Seismic Waveform Inversion by Stochastic Optimization

Table 1

Summary of bounds, adapted from Avron and Toledo [14].

Estimator	Distribution of 𝑤	Variance of one sample	Bound on 𝐾 for (𝜖,𝛿) bound
Hutchinson
𝐻𝐾=(1/𝐾)∑^𝐾𝑗=1𝑤^⊤𝑗𝐴𝑤𝑗	Pr(𝑤𝑗=±1)=1/2	2(‖𝐴‖^2𝐹−∑^𝑁𝑖=1𝐴^2𝑖𝑖)	6𝜖^−2ln(2rank(𝐴)/𝛿)
Gaussian
𝐺𝐾=(1/𝐾)∑^𝐾𝑗=1𝑤^⊤𝑗𝐴𝑤𝑗	𝑤𝑗∈𝑁(0,1)	2‖𝐴‖^2𝐹	20𝜖^−2ln(2/𝛿)
Phase encoded
𝐿𝐾=(𝑁/𝐾)∑^𝐾𝑗=1𝑤^𝑇𝑗ℱ𝐴ℱ^𝑇𝑤𝑗	𝑤𝑗 drawn uniformly from {𝑒1,…,𝑒𝑁}	n/a	2𝜖^−2ln(4𝑛^2/𝛿)ln(4/𝛿)