The electric
Tensor decompositions have proven to be effective in analyzing the structure of multidimensional data. However, most of these methods require a key parameter: the number of desired components. In the case of the CANDECOMP/PARAFAC decomposition (CPD), the ideal value for the number of components is known as the canonical rank and greatly affects the quality of the decomposition results. Existing methods use heuristics or Bayesian methods to estimate this value by repeatedly calculating the CPD, making them extremely computationally expensive. In this work, we propose
 NSFPAR ID:
 10544398
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Data Mining and Knowledge Discovery
 Volume:
 38
 Issue:
 6
 ISSN:
 13845810
 Format(s):
 Medium: X Size: p. 42174232
 Size(s):
 p. 42174232
 Sponsoring Org:
 National Science Foundation
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Abstract E 1 and magneticM 1 dipole responses of the nucleus$$N=Z$$ $N=Z$ Mg were investigated in an inelastic photon scattering experiment. The 13.0 MeV electrons, which were used to produce the unpolarised bremsstrahlung in the entrance channel of the$$^{24}$$ ${}^{24}$ Mg($$^{24}$$ ${}^{24}$ ) reaction, were delivered by the ELBE accelerator of the HelmholtzZentrum DresdenRossendorf. The collimated bremsstrahlung photons excited one$$\gamma ,\gamma ^{\prime }$$ $\gamma ,{\gamma}^{\prime}$ , four$$J^{\pi }=1^$$ ${J}^{\pi}={1}^{}$ , and six$$J^{\pi }=1^+$$ ${J}^{\pi}={1}^{+}$ states in$$J^{\pi }=2^+$$ ${J}^{\pi}={2}^{+}$ Mg. Deexcitation$$^{24}$$ ${}^{24}$ rays were detected using the four highpurity germanium detectors of the$$\gamma $$ $\gamma $ ELBE setup, which is dedicated to nuclear resonance fluorescence experiments. In the energy region up to 13.0 MeV a total$$\gamma $$ $\gamma $ is observed, but this$$B(M1)\uparrow = 2.7(3)~\mu _N^2$$ $B\left(M1\right)\uparrow =2.7\left(3\right)\phantom{\rule{0ex}{0ex}}{\mu}_{N}^{2}$ nucleus exhibits only marginal$$N=Z$$ $N=Z$E 1 strength of less than e$$\sum B(E1)\uparrow \le 0.61 \times 10^{3}$$ $\sum B\left(E1\right)\uparrow \le 0.61\times {10}^{3}$ fm$$^2 \, $$ ${}^{2}\phantom{\rule{0ex}{0ex}}$ . The$$^2$$ ${}^{2}$ branching ratios in combination with the expected results from the Alaga rules demonstrate that$$B(\varPi 1, 1^{\pi }_i \rightarrow 2^+_1)/B(\varPi 1, 1^{\pi }_i \rightarrow 0^+_{gs})$$ $B(\Pi 1,{1}_{i}^{\pi}\to {2}_{1}^{+})/B(\Pi 1,{1}_{i}^{\pi}\to {0}_{\mathrm{gs}}^{+})$K is a good approximative quantum number for Mg. The use of the known$$^{24}$$ ${}^{24}$ strength and the measured$$\rho ^2(E0, 0^+_2 \rightarrow 0^+_{gs})$$ ${\rho}^{2}(E0,{0}_{2}^{+}\to {0}_{\mathrm{gs}}^{+})$ branching ratio of the 10.712 MeV$$B(M1, 1^+ \rightarrow 0^+_2)/B(M1, 1^+ \rightarrow 0^+_{gs})$$ $B(M1,{1}^{+}\to {0}_{2}^{+})/B(M1,{1}^{+}\to {0}_{\mathrm{gs}}^{+})$ level allows, in a twostate mixing model, an extraction of the difference$$1^+$$ ${1}^{+}$ between the prolate groundstate structure and shapecoexisting superdeformed structure built upon the 6432keV$$\varDelta \beta _2^2$$ $\Delta {\beta}_{2}^{2}$ level.$$0^+_2$$ ${0}_{2}^{+}$ 
Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the highlevel model and the lens of submodularity in addressing this class of covering problems. 
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Abstract We study the sparsity of the solutions to systems of linear Diophantine equations with and without nonnegativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the
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Abstract In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying
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