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Math research shows fewer projections needed to define point sets in high dimensions

By PulseAugur Editorial · [1 sources] ·

Researchers have investigated the minimum number of sorted projections required to uniquely determine a multiset of n points in k dimensions. While previous work established bounds of over (k-1)log2(n) directions needed and (k-1)n+1 directions sufficient, new findings suggest that a significantly smaller number of directions can suffice. By leveraging lemmas from coding theory, the study demonstrates that almost every choice of approximately (4k-2)ln(2n)ln(k log2(2n)) + k + 1 directions is enough, outperforming linear bounds for k<=10 and n>100. Further analysis shows that 2k-1 directions are insufficient for determining even three points, indicating the logarithmic lower bound is not tight. AI

RANK_REASON Academic paper detailing mathematical findings on geometric projections. [lever_c_demoted from research: ic=1 ai=0.1]

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Math research shows fewer projections needed to define point sets in high dimensions

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Academic paper detailing mathematical findings on geometric projections. [lever_c_demoted from research: ic=1 ai=0.1]
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  1. Hugging Face Daily Papers TIER_1 English(EN) ·

    How many sorted projections determine n points in k dimensions?

    Let $d_1,\dots,d_m$ be directions in $\mathbb{R}^k$, where $k\ge 2$. A multiset $X$ of $n$ points of $\mathbb{R}^k$ projects to the $m$ multisets of numbers $\{\{x\cdot d_j : x\in X\}\}$, or equivalently to the $m$ sorted lists of these numbers. How many directions are needed for…

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