The -means algorithm is by far the most widely used method for discovering clusters in data. We show how to accelerate it dramatically, while still always computing exactly the same result as the standard algorithm. The accelerated algorithm avoids unnecessary distance calculations by applying the triangle inequality in two different ways, and by keeping track of lower and upper bounds for dist...

For a triangle in the hyperbolic plane, let α, β, γ denote the angles opposite the sides a, b, c, respectively. Also, let h be the height of the altitude to side c. Under the assumption that α, β, γ can be chosen uniformly in the interval (0, π) and it is given that α + β + γ < π, we show that the strong triangle inequality a + b > c + h holds approximately 79% of the time. To accomplish this, ...

We obtain refined estimates of the triangle inequality in a normed space using integrals and the Tapia semi-product. The particular case of an inner product space is discussed in more detail.

With the advent of both analytical and evolutionary models that specify complex character-transformation weights (or probabilities) and computer programs in which they are implemented, the use of multistate characters with elaborate state-transformation costs (matrix characters; Sankoff and Rousseau 1975; Sankoff and Cedergren 1983) is increasing. What has not yet appeared, however, is a genera...

Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed. The Jaccard index [8] is a classical similarity measure on sets with a lot of practical applications in information retrieval, data mining, machine learning, and many more (cf., e.g., [7]). Measuring the relative size of the overlap of two finite...

We will show that for any two bounded linear operators X, Y on a Hilbert space H, if they satisfy the triangle equality |X + Y | = |X | + |Y |, there exists a partial isometry U on H such that X = U |X | and Y = U |Y |. This is a generalization of Thompson’s theorem to the matrix case proved by using a trace. 47A05, 47A10, 47A12 operator theory, Hilbert space, triangle inequality

Facility Dispersion Problem, originally studied in Operations Research, has recently found important new applications in Result Diversification approach in information sciences. This optimisation problem consists in selecting a small set of p items out of a large set of candidates to maximise a given objective function. The function expresses the notion of dispersion of a set of selected items ...

1.1. Norms on Vector Spaces. We want to measure the “size” or “length” of a vector. The mathematical device for doing this in called a norm. Here is the definition. Definition 1.1. Let V be a vector space over K. A norm on V is a function ‖·‖ : V → R : v 7→ ‖v‖ that has the following properties. (1) ‖v‖ ≥ 0 for all v ∈ V , and ‖v‖ = 0 if and only if v = 0. (2) If λ ∈ K and v ∈ V , ‖λv‖ = |λ| ‖v...

The function ez shows that a complex entire function can indeed omit one value. Lately, it has become fashionable to prove p -adic versions of value distribution theorems, of which Picard’s Theorem is an example, though not a recent one. More recent examples can be found in the works listed in the references section. Recall that the p -adic absolute value | |p on the rational number field Q is ...