Computation and Applications of Euclidean and Normed Representations of Massive Data

2026-09-06

AI Summary

论文研究了大规模数据的欧几里得空间和$\ell_p$-范数表示，提出高效算法计算这些表示，并讨论了它们在数据压缩和洞察提取中的应用。

Abstract

This thesis investigates Euclidean-space and $\ell_p$-norm representations of different forms of data, with a focus on efficient algorithms for computing these representations in settings where the amount of data is very large. The applications of such representations are also discussed: they may be used to summarize the data in a more compact form for downstream tasks while preserving its most salient properties; and further, they may also be used to extract insights about the data that may not be apparent in its original form. This thesis presents novel algorithms for computing Euclidean representations in the case where the data comes with linear structure, as well as in the case where the data comes only with metric, or distance structure. In addition, we give algorithms for computing $\ell_p$-norm representations in linear structured cases. The analyses of these algorithms use tools from geometry, probability, and optimization, and these tools are used to illuminate other algorithms for similar problems.

Problem

• Euclidean representations
• ell_p-norm representations
• massive data
• efficient algorithms
• data summarization

Innovation

• Euclidean representations
• ℓp-norm representations
• large-scale data
• efficient algorithms
• geometric and probabilistic tools