Construct · 1 fields · 2 citations

curse of dimensionality: a standalone construct

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curse ofdimensionality1 CONSTRUCTCurse of dimensionsBELLMAN · DIMENSIONS

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Optimization and machine learning

The breakdown of methods as the number of data dimensions grows

Richard Bellman coined the phrase in his 1957 book Dynamic Programming, where he observed that the number of grid points needed to search a space grows exponentially with the number of variables, or dimensions, so that a problem that is tractable in a few dimensions becomes hopeless in many. The name now covers a family of related effects that appear as dimensions grow: the volume of the space expands so quickly that any fixed set of data points becomes sparse, the distances between points grow more alike, and estimates that must fill the space demand exponentially more data. Beyer, Goldstein, Ramakrishnan, and Shaft (1999) gave one of the sharpest statements, showing that under broad conditions the distance to the nearest data point and the distance to the farthest one converge as dimension increases, so that the idea of a nearest neighbor loses its meaning. The construct is distinctive as a general obstacle rather than the flaw of any one technique: it constrains optimization, search, density estimation, and any method that relies on filling or measuring distance within a high-dimensional space, and it is the reason many analyses reduce dimensions before proceeding.[1][2]

Unit
A named phenomenon describing how required data and computation scale with the number of dimensions
Recovery means
Not applicable
Measured by
Identified analytically by how sample requirements, volume, or distance concentration scale with dimension, and empirically by degrading accuracy or nearest-neighbor contrast as features are added

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Nearby constructs

manifold2 fields · 2 citationsjingle · one word, two constructsdimension4 fields · 5 citationsjingle · one word, four constructsbook1 fields · 1 citationsanchor · a standalone construct

References

  1. Bellman, R. (1957). Dynamic programming. Princeton University Press. ISBN 9780691079516
  2. Beyer, K., Goldstein, J., Ramakrishnan, R., & Shaft, U. (1999). When is “nearest neighbor” meaningful? In Database theory: ICDT’99 (Lecture Notes in Computer Science, Vol. 1540, pp. 217–235). Springer. https://doi.org/10.1007/3-540-49257-7_15

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