动态轨迹估计
Dynamic Trace Estimation
摘要：
我们研究了动态版本的隐式迹估计问题。由于访问一个oracle计算矩阵向量乘法与一个动态变化的矩阵A，我们的目标是保持一个准确的近似A的轨迹使用尽可能少的乘法。我们提出了一个实用的算法来解决这个问题，并证明，在一个自然的设置，其复杂性是二次优于标准的解决方案，反复应用哈钦森的随机跟踪估计。我们还提供了一种改进的算法，假设对动态矩阵A的假设略强。我们支持我们的理论与实证结果，显示显着的计算改进机器学习和网络科学中的三个应用程序：跟踪时刻的海森谱密度在神经网络优化，计数三角形，并估计自然连接的动态变化的图形。
英文摘要：
We study a dynamic version of the implicit trace estimation problem. Given access to an oracle for computing matrix-vector multiplications with a dynamically changing matrix A, our goal is to maintain an accurate approximation to A's trace using as few multiplications as possible. We present a practical algorithm for solving this problem and prove that, in a natural setting, its complexity is quadratically better than the standard solution of repeatedly applying Hutchinson's stochastic trace estimator. We also provide an improved algorithm assuming slightly stronger assumptions on the dynamic matrix A. We support our theory with empirical results, showing significant computational improvements on three applications in machine learning and network science: tracking moments of the Hessian spectral density during neural network optimization, counting triangles, and estimating natural connectivity in a dynamically changing graph.
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