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

Equation 6 · AI Agents & Systems

Embeddings and the Geometry of Similarity

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

Beyer and colleagues proved the canonical result. Under broad conditions on the data and query distributions — much broader than independence 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 everything else. The authors call such a query…

Meanings in this article

• $d_m$: the writing.
• $\mathbb{E}$: The expected value operator: the probability-weighted average of the quantity inside its brackets.

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