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Mathematics & StatisticsMachine Learning

Euclidean distance

nounAlso: L2 distance, Euclidean metric, straight-line distance

In plain wordsHow far apart two points are in a straight line, as a ruler would measure it, extended from two or three dimensions to lists of numbers of any length.

  1. The distance between two points in n-dimensional space computed as the square root of the sum of squared differences of their coordinates — the L2 normnorm, n.: A way of measuring the size, or length, of a list of numbers — for example the straight-line distance from the origin to the point the list describes. of their difference, generalizing the Pythagorean theorem. It is the default distance in k-means clusteringk-means clustering, n.: A simple way to split data into a chosen number of groups: guess group centers, assign every point to its nearest center, move each center to the middle of its points, and repeat., k-nearest neighborsk-nearest neighbors, n.: Predicting something about a new example by finding the few most similar examples already known and going with their majority answer, or their average. and many nearest-neighbor indexesapproximate nearest neighbor search, n.: A family of shortcuts for finding the most similar items in a huge collection without checking every one — accepting an occasional miss in exchange for answers thousands of times faster., and its square is what mean squared errormean squared error, n.: A way of scoring numerical predictions: square each prediction's error, then average them, so big misses count much more than small ones. Lower is better. averages. Because it depends on the units of each coordinate, features are usually standardized first (see feature scalingfeature scaling, n.: Putting input measurements on comparable scales, so that income in dollars and age in years, say, both span similar ranges and no feature dominates just because of its units.). In high dimensions the distances between random points crowd around a common value, one face of the curse of dimensionalitycurse of dimensionality, n.: The problems that pile up when each item is described by very many measurements: even huge datasets end up thinly scattered across the possibilities, and 'near' and 'far' stop meaning much., so neighbor rankings can become fragile; for embeddings, practitioners often use cosine similaritycosine similarity, n.: A score from −1 to 1 for how closely two lists of numbers point the same way, ignoring their size. It is the standard test of whether two texts mean similar things., which gives the same ranking for unit-length vectors.

    We normalize the vectors so that Euclidean distance and cosine similarity agree.

d(x,y)=∑i=1n(xi−yi)2d(\mathbf{x},\mathbf{y}) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}

Origin

Named for Euclid of Alexandria (c. 300 BCE), whose Elements contains the plane geometry from which the formula follows by the Pythagorean theorem. Arthur Cayley (1843) and Hermann Grassmann (1844) extended such geometry beyond three dimensions, and Maurice Fréchet (1906) and Felix Hausdorff (1914) abstracted distance itself into the metric space.

Related

See also
  • normnorm, n.: A way of measuring the size, or length, of a list of numbers — for example the straight-line distance from the origin to the point the list describes.
  • cosine similaritycosine similarity, n.: A score from −1 to 1 for how closely two lists of numbers point the same way, ignoring their size. It is the standard test of whether two texts mean similar things.
  • k-nearest neighborsk-nearest neighbors, n.: Predicting something about a new example by finding the few most similar examples already known and going with their majority answer, or their average.
  • k-means clusteringk-means clustering, n.: A simple way to split data into a chosen number of groups: guess group centers, assign every point to its nearest center, move each center to the middle of its points, and repeat.
  • curse of dimensionalitycurse of dimensionality, n.: The problems that pile up when each item is described by very many measurements: even huge datasets end up thinly scattered across the possibilities, and 'near' and 'far' stop meaning much.
  • mean squared errormean squared error, n.: A way of scoring numerical predictions: square each prediction's error, then average them, so big misses count much more than small ones. Lower is better.
Contrast with
  • cosine similaritycosine similarity, n.: A score from −1 to 1 for how closely two lists of numbers point the same way, ignoring their size. It is the standard test of whether two texts mean similar things.

Mentioned in

cosine similaritycosine similarity, n.: A score from −1 to 1 for how closely two lists of numbers point the same way, ignoring their size. It is the standard test of whether two texts mean similar things., hierarchical clusteringhierarchical clustering, n.: A way of grouping data that builds a family tree of clusters, joining the most similar items step by step, so you can cut the tree at whatever level of detail you need., k-means clusteringk-means clustering, n.: A simple way to split data into a chosen number of groups: guess group centers, assign every point to its nearest center, move each center to the middle of its points, and repeat., k-nearest neighborsk-nearest neighbors, n.: Predicting something about a new example by finding the few most similar examples already known and going with their majority answer, or their average., normnorm, n.: A way of measuring the size, or length, of a list of numbers — for example the straight-line distance from the origin to the point the list describes., silhouette scoresilhouette score, n.: A number between −1 and 1 that grades a grouping of data by checking, for each point, whether it sits closer to its own group than to the next-nearest group., vector databasevector database, n.: A database built to store embeddings — lists of numbers that represent meaning — and to quickly find the stored items most similar to a new one. It is the memory behind many AI search tools., vector searchvector search, n.: Searching by closeness instead of exact match: the query and all stored items are lists of numbers, and the search returns the items whose numbers are most similar to the query's., Johnson–Lindenstrauss lemmaJohnson–Lindenstrauss lemma, n.: A surprising fact about high-dimensional data: a random projection can squash a large set of points into far fewer dimensions while distorting every distance between them only slightly., locality-sensitive hashinglocality-sensitive hashing, n.: A trick for finding look-alikes in a huge collection without comparing every pair: items get short codes designed so that similar items tend to share a code, and only items sharing one are compared., metric learningmetric learning, n.: Teaching a model to measure similarity: it learns to place items so that ones that should count as alike end up close together and different ones far apart.