基数稀疏性：在矩阵-矩阵乘法和机器学习中的应用
Cardinality Sparsity: Applications in Matrix-Matrix Multiplications and Machine Learning
AI总结：
本文提出“基数稀疏性”概念，以唯一值数量刻画张量稀疏性，并证明其可改进深度学习与张量回归、大幅加速二进制矩阵乘法及降低内存占用。
摘要：
高维数据已在各科学领域无处不在，但也带来了计算和统计方面的挑战。应对这些挑战的一种常见方法是利用稀疏性。本文提出了一种称为基数稀疏性的稀疏性新概念。概括而言，若一个张量仅包含少量唯一值，我们就将其定义为稀疏张量。我们证明，基数稀疏性能够在统计和计算两方面改进深度学习与张量回归。在此过程中，我们推广了这些领域中近期的统计理论。最重要的是，我们表明基数稀疏性在高维数据分析之外具有一个极其强大的应用：它可以显著加速矩阵-矩阵乘法。例如，我们证明基数稀疏性催生了用于二进制矩阵乘法的算法，这些算法以相当大的幅度优于最先进算法。此外，这种稀疏性的另一个关键方面是最小化内存使用。通过在压缩域中执行矩阵乘法，我们可以显著降低存储输入数据所需的内存量。
英文摘要：
High-dimensional data has become ubiquitous across the sciences but presents computational and statistical challenges. A common approach to addressing these challenges is through sparsity. In this paper, we introduce a new concept of sparsity, called cardinality sparsity. Broadly speaking, we define a tensor as sparse if it contains only a small number of unique values. We demonstrate that cardinality sparsity can improve deep learning and tensor regression both statistically and computationally. Along the way, we generalize recent statistical theories in these fields. Most importantly, we show that cardinality sparsity has a strikingly powerful application beyond high-dimensional data analysis: it can significantly speed up matrix-matrix multiplications. For instance, we demonstrate that cardinality sparsity leads to algorithms for binary-matrix multiplication that outperform state-of-the-art algorithms by a substantial margin. Additionally, another crucial aspect of this sparsity is minimizing memory usage. By executing matrix multiplication in the compressed domain, we can significantly lower the amount of memory needed to store the input data.
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