stars is (at least) bimodal, with newly determined peaks centered on $A$(C)$=7.96$ (the high-C region) and $A$(C)$=6.28$ (the low-C region). A very high fraction of CEMP-$s$ (and CEMP-r/s) stars belong to the high-C region, while the great majority of CEMP-no stars reside in the low-C region. However, there exists complexity in the morphology of the $A$(C)-[Fe/H] space for the CEMP-no stars, a first indication that more than one class of first-generation stellar progenitors may be required to account for their observed abundances. The two groups of CEMP-no stars we identify exhibit clearly different locations in the $A$(Na)-$A$(C) and $A$(Mg)-$A$(C) spaces, also suggesting multiple progenitors. The clear distinction in $A$(C) between the CEMP-$s$ (and CEMP-$r/s$) stars and the CEMP-no stars appears to be $as\ successful$, and$likely\ more\ astrophysically\ fundamental$, for the separation of these sub-classes as the previously recommended criterion based on [Ba/Fe] (and [Ba/Eu]) abundance ratios. This result opens the window for its application to present and future large-scale low- and medium-resolution spectroscopic surveys.Comment: 26pages, 7 figures, and 3 Tables ; Accepted for publication in ApJ; added more data and corrected minor inconsistencies existed in the compiled data of the previous studie

An attractor for dark matter structures

Cosmological simulations of dark matter structures have identified a set of universal profiles, and similar characteristics have been seen in non-cosmological simulations. It has therefore been speculated whether these profiles of collisionless systems relate to accretion and merger history, or if there is an attractor for the dark matter systems. Here we identify such a 1-dimensional attractor in the 3-dimensional space spanned by the 2 radial slopes of the density and velocity dispersion, and the velocity anisotropy. This attractor effectively removes one degree of freedom from the Jeans equation. It also allows us to speculate on a new fluid interpretation for the Jeans equation, with an effective polytropic index for the dark matter particles between 1/2 and 3/4. If this attractor solution holds for other collisionless structures, then it may hold the key to break the mass-anisotropy degeneracy, which presently prevents us from measuring the mass profiles in dwarf galaxies uniquely.Comment: 7 pages, 2 figures, comments welcom

Distribution Function in Center of Dark Matter Halo

N-body simulations of dark matter halos show that the density profiles of halos behave as $\rho(r)\propto r^{-\alpha(r)}$, where the density logarithmic slope $\alpha \simeq 1\sim1.5$ in the center and $\alpha \simeq 3\sim 4$ in the outer parts of halos. However, some observations are not in agreement with simulations in the very central region of halos. The simulations also show that velocity dispersion anisotropy parameter $\beta\approx 0$ in the inner part of the halo and the so called "pseudo phase-space density" $\rho/\sigma^3$ behaves as a power-law in radius $r$. With these results in mind, we study the distribution function and the pseudo phase-space density $\rho/\sigma^3$ of the center of dark matter halos and find that they are closely-related.Comment: 8 pages, accepted by Int. J. Mod. Phys. D, add a quotation and remove the appendi

A combined SEM, CV and EIS study of multi-layered porous ceramic reactors for flue gas purification

The effect of sintering temperature of 12-layered porous ceramic reactors (comprising 5 cells) was studied using scanning electron microscopy (SEM), cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS). The difference in microstructures of the reactors was evaluated by SEM. Additional information on the influence of sintering temperature on the properties of the reactors could be gained by the use of EIS. The present work has provided the first set of fundamental electrochemical data and their interpretation in terms of fabrication conditions, for the multi-layered porous ceramic reactors

Limits on the parameters of the equation of state for interacting dark energy

Under the assumption that cold dark matter and dark energy interact with each other through a small coupling term, $Q$, we constrain the parameter space of the equation of state $w$ of those dark energy fields whose variation of the field since last scattering do not exceed Planck's mass. We use three parameterizations of $w$ and two different expressions for $Q$. Our work extends previous ones.Comment: 18 pages, 11 figures, accepted for publication on Physics Letters