Difficulties arising when analyzing data with many aspects ("dimensions")
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in dynamic programming.[1][2] The curse generally refers to issues that arise when the number of datapoints is small (in a suitably defined sense) relative to the intrinsic dimension of the data.
Dimensionally cursed phenomena occur in domains such as numerical analysis, sampling, combinatorics, machine learning, data mining and databases. The common theme of these problems is that when the dimensionality increases, the volume of the space increases so fast that the available data become sparse. In order to obtain a reliable result, the amount of data needed often grows exponentially with the dimensionality. Also, organizing and searching data often relies on detecting areas where objects form groups with similar properties; in high dimensional data, however, all objects appear to be sparse and dissimilar in many ways, which prevents common data organization strategies from being efficient.
^Bellman, Richard Ernest; Rand Corporation (1957). Dynamic programming. Princeton University Press. p. ix. ISBN 978-0-691-07951-6., Republished: Bellman, Richard Ernest (2003). Dynamic Programming. Courier Dover Publications. ISBN 978-0-486-42809-3.
^Bellman, Richard Ernest (1961). Adaptive control processes: a guided tour. Princeton University Press. ISBN 9780691079011.
and 25 Related for: Curse of dimensionality information
Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curseofdimensionality, and analyzing...
dimensions. The curse of dimensionality is expressed in the order of the integration error that is made by a quadrature of level l {\displaystyle l}...
problem intractable due to the “curseofdimensionality”. Alternatively, it has been shown that if the cost function of the multi-stage optimization problem...
in art Fourth dimension in literature Higher dimensions in mathematics Octonion Vector space Plane of rotation Curseofdimensionality in physics Kaluza–Klein...
high-dimensional data, tree-based exact vector search techniques such as the k-d tree and R-tree do not perform well enough because of the curseof dimensionality...
computational cost, the curseofdimensionality, the reliability of random number generators, and the verification and validation of the results. Monte Carlo...
M−1 dimensional space where M is the number of categories. MDA is useful because most classifiers are strongly affected by the curseofdimensionality. In...
number of features that are descriptive of the object. The number of features should not be too large, because of the curseofdimensionality; but should...
models. Because of this, it is less affected by the curseofdimensionality than e.g. a p-dimensional smoother. Furthermore, the AM is more flexible than...
required number of layer and their width can be exponentially smaller. Even if f {\displaystyle f} is not smooth, the curseofdimensionality can be broken...
at each iteration, it helps address the curseofdimensionality. Reversible-jump: This method is a variant of the Metropolis–Hastings algorithm that allows...
Coinage of the term 'ridge function' is often attributed to B.F. Logan and L.A. Shepp. A ridge function is not susceptible to the curseofdimensionality[clarification...
clustering, correlation clustering, and the curseofdimensionality. He is one of the founders and core developers of the open-source ELKI data mining framework...
usually referred to as the curseofdimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using...
traction over large collections, due to a manifestation of the so-called curseofdimensionality, and there are still many unsolved problems. Unfortunately...
results in a very sparse vector space of high dimensionality (cf. curseofdimensionality). Reducing the number of dimensions using linear algebraic methods...
dimensionality. The intrinsic dimension can be used as a lower bound of what dimension it is possible to compress a data set into through dimension reduction...
The CurseofDimensionality, Springer, 1997, pp. 282–294. Donoho, D., Tanner, J. (2009) Observed universality of phase transitions in high-dimensional geometry...
growth Combinatorial explosion Branching factor Granularity Curseofdimensionality Concentration of measure Noga Alon George Andrews József Beck Eric Temple...
superficially easier approach of just understanding the other. Birthday problem Exponential growth Metcalfe's law Curseofdimensionality Information explosion...
Global Information