Equation 6 · Part 3 · Embeddings and the Geometry of Similarity

Symbol P_m

\lim_{m \to \infty} \operatorname{var}\!\left( \frac{d_m(P_m, Q_m)^p}{\mathbb{E}\left[ d_m(P_m, Q_m)^p \right]} \right) = 0 \;\;\Longrightarrow\;\; \lim_{m \to \infty} \Pr\!\left[ D_{\max}^{(m)} \le (1 + \varepsilon)\, D_{\min}^{(m)} \right] = 1

P_m

What this part means

$P_m$ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

$P_m$ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Full expressionSymbol P_mArticle meaning

The passage around this formula

…and identical distribution across dimensions — as dimensionality rises the distance to the nearest data point approaches the distance to the farthest. Formally, writing $d_m$ for the distance function in m dimensions, $P_m$ for a data point and $Q_m$ for a query point: $\lim_{m \to \infty} \operatorname{var}\!\left( \frac{d_m(P_m, Q_m)^p}{\mathbb{E}\left[ d_m(P_m, Q_m)^p \right]} \right) = 0 \;\;\Longrightarrow\;\; \lim_{m \to \infty} \Pr\!\left[ D_{\max}^{(m)} \le (1 + \varepsilon)\, D_{\min}^{(m)} \right] = 1$ . for every $\varepsilon$ > 0 [ 11 ] . The condition is on the relative variance of the distance distribution: when distances stop varying much relative to their own mean, the nearest neighbour stops being distinguishable from…

Read this part in the article →

Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

See this notation across published equations →

Sources cited in the surrounding passage

[11] When Is "Nearest Neighbor" Meaningful? ↗

These citations provide research context; check each source for the exact claim it supports.