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Journal articles on the topic 'Optimal embeddings'

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Published: 4 June 2021

Last updated: 12 February 2022

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1

BOZKURT, ILKER NADI, HAI HUANG, BRUCE MAGGS, ANDRÉA RICHA, and MAVERICK WOO. "Mutual Embeddings." Journal of Interconnection Networks 15, no. 01n02 (March 2015): 1550001. http://dx.doi.org/10.1142/s0219265915500012.

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This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

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2

Vernaeve, H. "OPTIMAL EMBEDDINGS OF DISTRIBUTIONS INTO ALGEBRAS." Proceedings of the Edinburgh Mathematical Society 46, no. 2 (June 2003): 373–78. http://dx.doi.org/10.1017/s0013091500001188.

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AbstractLet $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

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3

Cianchi, Andrea, and Luboš Pick. "Optimal Sobolev trace embeddings." Transactions of the American Mathematical Society 368, no. 12 (January 19, 2016): 8349–82. http://dx.doi.org/10.1090/tran/6606.

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4

Cianchi, Andrea, and Luboš Pick. "Optimal Gaussian Sobolev embeddings." Journal of Functional Analysis 256, no. 11 (June 2009): 3588–642. http://dx.doi.org/10.1016/j.jfa.2009.03.001.

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5

Kim, Sook-Yeon, Oh-Heum Kwon, and Kyung-Yong Chwa. "Embeddings of Butterflies into Hypermeshes." Parallel Processing Letters 08, no. 03 (September 1998): 337–50. http://dx.doi.org/10.1142/s0129626498000353.

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Hypermeshes have been given much attention as a versatile interconnection network of parallel computers. A hypermesh is obtained from a mesh by replacing each linear connection with a hyperedge. In this paper, we show how to embed a butterfly or multiple copies of a butterfly into a hypermesh. First, a butterfly B(s) of (s + 1)2s nodes is embedded into a 2s × X hypermesh where X = 2⌊ log 2 s ⌋+ 1. Second, the butterfly B(s) is embedded into a square hypermesh. Third, multiple copies of the butterfly B(s) are embedded into a hypermesh of variable aspect ratio. The efficiency of these embeddings is measured by alignment cost, congestion, and expansion. The alignment cost of all of these embeddings is optimal. The congestion of the first and third embedding is optimal. The expansion of the first and third embedding is one if s = 2k - 1 for some integer k, otherwise, less than two. The expansion of the second embedding is 2 + ∊ (s) where ∊(s) = (2 log (s + 1) + 2)/(s + 1).

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6

CASSANI, DANIELE, BERNHARD RUF, and CRISTINA TARSI. "GROUP INVARIANCE AND POHOZAEV IDENTITY IN MOSER-TYPE INEQUALITIES." Communications in Contemporary Mathematics 15, no. 02 (March 7, 2013): 1250054. http://dx.doi.org/10.1142/s021919971250054x.

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We study the so-called limiting Sobolev cases for embeddings of the spaces [Formula: see text], where Ω ⊂ ℝn is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler–Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the corresponding embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a Pohozaev-type identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.

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7

Parini, Enea, Bernhard Ruf, and Cristina Tarsi. "Limiting Sobolev inequalities and the 1-biharmonic operator." Advances in Nonlinear Analysis 3, S1 (September 1, 2014): s19—s36. http://dx.doi.org/10.1515/anona-2014-0007.

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AbstractIn this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L1. We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L1-data by Maz'ya (N ≥ 3) and Brezis–Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L1, for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1-biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber–Krahn type inequalities.

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8

Maclachlan, C. "Optimal embeddings in quaternion algebras." Journal of Number Theory 128, no. 10 (October 2008): 2852–60. http://dx.doi.org/10.1016/j.jnt.2007.12.007.

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9

Arenas, Manuel, Luis Arenas-Carmona, and Jaime Contreras. "On optimal embeddings and trees." Journal of Number Theory 193 (December 2018): 91–117. http://dx.doi.org/10.1016/j.jnt.2018.04.004.

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10

Chandran, Nishanth, Ryan Moriarty, Rafail Ostrovsky, Omkant Pandey, Mohammad Ali Safari, and Amit Sahai. "Improved algorithms for optimal embeddings." ACM Transactions on Algorithms 4, no. 4 (August 2008): 1–14. http://dx.doi.org/10.1145/1383369.1383376.

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11

Gogatishvili, Amiran, Júlio S. Neves, and Bohumír Opic. "Optimal embeddings and compact embeddings of Bessel-potential-type spaces." Mathematische Zeitschrift 262, no. 3 (July 15, 2008): 645–82. http://dx.doi.org/10.1007/s00209-008-0395-5.

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12

Hind, R. "Some optimal embeddings of symplectic ellipsoids." Journal of Topology 8, no. 3 (July 2, 2015): 871–83. http://dx.doi.org/10.1112/jtopol/jtv016.

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13

Gogatishvili, Amiran, and Vladimir I. Ovchinnikov. "Interpolation orbits and optimal Sobolev's embeddings." Journal of Functional Analysis 253, no. 1 (December 2007): 1–17. http://dx.doi.org/10.1016/j.jfa.2007.08.008.

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14

Ahmed, Irshaad, and Georgi Eremiev Karadzhov. "Optimal embeddings of generalized homogeneous Sobolev spaces." Colloquium Mathematicum 123, no. 1 (2011): 1–20. http://dx.doi.org/10.4064/cm123-1-1.

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15

Vybíral, J. "Optimal Sobolev embeddings on $\mathbb{R}^n$." Publicacions Matemàtiques 51 (January 1, 2007): 17–44. http://dx.doi.org/10.5565/publmat_51107_02.

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16

Caha, R., and V. Koubek. "Optimal embeddings of generalized ladders into hypercubes." Discrete Mathematics 233, no. 1-3 (April 2001): 65–83. http://dx.doi.org/10.1016/s0012-365x(00)00227-2.

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17

Cianchi, Andrea, and Vit Musil. "Optimal domain spaces in Orlicz-Sobolev embeddings." Indiana University Mathematics Journal 68, no. 3 (2019): 925–66. http://dx.doi.org/10.1512/iumj.2019.68.7649.

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18

Boman, Erik G., Stephen Guattery, and Bruce Hendrickson. "Optimal Embeddings and Eigenvalues in Support Theory." SIAM Journal on Matrix Analysis and Applications 29, no. 2 (January 2007): 596–605. http://dx.doi.org/10.1137/050642174.

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19

Bläsius, Thomas, Tobias Friedrich, Maximilian Katzmann, and Anton Krohmer. "Hyperbolic Embeddings for Near-Optimal Greedy Routing." ACM Journal of Experimental Algorithmics 25 (November 8, 2020): 1–18. http://dx.doi.org/10.1145/3381751.

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20

Debrouwere, Andreas, Hans Vernaeve, and Jasson Vindas. "Optimal embeddings of ultradistributions into differential algebras." Monatshefte für Mathematik 186, no. 3 (May 28, 2017): 407–38. http://dx.doi.org/10.1007/s00605-017-1066-6.

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21

Tsitsulin, Anton, Marina Munkhoeva, Davide Mottin, Panagiotis Karras, Ivan Oseledets, and Emmanuel Müller. "FREDE." Proceedings of the VLDB Endowment 14, no. 6 (February 2021): 1102–10. http://dx.doi.org/10.14778/3447689.3447713.

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Low-dimensional representations, or embeddings , of a graph's nodes facilitate several practical data science and data engineering tasks. As such embeddings rely, explicitly or implicitly, on a similarity measure among nodes, they require the computation of a quadratic similarity matrix, inducing a tradeoff between space complexity and embedding quality. To date, no graph embedding work combines (i) linear space complexity, (ii) a nonlinear transform as its basis, and (iii) nontrivial quality guarantees. In this paper we introduce FREDE ( FREquent Directions Embedding ), a graph embedding based on matrix sketching that combines those three desiderata. Starting out from the observation that embedding methods aim to preserve the covariance among the rows of a similarity matrix, FREDE iteratively improves on quality while individually processing rows of a nonlinearly transformed PPR similarity matrix derived from a state-of-the-art graph embedding method and provides, at any iteration , column-covariance approximation guarantees in due course almost indistinguishable from those of the optimal approximation by SVD. Our experimental evaluation on variably sized networks shows that FREDE performs almost as well as SVD and competitively against state-of-the-art embedding methods in diverse data science tasks, even when it is based on as little as 10% of node similarities.

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22

Wadade, Hidemitsu. "Optimal embeddings of critical Sobolev–Lorentz–Zygmund spaces." Studia Mathematica 223, no. 1 (2014): 77–95. http://dx.doi.org/10.4064/sm223-1-5.

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23

Caha, R., and V. Koubek. "Optimal embeddings of odd ladders into a hypercube." Discrete Applied Mathematics 116, no. 1-2 (January 2002): 73–102. http://dx.doi.org/10.1016/s0166-218x(00)00329-2.

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24

Kovalev, Leonid V. "Optimal extension of Lipschitz embeddings in the plane." Bulletin of the London Mathematical Society 51, no. 4 (May 3, 2019): 622–32. http://dx.doi.org/10.1112/blms.12255.

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25

Hochberg, Robert A., and Matthias F. Stallmann. "Optimal one-page tree embeddings in linear time." Information Processing Letters 87, no. 2 (July 2003): 59–66. http://dx.doi.org/10.1016/s0020-0190(03)00261-8.

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26

Liebert, W., K. Pawelzik, and H. G. Schuster. "Optimal Embeddings of Chaotic Attractors from Topological Considerations." Europhysics Letters (EPL) 14, no. 6 (March 15, 1991): 521–26. http://dx.doi.org/10.1209/0295-5075/14/6/004.

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27

Goldman, Mikhail L. "Some constructive criteria of optimal embeddings for potentials." Complex Variables and Elliptic Equations 56, no. 10-11 (October 2011): 885–903. http://dx.doi.org/10.1080/17476933.2011.592577.

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28

Mörters, Peter, and István Redl. "Optimal embeddings by unbiased shifts of Brownian motion." Bulletin of the London Mathematical Society 49, no. 2 (February 9, 2017): 331–41. http://dx.doi.org/10.1112/blms.12030.

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29

Goldman, M. L. "Optimal embeddings of generalized Bessel and Riesz potentials." Proceedings of the Steklov Institute of Mathematics 269, no. 1 (July 2010): 85–105. http://dx.doi.org/10.1134/s0081543810020082.

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30

Garrigós, Gustavo, Eugenio Hernández, and Maria de Natividade. "Democracy functions and optimal embeddings for approximation spaces." Advances in Computational Mathematics 37, no. 2 (September 23, 2011): 255–83. http://dx.doi.org/10.1007/s10444-011-9197-0.

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31

Bashir, Zia, Fernando Cobos, and Georgi E. Karadzhov. "Optimal Embeddings of Calderón Spaces in Hölder-Zygmund Spaces." MATHEMATICA SCANDINAVICA 114, no. 1 (January 17, 2014): 120. http://dx.doi.org/10.7146/math.scand.a-16642.

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We prove optimal embeddings of Calderón spaces built-up over function spaces defined in $\mathbf{R}^n$ with the Lebesgue measure into generalized Hölder-Zygmund spaces in the super-critical and critical cases.

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32

Heun, Volker, and Ernst W. Mayr. "Optimal Dynamic Embeddings of Complete Binary Trees into Hypercubes." Journal of Parallel and Distributed Computing 61, no. 8 (August 2001): 1110–25. http://dx.doi.org/10.1006/jpdc.2001.1734.

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33

Huang, Shou-Hsuan S., Hongfei Liu, and Rakesh M. Verma. "A New combinatorial approach to optimal embeddings of rectangles." Algorithmica 16, no. 2 (August 1996): 161–80. http://dx.doi.org/10.1007/bf01940645.

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34

Clavero, Nadia, and Javier Soria. "Optimal Rearrangement Invariant Sobolev Embeddings in Mixed Norm Spaces." Journal of Geometric Analysis 26, no. 4 (November 13, 2015): 2930–54. http://dx.doi.org/10.1007/s12220-015-9655-x.

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35

Czerwinski, Paul, and Vijaya Ramachandran. "Optimal VLSI graph embeddings in variable aspect ratio rectangles." Algorithmica 3, no. 1-4 (November 1988): 487–510. http://dx.doi.org/10.1007/bf01762128.

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36

Musil, Vít. "Optimal Orlicz domains in Sobolev embeddings into Marcinkiewicz spaces." Journal of Functional Analysis 270, no. 7 (April 2016): 2653–90. http://dx.doi.org/10.1016/j.jfa.2016.01.019.

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37

Fontana, Luigi, and Carlo Morpurgo. "Optimal limiting embeddings for Δ-reduced Sobolev spaces inL1." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 31, no. 2 (March 2014): 217–30. http://dx.doi.org/10.1016/j.anihpc.2013.02.007.

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38

Greenberg, David S., Lenwood S. Heath, and Arnold L. Rosenberg. "Optimal embeddings of butterfly-like graphs in the hypercube." Mathematical Systems Theory 23, no. 1 (December 1990): 61–77. http://dx.doi.org/10.1007/bf02090766.

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39

Oniani, David, Guoqian Jiang, Hongfang Liu, and Feichen Shen. "Constructing co-occurrence network embeddings to assist association extraction for COVID-19 and other coronavirus infectious diseases." Journal of the American Medical Informatics Association 27, no. 8 (May 27, 2020): 1259–67. http://dx.doi.org/10.1093/jamia/ocaa117.

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Abstract Objective As coronavirus disease 2019 (COVID-19) started its rapid emergence and gradually transformed into an unprecedented pandemic, the need for having a knowledge repository for the disease became crucial. To address this issue, a new COVID-19 machine-readable dataset known as the COVID-19 Open Research Dataset (CORD-19) has been released. Based on this, our objective was to build a computable co-occurrence network embeddings to assist association detection among COVID-19–related biomedical entities. Materials and Methods Leveraging a Linked Data version of CORD-19 (ie, CORD-19-on-FHIR), we first utilized SPARQL to extract co-occurrences among chemicals, diseases, genes, and mutations and build a co-occurrence network. We then trained the representation of the derived co-occurrence network using node2vec with 4 edge embeddings operations (L1, L2, Average, and Hadamard). Six algorithms (decision tree, logistic regression, support vector machine, random forest, naïve Bayes, and multilayer perceptron) were applied to evaluate performance on link prediction. An unsupervised learning strategy was also developed incorporating the t-SNE (t-distributed stochastic neighbor embedding) and DBSCAN (density-based spatial clustering of applications with noise) algorithms for case studies. Results The random forest classifier showed the best performance on link prediction across different network embeddings. For edge embeddings generated using the Average operation, random forest achieved the optimal average precision of 0.97 along with a F1 score of 0.90. For unsupervised learning, 63 clusters were formed with silhouette score of 0.128. Significant associations were detected for 5 coronavirus infectious diseases in their corresponding subgroups. Conclusions In this study, we constructed COVID-19–centered co-occurrence network embeddings. Results indicated that the generated embeddings were able to extract significant associations for COVID-19 and coronavirus infectious diseases.

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40

Fan, Jianxi, Xiaohua Jia, and Xiaola Lin. "Optimal Embeddings of Paths with Various Lengths in Twisted Cubes." IEEE Transactions on Parallel and Distributed Systems 18, no. 4 (April 2007): 511–21. http://dx.doi.org/10.1109/tpds.2007.1003.

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41

Gol’dman, M. L. "Local growth envelopes and optimal embeddings of generalized Sobolev spaces." Doklady Mathematics 74, no. 2 (October 2006): 692–95. http://dx.doi.org/10.1134/s106456240605019x.

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42

Ahmed, Irshaad, and Georgi E. Karadzhov. "Optimal embeddings of generalized inhomogeneous Sobolev spaces on R^n." Mathematical Inequalities & Applications, no. 3 (2011): 737–45. http://dx.doi.org/10.7153/mia-14-62.

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43

Vallentin, Frank. "Optimal distortion embeddings of distance regular graphs into Euclidean spaces." Journal of Combinatorial Theory, Series B 98, no. 1 (January 2008): 95–104. http://dx.doi.org/10.1016/j.jctb.2007.06.002.

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44

O'Keeffe, Michael, and Michael M. J. Treacy. "Isogonal weavings on the sphere: knots, links, polycatenanes." Acta Crystallographica Section A Foundations and Advances 76, no. 5 (August 21, 2020): 611–21. http://dx.doi.org/10.1107/s2053273320010669.

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Mathematical knots and links are described as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'-transitive). Corner- and stick-transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.

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45

Kao, Ming-Yang, Martin Fürer, Xin He, and Balaji Raghavachari. "Optimal Parallel Algorithms for Straight-Line Grid Embeddings of Planar Graphs." SIAM Journal on Discrete Mathematics 7, no. 4 (November 1994): 632–46. http://dx.doi.org/10.1137/s0895480191221453.

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46

Games, Richard A. "Optimal book embeddings of the FFT, benes, and barrel shifter networks." Algorithmica 1, no. 1-4 (November 1986): 233–50. http://dx.doi.org/10.1007/bf01840445.

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47

Vysotskiy, L. I., and S. A. Lozhkin. "Optimal Two-Sided Embeddings of Complete Binary Trees in Rectangular Grids." Computational Mathematics and Modeling 30, no. 2 (April 2019): 115–28. http://dx.doi.org/10.1007/s10598-019-09440-3.

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48

Moura, Susana D., Júlio S. Neves, and Cornelia Schneider. "Optimal Embeddings of Spaces of Generalized Smoothness in the Critical Case." Journal of Fourier Analysis and Applications 17, no. 5 (November 18, 2010): 777–800. http://dx.doi.org/10.1007/s00041-010-9155-0.

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49

Neves, Júlio S., and Bohumír Opic. "Optimal local embeddings of Besov spaces involving only slowly varying smoothness." Journal of Approximation Theory 254 (June 2020): 105393. http://dx.doi.org/10.1016/j.jat.2020.105393.

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50

Liu, Yuzhong, Michael O'Keeffe, Michael M. J. Treacy, and Omar M. Yaghi. "The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry." Chemical Society Reviews 47, no. 12 (2018): 4642–64. http://dx.doi.org/10.1039/c7cs00695k.

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