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Published equation contexts

lim⁡d→∞E ⁣[Dmax⁡k−Dmin⁡kd1/k−1/2]=C⋅1(k+1)1/k12k+1\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}}

Why this formula appears here

Aggarwal, Hinneburg and Keim then showed the rate depends on the norm. For uniform data under an LkL_k metric, the expected gap between the farthest and nearest distances scales as lim⁡d→∞E ⁣[Dmax⁡k−Dmin⁡kd1/k−1/2]=C⋅1(k+1)1/k12k+1\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}}. with C constant, while typical distances themselves grow like d1/kd^{1/k} . Relative contrast therefore decays with dimension, and the constant shrinks as k rises — which is why they conclude that L1L_1 is preferable to L2L_2 in high dimensions, L2L_2 to L3L_3 , and that fractional norms preserve contrast better still [ 12 ] .

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dd

Symbol d

d occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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Dmax⁡kD_{\max}^{k}

Symbol D_max^k

DmD_maxkx^k occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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Dmin⁡kD_{\min}^{k}

Symbol D_min^k

DmD_minkn^k occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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d1/k−1/2d^{1/k - 1/2}

Symbol d^1/k - 1/2

d1d^1/k - 1/2 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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Dmax⁡k−Dmin⁡kD_{\max}^{k} - D_{\min}^{k}

Numerator: D_max^k - D_min^k

The complete quantity above the fraction bar.

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(k+1)1/k(k+1)^{1/k}

Denominator: (k+1)^1/k

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

lim⁡d→∞E ⁣[Dmax⁡k−Dmin⁡kd1/k−1/2]=C⋅1(k+1)1/k12k+1,\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}},

Equation 9 · AI Agents & Systems

Embeddings and the Geometry of Similarity

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Aggarwal, Hinneburg and Keim then showed the rate depends on the norm. For uniform data under an LkL_k metric, the expected gap between the farthest and nearest distances scales as lim⁡d→∞E ⁣[Dmax⁡k−Dmin⁡kd1/k−1/2]=C⋅1(k+1)1/k12k+1\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}}. with C constant, while typical distances themselves grow like d1/kd^{1/k} . Relative contrast therefore decays with dimension, and the constant shrinks as k rises — which is why they conclude that L1L_1 is preferable to L2L_2 in high dimensions, L2L_2 to L3L_3 , and that fractional norms preserve contrast better still [ 12 ] .

Meanings in this article

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