General Metric properties

, $d(x,y) = 0 \iff x=y$, $d(x,y) = d(y,x)$

Deep Breakdown

What it tells me (Situation Awareness): A metric (or distance function) $d(x, y)$ is a function that satisfies three properties: non-negativity and identity ($d(x,y) \geq 0$, with $d(x,y)=0$ iff $x=y$), symmetry ($d(x,y) = d(y,x)$), and the triangle inequality ($d(x,z) \leq d(x,y) + d(y,z)$). These three axioms define what it means to be a valid "distance." Any function satisfying all three is a metric, and you can use it to define k-NN, clustering, and any distance-based algorithm. If a proposed "distance" fails any axiom, it is not a valid metric and may produce nonsensical results.

Flowchart

Real-world use

Where it is used

When choosing or defining a distance measure for ML algorithms (k-NN, k-means, DBSCAN, hierarchical clustering), verifying that a custom similarity/distance function is mathematically valid, and when reading papers that define novel similarity measures that claim to be metrics.

Problem it solves

Not every function that "looks like" a distance is a valid metric. For example, if symmetry fails ($d(A,B) \neq d(B,A)$), the nearest-neighbor relationship becomes asymmetric and direction-dependent — which breaks many algorithms. The metric axioms guarantee a mathematically consistent notion of distance.

How it solves the problem

Each axiom prevents a specific failure mode: non-negativity prevents negative "distances" (meaningless as a length), identity prevents two distinct points from having zero distance (they would be indistinguishable), symmetry ensures "distance from A to B" equals "distance from B to A," and the triangle inequality ensures no shortcut is shorter than the direct path.

Example in a business

A DS implements a custom "business distance" for CRM segmentation: $d_{\text{biz}}(A,B) = $ weighted combination of customer feature differences. They verify all three metric properties before using it in k-NN: (1) can it be negative? (2) does $d_{\text{biz}}(A,A) = 0$? (3) is it symmetric? (4) does the triangle inequality hold? Only after all checks pass do they deploy it.

The formula, piece by piece

Three separate properties — all must hold simultaneously:

1. $d(x,y) \geq 0$ (non-negativity) and $d(x,y) = 0 \Leftrightarrow x = y$ (identity of indiscernibles)
2. $d(x,y) = d(y,x)$ (symmetry)
3. $d(x,z) \leq d(x,y) + d(y,z)$ (triangle inequality — covered in its own file)

The formula in words

1. "Distance is never negative. If the distance between two points is zero, they must be the same point."
2. "The distance from A to B equals the distance from B to A."
3. "Going directly from x to z is never longer than going via y." Tiny example with Euclidean distance in 1D ($d(x,y) = |x-y|$):

• $d(3, 7) = 4 \geq 0$. $d(5,5) = 0$, and $5=5$. $\checkmark$
• $d(3,7) = 4 = d(7,3)$. $\checkmark$
• $d(3,9) = 6 \leq d(3,7) + d(7,9) = 4 + 2 = 6$. $\checkmark$

Critical Concepts

• All three must hold: a function failing even one axiom is not a metric.
• Euclidean, Manhattan, Chebyshev: all satisfy all three properties and are valid metrics.
• Cosine similarity is NOT a metric: $\text{sim}(x,x) = 1 \neq 0$. But cosine distance ($1 - \text{cosine similarity}$) does satisfy all metric properties.
• KL divergence is NOT a metric: it is asymmetric ($KL(P\|Q) \neq KL(Q\|P)$ in general) and does not satisfy the triangle inequality.
• Near-zero distances: in floating point, $d(x,y)$ may not be exactly 0 even for identical points. Use np.isclose rather than exact equality.
• Metric space: a set equipped with a metric is called a metric space.

Many ML convergence proofs require a metric space structure.

Step-by-step calculation (tiny example)

Step 1: Verify non-negativity and identity Points $A=(1,2)$, $B=(4,6)$ using $d = \|A-B\|_2$. $d(A,B) = \sqrt{9+16} = 5 \geq 0$. $\checkmark$ $d(A,A) = 0$ and $A=A$. $\checkmark$

Step 2: Verify symmetry $d(A,B) = 5$. $d(B,A) = \sqrt{(1-4)^2+(2-6)^2} = \sqrt{9+16} = 5$. Equal. $\checkmark$

Step 3: Verify triangle inequality with a third point $C = (1,6)$. $d(A,C) = \sqrt{0+16} = 4$. $d(A,B) + d(B,C) = 5 + \sqrt{9+0} = 5+3 = 8 \geq 4$. $\checkmark$

Code examples

Beginner example (Python)

import numpy as np
from itertools import combinations   # -- Verify metric properties for L2 distance ---

def euclidean(a, b):
    return np.linalg.norm(np.array(a) - np.array(b))   # Test points
points = [(1,2), (4,6), (1,6), (4,2)]
n = len(points)   # 1. Non-negativity and identity
print("=== Non-negativity and identity ===")
for p in points:
    same = euclidean(p, p)
print(f"d({p},{p}) = {same:.4f} (should be 0: {np.isclose(same, 0)})")
for a, b in combinations(points, 2):
    d = euclidean(a, b)
print(f"d({a},{b}) = {d:.4f} >= 0: {d >= 0}")   # 2. Symmetry
print("\n=== Symmetry ===")
for a, b in combinations(points, 2):
    print(f"d({a},{b}) = {euclidean(a,b):.4f}, d({b},{a}) = {euclidean(b,a):.4f}", f"| sym: {np.isclose(euclidean(a,b), euclidean(b,a))}")   # 3. Triangle inequality
print("\n=== Triangle Inequality ===")
for a, b, c in [(points[0], points[1], points[2])]:
    lhs = euclidean(a, c)
rhs = euclidean(a, b) + euclidean(b, c)
print(f"d({a},{c}) = {lhs:.4f} <= d({a},{b})+d({b},{c}) = {rhs:.4f}: {lhs <= rhs + 1e-10}")

Beginner example (R)

# -- Verify metric properties for L2 ---
euclidean <- function(a, b) sqrt(sum((a - b)^2))
points <- list(c(1,2), c(4,6), c(1,6), c(4,2))  # Non-negativity and identity
cat("=== Non-negativity and identity ===\n")
for (p in points)
cat(sprintf("d(p,p) = %.4f\n", euclidean(p, p)))  # Symmetry
cat("\n=== Symmetry ===\n")
for (i in 1:3)
for (j in (i+1):4) { a <- pointsi; b <- pointsj
  cat(sprintf("d(A,B)=%.4f, d(B,A)=%.4f, sym=%s\n", euclidean(a,b), euclidean(b,a), isTRUE(all.equal(euclidean(a,b), euclidean(b,a)))))
}

Professional example (Python)

import numpy as np
import pandas as pd   # -- Business scenario: validate a custom business distance ---
# A DS proposes a weighted Euclidean distance for CRM segmentation.
# Weights: [2.0, 1.0, 0.5] for [recency, frequency, monetary].

def weighted_euclidean(a: np.ndarray, b: np.ndarray, weights: np.ndarray) -> float:
    """Weighted Euclidean distance: sqrt(sum(w_i*(a_i-b_i)^2))."""
    return float(np.sqrt(np.sum(weights * (a - b)**2)))
np.random.seed(42)
N = 20
weights = np.array([2.0, 1.0, 0.5])
customers = np.random.uniform(0, 100, (N, 3))   # RFM features # Property 1: non-negativity and identity
neg_violations = 0
ident_violations = 0
for i in range(N):
    for j in range(N):
        d = weighted_euclidean(customers[i], customers[j], weights)
    if d < 0:
        neg_violations += 1
    if i == j and not np.isclose(d, 0):
        ident_violations += 1   # Property 2: symmetry
sym_violations = 0
for i in range(N):
    for j in range(i+1, N):
        dij = weighted_euclidean(customers[i], customers[j], weights)
    dji = weighted_euclidean(customers[j], customers[i], weights)
if not np.isclose(dij, dji):
    sym_violations += 1   # Property 3: triangle inequality
tri_violations = 0
for i in range(min(N, 10)):
    for j in range(min(N, 10)):
        for k in range(min(N, 10)):
            dik = weighted_euclidean(customers[i], customers[k], weights)
        dij = weighted_euclidean(customers[i], customers[j], weights)
    djk = weighted_euclidean(customers[j], customers[k], weights)
if dik > dij + djk + 1e-10:
    tri_violations += 1
print("Weighted Euclidean metric validation:")
print(f" Non-negativity/identity violations : {neg_violations + ident_violations}")
print(f" Symmetry violations : {sym_violations}")
print(f" Triangle inequality violations : {tri_violations}")
print(f"\n VALID METRIC: {(neg_violations + ident_violations + sym_violations + tri_violations) == 0}")

Source: MML2 p. 63

#linear-algebra#spatial-distance-norms-and-analytic-geometry#distance-metrics-and-norms