58:
1206:
1226:. Many popular machine learning algorithms use specific distance metrics such as the aforementioned to compare the similarity of two data points. Depending on the nature of the data being analyzed, various metrics can be used. The Minkowski metric is most useful for numerical datasets where you want to determine the similarity of size between multiple datapoint vectors.
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1354:
Zezula, Pavel; Amato, Giuseppe; Dohnal, Vlastislav; Batko, Michal (2006), "Chapter 1, Foundations of Metric Space
Searching, Section 3.1, Minkowski Distances",
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1117:{\displaystyle \lim _{p\to -\infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\min _{i=1}^{n}|x_{i}-y_{i}|.}
1845:
910:{\displaystyle \lim _{p\to \infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\max _{i=1}^{n}|x_{i}-y_{i}|.}
1708:
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61:
Comparison of
Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard
1739:
1426:
1223:
248:{\displaystyle X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n}}
1389:
1179:
of the distance function where all points are at the unit distance from the center) with various values of
1953:
1754:
1683:
1576:
1556:
1395:
645:
it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of
1581:
1473:
380:{\displaystyle D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.}
1759:
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419:
35:
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522:
39:
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1419:
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704:
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51:
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20:
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1571:
1485:
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1238: – N-th root of the arithmetic mean of the given numbers raised to the power n
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677:
1301: – Function spaces generalizing finite-dimensional p norm spaces
1279:
1270: – Function spaces generalizing finite-dimensional p norm spaces
56:
1415:
1126:
The
Minkowski distance can also be viewed as a multiple of the
1411:
19:
Not to be confused with the pseudo-Euclidean metric of the
42:
which can be considered as a generalization of both the
1322:Ĺžuhubi, ErdoÄźan S. (2003), "Chapter V: Metric Spaces",
1303:
Pages displaying short descriptions of redirect targets
1358:, Advances in Database Systems, Springer, p. 10,
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The
Minkowski metric is very useful in the field of
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from both of these points. Since this violates the
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1390:Unit Balls for Different p-Norms in 2D and 3D
1175:The following figure shows unit circles (the
50:. It is named after the Polish mathematician
8:
1356:Similarity Search: The Metric Space Approach
1851:Vitale's random Brunn–Minkowski inequality
1808:
1434:
1420:
1412:
1326:, Springer Netherlands, pp. 261–356,
1130:of the component-wise differences between
683:Minkowski distance is typically used with
16:Mathematical metric in normed vector space
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1606:
1396:Unit-Norm Vectors under Different p-Norms
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711:, respectively. In the limiting case of
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703:being 1 or 2, which correspond to the
1402:Simple IEEE 754 implementation in C++
939:reaching negative infinity, we have:
7:
1864:Applications & related
1783:Marcinkiewicz interpolation theorem
1709:Symmetric decreasing rearrangement
1613:
960:
753:
110:is an integer) between two points
14:
731:reaching infinity, we obtain the
1276: – Length in a vector space
1204:
676:The resulting metric is also an
70:The Minkowski distance of order
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954:
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582:
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1:
1679:Convergence almost everywhere
1407:NPM JavaScript Package/Module
556:{\displaystyle 2^{1/p}>2,}
418:the Minkowski distance is a
1846:Prékopa–Leindler inequality
1699:Locally integrable function
1621:{\displaystyle L^{\infty }}
1332:10.1007/978-94-017-0141-9_5
1970:
1592:Square-integrable function
18:
1841:Minkowski–Steiner formula
1824:Isoperimetric inequality
411:{\displaystyle p\geq 1,}
1829:Brunn–Minkowski theorem
448:{\displaystyle p<1,}
1684:Convergence in measure
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638:{\displaystyle p<1}
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1261:{\displaystyle L^{p}}
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588:{\displaystyle (0,1)}
558:
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512:{\displaystyle (1,1)}
482:
480:{\displaystyle (0,0)}
455:the distance between
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1803:Riesz–Thorin theorem
1646:Infimum and supremum
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1531:Lebesgue integration
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669:{\displaystyle 1/p.}
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424:Minkowski inequality
393:
259:
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1765:Young's convolution
1704:Measurable function
1587:Pythagorean theorem
1577:Parseval's identity
1526:Integrable function
1324:Functional Analysis
617:triangle inequality
422:as a result of the
40:normed vector space
1886:Probability theory
1788:Plancherel theorem
1694:Integral transform
1641:Chebyshev distance
1618:
1567:Euclidean distance
1500:Minkowski distance
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1274:Norm (mathematics)
1258:
1189:
1166:{\displaystyle Q.}
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733:Chebyshev distance
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705:Manhattan distance
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48:Manhattan distance
44:Euclidean distance
28:Minkowski distance
1944:Hermann Minkowski
1926:
1925:
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1858:
1674:Almost everywhere
1459: &
1292:{\displaystyle p}
1192:{\displaystyle p}
1143:{\displaystyle P}
1050:
946:
932:{\displaystyle p}
843:
742:
724:{\displaystyle p}
696:{\displaystyle p}
608:{\displaystyle 1}
595:is at a distance
371:
174:
103:{\displaystyle p}
83:{\displaystyle p}
52:Hermann Minkowski
1961:
1876:Fourier analysis
1834:Milman's reverse
1817:
1815:Lebesgue measure
1809:
1793:Riemann–Lebesgue
1636:Bounded function
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1536:Taxicab geometry
1491:Measurable space
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1745:Hausdorff–Young
1725:Babenko–Beckner
1713:
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1602:
1596:
1540:
1509:
1505:Sequence spaces
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255:is defined as:
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21:Minkowski space
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1689:Function space
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1562:Cauchy–Schwarz
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1461:Hilbert spaces
1453:
1451:
1450:Basic concepts
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1399:
1398:at wolfram.com
1393:
1392:at wolfram.com
1385:
1384:External links
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1881:Lorentz space
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1572:Hilbert space
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1186:
1178:
1173:
1160:
1157:
1137:
1129:
1124:
1111:
1101:
1097:
1093:
1088:
1084:
1073:
1068:
1065:
1062:
1054:
1047:
1044:
1029:
1017:
1013:
1009:
1004:
1000:
989:
984:
981:
978:
974:
957:
951:
926:
917:
904:
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890:
886:
881:
877:
866:
861:
858:
855:
847:
840:
837:
822:
810:
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802:
797:
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782:
777:
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767:
747:
734:
718:
710:
706:
690:
681:
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663:
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652:
632:
629:
626:
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387:
374:
368:
365:
350:
338:
334:
330:
325:
321:
310:
305:
302:
299:
295:
284:
280:
276:
273:
270:
266:
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240:
230:
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117:
97:
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53:
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37:
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1718:Inequalities
1658:Uniform norm
1546:
1515:
1499:
1466:
1355:
1349:
1323:
1317:
1217:
1214:Applications
1174:
1125:
918:
682:
388:
69:
31:
27:
25:
1916:Von Neumann
1730:Chebyshev's
1933:Categories
1911:C*-algebra
1735:Clarkson's
1309:References
1128:power mean
66:Definition
1906:*-algebra
1891:Quasinorm
1760:Minkowski
1651:Essential
1614:∞
1443:Lp spaces
1177:level set
1094:−
1010:−
975:∑
961:∞
958:−
955:→
887:−
803:−
768:∑
754:∞
751:→
400:≥
331:−
296:∑
231:∈
212:…
153:…
1939:Distance
1755:Markov's
1750:Hölder's
1740:Hanner's
1557:Bessel's
1495:function
1479:Lebesgue
1230:See also
707:and the
46:and the
1775:Results
1474:Measure
426:. When
90:(where
1628:spaces
1549:spaces
1518:spaces
1469:spaces
1457:Banach
1370:
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678:F-norm
619:, for
420:metric
36:metric
1299:-norm
1268:space
38:in a
34:is a
1813:For
1667:Maps
1368:ISBN
1336:ISBN
1222:and
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