大规模数据的欧几里得与范数表示的计算及应用
Computation and Applications of Euclidean and Normed Representations of Massive Data
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中文总结 AI 辅助
本论文研究大规模数据的欧几里得与范数表示的高效计算算法及其在数据压缩和洞察提取中的应用，覆盖线性和度量结构情形。
中文摘要 AI 辅助
本论文研究了不同形式数据的欧几里得空间表示和$\ell_p$-范数表示，重点关注在数据量非常大的情况下计算这些表示的高效算法。同时讨论了此类表示的应用：它们可用于以更紧凑的形式汇总数据以供下游任务使用，同时保留数据最显著的性质；此外，它们还可用于提取原始形式中可能不明显的数据洞察。本论文提出了在数据具有线性结构以及数据仅具有度量或距离结构的情况下计算欧几里得表示的新算法。此外，我们给出了在线性结构情形下计算$\ell_p$-范数表示的算法。这些算法的分析使用了几何、概率和优化工具，并且这些工具被用于阐明解决类似问题的其他算法。
英文摘要
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.
发表机构
- TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO（芝加哥丰田技术研究所）
机构由 AI 辅助整理，请以论文原文为准。
