Remove Metric <: SemiMetric <: PreMetric

README.md

@@ -141,31 +141,36 @@ pairwise!(R, dist, X, dims=i)
 Please pay attention to the difference, the functions for inplace computation are
 `colwise!` and `pairwise!` (instead of `colwise` and `pairwise`).
 
-## Distance type hierarchy
+## Distance properties
 
-The distances are organized into a type hierarchy.
+Distances is an abstract class for function `d` that satisfies
 
-At the top of this hierarchy is an abstract class **PreMetric**, which is defined to be a function `d` that satisfies
+
+`MetricType` indicates the particular properties of the distance
+
+`IsPreMetric` indicates that the distance is a premetric; i.e.,
 
     d(x, x) == 0  for all x
     d(x, y) >= 0  for all x, y
 
-**SemiMetric** is a abstract type that refines **PreMetric**. Formally, a *semi-metric* is a *pre-metric* that is also symmetric, as
+`IsSemiMetric` indicates that the distance is also symmetric; i.e.,
 
     d(x, y) == d(y, x)  for all x, y
 
-**Metric** is a abstract type that further refines **SemiMetric**. Formally, a *metric* is a *semi-metric* that also satisfies triangle inequality, as
+`IsMetric` indicates that the distance also satisfies the triangle inequality; i.e.,
 
     d(x, z) <= d(x, y) + d(y, z)  for all x, y, z
 
-This type system has practical significance. For example, when computing pairwise distances
+Note that we have `IsMetric <: IsSemiMetric <: IsPreMetric`.
+
+These properties have practical significance. For example, when computing pairwise distances
 between a set of vectors, you may only perform computation for half of the pairs, derive the
-values immediately for the remaining half by leveraging the symmetry of *semi-metrics*. Note
-that the types of `SemiMetric` and `Metric` do not completely follow the definition in
-mathematics as they do not require the "distance" to be able to distinguish between points:
-for these types `x != y` does not imply that `d(x, y) != 0` in general compared to the
-mathematical definition of semi-metric and metric, as this property does not change
-computations in practice.
+values immediately for the remaining half by leveraging the symmetry of the distance. 
+
+Note that a subadditive distance is not necessarly a metric in the mathematical since 
+it is not required to be able to distinguish between points:
+`x != y` does not imply that `d(x, y) != 0` in general compared to the
+mathematical definition of metric.
 
 Each distance corresponds to a distance type. The type name and the corresponding mathematical
 definitions of the distances are listed in the following table.

src/Distances.jl

@@ -1,16 +1,23 @@
 module Distances
 
 using LinearAlgebra
+import LinearAlgebra: issymmetric
 using Statistics
 import StatsAPI: pairwise, pairwise!
 
-export
-    # generic types/functions
-    PreMetric,
-    SemiMetric,
-    Metric,
 
+
+
+export 
+    Distance,
+    # traits
+    MetricType,
+    IsPreMetric,
+    IsSemiMetric,
+    IsMetric,
     # generic functions
+    issymmetric,
+    issubadditive,
     result_type,
     colwise,
     pairwise,

src/bhattacharyya.jl

@@ -2,10 +2,11 @@
 # be compared are probability distributions, frequencies or counts rather than
 # vectors of samples. Pre-calc accordingly if you have samples.
 
-struct BhattacharyyaDist <: SemiMetric end
-
-struct HellingerDist <: Metric end
+struct BhattacharyyaDist <: Distance end
+MetricType(::Type{BhattacharyyaDist}) = IsSemiMetric
 
+struct HellingerDist <: Distance end
+MetricType(::Type{HellingerDist}) = IsMetric
 
 # Bhattacharyya coefficient
 

src/bregman.jl

@@ -12,7 +12,7 @@ of good automatic differentiation packages.
 
 function evaluate(dist::Bregman, p::AbstractVector, q::AbstractVector)
 """
-struct Bregman{T1 <: Function, T2 <: Function, T3 <: Function} <: PreMetric
+struct Bregman{T1 <: Function, T2 <: Function, T3 <: Function} <: Distance
     F::T1
     ∇::T2
     inner::T3