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On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics)

1998
*
SIAM journal on computing (Print)
*

We consider

doi:10.1137/s0097539795296334
fatcat:xcdfmrgia5blxkgczaxt4cxyam
*the*problem*of**fitting*an n × n*distance*matrix D*by*a*tree**metric*T . Let ε be*the**distance*to*the*closest*tree**metric*under*the*L∞ norm; that is, ε = min T { T − D ∞}. ... First we present an O(n 2 ) algorithm for finding a*tree**metric*T such that T − D ∞ ≤ 3ε. Second we show that it is N P-hard to find a*tree**metric*T such that T − D ∞ < 9 8 ε. ...*One**of**the*most common methods for clustering*numeric*data involves*fitting**the*data to a*tree**metric*, which is defined*by*a weighted*tree*spanning*the*points*of**the**metric*,*the**distance*between two points ...##
###
Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor
[article]

2021
*
arXiv
*
pre-print

We consider

arXiv:2110.02807v1
fatcat:sxz5iov6xzaotmimokwvdjon6i
*the**numerical**taxonomy*problem*of**fitting*a positive*distance*function D:S 2→ℝ_>0*by*a*tree**metric*. ... We can do this both for general*trees*, and for*the*special case*of*ultrametrics with a root having*the*same*distance*to all vertices in S. ...*The*best previous*approximation*factor was O((log n)(log log n))*by*who wrote "Determining whether an O(1)*approximation*can be obtained is a fascinating question". ...##
###
Approximating Additive Distortion of Embeddings into Line Metrics
[chapter]

2004
*
Lecture Notes in Computer Science
*

We consider

doi:10.1007/978-3-540-27821-4_9
fatcat:3b7ofxe36neobe5kabuqxq3oru
*the*problem*of**fitting**metric*data*on*n points to a path (line)*metric*. Our objective is to minimize*the*total additive distortion*of*this mapping. ...*The*total additive distortion is*the*sum*of*errors in all pairwise*distances*in*the*input data. This problem has been shown to be NP-hard*by*[13] . ... Introduction*One**of**the*most common methods for clustering*numerical*data is to*fit**the*data to*tree**metrics*. A*tree**metric*is defined*on*vertices*of*a weighted*tree*. ...##
###
Page 2049 of Mathematical Reviews Vol. , Issue 97C
[page]

1997
*
Mathematical Reviews
*

*the*

*approximability*

*of*

*numerical*

*taxonomy*(

*fitting*

*distances*

*by*

*tree*

*metrics*). ... Summary: “We consider

*the*problem

*of*

*fitting*an n x n

*distance*matrix D

*by*a

*tree*

*metric*7. Let € be

*the*

*distance*to

*the*closest

*tree*

*metric*under

*the*L.. norm, that is, e = miny{||7, D||,..}. ...

##
###
Improving the Reliability of Decision-Support Systems for Nuclear Emergency Management by Leveraging Software Design Diversity

english

2016
*
Journal of Computing and Information Technology
*

english

*The*acceptance test and

*the*voter are used in a new scheme, which extends

*the*Consensus Recovery Block method

*by*a database

*of*result

*taxonomies*to support machine-learning. ... and

*the*trustworthiness

*of*

*the*simulation results used

*by*emergency managers in

*the*decision making process. ... Figure 3 . 3 Taxonomic

*trees*corresponding to

*the*three dispersion codes implemented in RODOS.

*The*

*trees*were constructed

*by*means

*of*

*the*GRSS

*distance*with p = 0.6. ...

##
###
Page 7336 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

[Farach-Colton, Martin]
(1-RTG-C; Piscataway, NJ);
Paterson, Mike (4-WARW-C; Coventry);
Thorup, Mikkel (DK-CPNH-CS; Copenhagen)

*On**the**approximability**of**numerical**taxonomy*(*fitting**distances**by**tree*... Summary: “We consider*the*problem*of**fitting*an n x n*distance*matrix D*by*a*tree**metric*T. Let € be*the**distance*to*the*closest*tree**metric*under*the*L,, norm; that is, e = miny{||T — D||.}. ...##
###
l∞-Approximation via Subdominants

2000
*
Journal of Mathematical Psychology
*

algorithm for

doi:10.1006/jmps.1999.1270
pmid:11133300
fatcat:mdlbjucod5c5no4w2yfpyistju
*the*problem*of**fitting*a*distance**by*a*tree**metric*). ... This leads to simple optimal algorithms for*the*problem*of*best l -*fitting**of**distances**by*ultrametrics and*by**tree**metrics*preserving*the**distances*to a fixed vertex (*the*latter provides a 3-*approximation*... In*numerical**taxonomy*, u is a*distance*(more generally, a dissimilarity)*on*a finite set X and K is*the*cone*of*all ultrametrics or*tree**metrics*defined*on*X; see Barthe lemy and Gue noche (1991) and ...##
###
Alignment-Free Genome Tree Inference by Learning Group-Specific Distance Metrics

2013
*
Genome Biology and Evolution
*

We propose a method to improve genome

doi:10.1093/gbe/evt105
pmid:23843191
pmcid:PMC3762195
fatcat:5ow74suvefbxvgthiyzna3rpji
*tree*inference*by*learning specific*distance**metrics*over*the*genome signature for groups*of*organisms with similar phylogenetic, genomic, or ecological properties ...*By*applying this method to more than a thousand prokaryotic genomes, we showed that, indeed, better*distance**metrics*could be learned for most*of**the*18 groups*of*organisms tested here. ... Acknowledgments*The*authors thank Lars Steinbrü ck (MPI Informatik, HHU) for critical reading*of**the*manuscript and useful comments. ...##
###
Fitting Tree Metrics: Hierarchical Clustering and Phylogeny

2011
*
SIAM journal on computing (Print)
*

Partially supported

doi:10.1137/100806886
fatcat:6w6nnelhene3pedbpl4zx7c2wa
*by*a Charlotte Elizabeth Procter Fellowship. ...*The*problem*of**fitting**tree**metrics*also arises in phylogeny where*the*objective is to learn*the*evolution*tree**by**fitting*a*tree*to dissimilarity data*on*taxa. ...*The*quality*of**the**fit*is measured*by*taking*the*p norm*of**the*difference between*the**tree**metric*constructed and*the*given data. ...##
###
Seeded Hierarchical Clustering for Expert-Crafted Taxonomies
[article]

2022
*
arXiv
*
pre-print

In this work, we study Seeded Hierarchical Clustering (SHC):

arXiv:2205.11602v1
fatcat:jmdwans4jnejlp5rxpwlyd3dbe
*the*task*of*automatically*fitting*unlabeled data to such*taxonomies*using only a small set*of*labeled examples. ... It outperforms both unsupervised and supervised baselines for*the*SHC task*on*three real-world datasets. ...*The*magnitude*of*c (i) other (i.e., c (i) other ) is*approximated*to be*the*magnitude*of**the*centroid*of**the*subtopics. Its representation is given*by*Equation 2 (see Appendix A for derivation). ...##
###
Page 6223 of Mathematical Reviews Vol. , Issue 96j
[page]

1996
*
Mathematical Reviews
*

Farach, Babu Narayanan, Mike Paterson and Mikkel Thorup,

*On**the**approximability**of**numerical**taxonomy*(*fitting**distances**by**tree**metrics*) (365-372); Paolo Ferragina and Roberto Grossi, Fast string searching ... Proceedings*of**the*conference held at*the*University*of*Leeds, Leeds, September 1993. Edited*by*D. M. Titterington.*The*Institute*of*Mathematics and its Applications cuaieines Series. New Series, 54. ...##
###
Many-to-Many Feature Matching Using Spherical Coding of Directed Graphs
[chapter]

2004
*
Lecture Notes in Computer Science
*

*The*algorithm was based

*on*a

*metric*-

*tree*representation

*of*labeled graphs and their

*metric*embedding into normed vector spaces, using

*the*embedding algorithm

*of*Matousek [13] . ... This reduces

*the*problem

*of*directed graph matching to

*the*problem

*of*geometric point matching, for which efficient many-to-many matching algorithms exist, such as

*the*Earth Mover's

*Distance*. ...

*The*work

*of*Yakov Keselman is supported, in part,

*by*

*the*NSF grant No. 0125068. Sven Dickinson acknowledges

*the*support

*of*NSERC, CITO, IRIS, PREA, and

*the*NSF. ...

##
###
Tumor classification using phylogenetic methods on expression data

2004
*
Journal of Theoretical Biology
*

To solve

doi:10.1016/j.jtbi.2004.02.021
pmid:15178197
fatcat:7urg6jw34bfopkopuu2bwgybtq
*the*class discovery problem, we impose a*metric**on*a set*of*tumors as a function*of*their gene expression levels, and impose a*tree*structure*on*this*metric*, using standard*tree**fitting*methods ... To solve*the*class prediction problem, we built a classification*tree**on**the*learning set, and then sought*the*optimal placement*of*each test sample within*the*classification*tree*. ... Also, we thank three anonymous referees for many helpful suggestions that led to substantial improvements in*the*manuscript. ...##
###
Statistical Object Data Analysis of Taxonomic Trees from Human Microbiome Data

2012
*
PLoS ONE
*

*The*contribution

*of*our work is threefold: first, a weighted

*tree*structure to analyze RDP data is introduced; second, using a probability measure to model a set

*of*taxonomic

*trees*, we introduce an

*approximate*...

*The*data objects that pertain to this work are taxonomic

*trees*

*of*bacteria built from analysis

*of*16S rRNA gene sequences (e.g. using RDP); there is

*one*such object for each biological sample analyzed. ... R z an arbitrary

*metric*

*of*

*distance*

*on*G. ...

##
###
Automatic Taxonomy Construction from Keywords via Scalable Bayesian Rose Trees

2015
*
IEEE Transactions on Knowledge and Data Engineering
*

,

doi:10.1109/tkde.2015.2397432
fatcat:qcxwfsed3vgwroyklu6h5ip2ma
*the*domain*of*interest is already represented*by*a set*of*keywords. ... We reduce*the*complexity*of*previous hierarchical clustering approaches from O(n 2 log n) to O(n log n) using a nearest-neighbor-based*approximation*, so that we can derive a domain-specific*taxonomy*from ... ACKNOWLEDGEMENTS We would like to thank Charles Blundell and Yee Whye Teh for their help*on**the*implementation*of**the*Bayesian rose*tree*and thanks to Ting Liu for help*on**the*implementation*of*Spilltree ...
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