Seonglae ChoNumber of vectors you can cram into a space nearly perpendicular grows exponentially with the number of dimensions
The Johnson-Lindenstrauss lemma is a fundamental result in high-dimensional geometry that demonstrates a remarkable property of high-dimensional spaces: the maximum number of vectors that can be placed in a space while remaining nearly perpendicular to each other grows exponentially with the number of dimensions.
The lemma reveals that as dimensionality increases, the capacity to accommodate nearly orthogonal vectors expands dramatically, following an exponential relationship:
where
N represents the number of dimensions and ε (epsilon) is a small positive constant that controls the degree of near-perpendicularity.- Dimensionality reduction: The lemma enables efficient projection of high-dimensional data into lower dimensions while approximately preserving pairwise distances between points.
- Computational efficiency: By reducing dimensions, algorithms can process data more quickly without significant loss of structural information.
- Connection to the curse of dimensionality: While high dimensions present computational challenges, the Johnson-Lindenstrauss lemma provides a powerful tool for mitigating these issues through controlled dimensionality reduction.
