高维模拟数据低秩压缩的确定性矩阵草图
Deterministic matrix sketches for low-rank compression of high-dimensional simulation data
摘要：
在科学应用中出现的矩阵经常承认线性低秩近似，由于平滑的物理和/或时间域的问题。在大规模问题中，计算最优低秩近似值可能非常昂贵。矩阵草图通过将输入矩阵减少到较小但具有代表性的矩阵来解决这个问题，该矩阵通过低维线性嵌入。如果通过嵌入产生的草图矩阵捕获了原始矩阵的足够的几何性质，则可以获得接近最优逼近。在矩阵素描中所做的大部分工作都集中在随机投影上。另外，在这项工作中，确定性矩阵草图生成粗糙的表示，兼容相应的PDE解决，被认为是在计算的奇异值分解和矩阵插值分解。在这项工作中的确定性草图的方法有许多优势，随机草图。一般来说，随机草图是数据不可知的，而所提出的草图绘制方法利用复杂PDE系统中生成的数据内的结构。这些确定性草图通常更快，需要访问输入矩阵的一小部分，并且不需要显式构造。一种新颖的单程，即，还提出了需要对输入进行一次读取的功率迭代算法。当数据的奇异值衰减较慢时，幂迭代法在改善低秩近似方面特别有效。最后，理论误差界和估计，以及在三个应用问题的数值结果，提供。
英文摘要：
Matrices arising in scientific applications frequently admit linear low-rank approximations due to smoothness in the physical and/or temporal domain of the problem. In large-scale problems, computing an optimal low-rank approximation can be prohibitively expensive. Matrix sketching addresses this by reducing the input matrix to a smaller, but representative matrix via a low-dimensional linear embedding. If the sketch matrix produced by the embedding captures sufficient geometric properties of the original matrix, then a near-optimal approximation may be obtained. Much of the work done in matrix sketching has centered on random projection. Alternatively, in this work, deterministic matrix sketches which generate coarse representations, compatible with the corresponding PDE solve, are considered in the computation of the singular value decomposition and matrix interpolative decomposition. The deterministic sketching approaches in this work have many advantages over randomized sketches. Broadly, randomized sketches are data-agnostic, whereas the proposed sketching methods exploit structures within data generated in complex PDE systems. These deterministic sketches are often faster, require access to a small fraction of the input matrix, and do not need to be explicitly constructed. A novel single-pass, i.e., requiring one read over the input, power iteration algorithm is also presented. The power iteration method is particularly effective in improving low-rank approximations when the singular value decay of data is slow. Finally, theoretical error bounds and estimates, as well as numerical results across three application problems, are provided.
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