2014/05/28 by Karl Wimmer, Yi Wu, Wimmer, Karl +3
Computer Science · Mathematics · #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.1405.7112
openalex publication_date 2014/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an implicit n× n matrix A with oracle access xTA x for any x∈ ℝn, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form (1)/(k)∑i=1k xiT A xi for xi∈ ℝn being i.i.d. for some special distribution. Our main results are summarized as follows. We give an exact characterization of the minimum variance unbiased estimator in the broad class of linear nonadaptive estimators (which subsumes all the existing known estimators). We also consider the query complexity lower bounds for any (possibly nonlinear and adaptive) estimators: (1) We show that any estimator requires Ω(1/ε) queries to have a guarantee of variance at most ε. (2) We show that any estimator requires Ω((1)/(ε2)log \frac1δ) queries to achieve a (1±ε)-multiplicative approximation guarantee with probability at least 1 - δ. Both above lower bounds are asymptotically tight. As a corollary, we also resolve a conjecture in the seminal work of Avron and Toledo (Journal of the ACM 2011) regarding the sample complexity of the Gaussian Estimator.