Estimation under group actions: Recovering orbits from invariants

Moment-based metrics for molecules computable from cryogenic electron microscopy images

“…In (12), the first moment m 1 is indexed by r and is thus an N =2 length vector. Note that in (5), Δϕ j = 2πj=N À 2π for j = 1,…,2N, but since e inΔϕ in ( 13) is 2π=n periodic, we have that e in 2πÀΔϕ ð Þ = e ÀinΔϕ , and hence, m 2 r,r 0 , Δϕ j = m 2 r,r 0 ,Δϕ j + N . Thus, Δϕ j for j = 1,…,N is redundant and we consider only Δϕ j for j = N + 1,…,2N, which enumerates 0,2π ½ .…”

“…To better approximate the L 2 norm in the continuous case, we scale the difference of each entry m 2 j,k 1 ,k 2 ð Þby ffiffiffiffiffiffiffiffiffiffiffi r k1 r k2 p so that the squared norm is scaled by r k1 r k2 . More precisely, we define weighted ℓ 2 norms ∥ Á ∥ w1 and ∥ Á ∥ w2 on ℝ N =2 and ℝ N × N =2 × N =2 , as described in (5). Let Φ and Φ 0 be two different molecules, and m 1 ,m 2 and m 1 0 , m 2 0 be the first and second moment tensors, respectively, from two different molecules.…”

“…This needs only to be done once and does not need to be recomputed for each molecule. Calculating this matrix takes O PL 5 À Á time and uses O PL 5 À Á space. For the calculation of the least-squares matrix itself, we precompute A ℓ 0 ⊗A ℓ ð Þ U n ℓ,ℓ 0 , for ℓ,ℓ 0 ,n as described in (25).…”

“…Kam's method has helped push the theoretical understanding of the reconstruction processunder certain conditions, it is a provable algorithm and provides bounds for the estimated structure's quality in terms of the noise level and the number of images. (2)(3)(4)(5)(6)(7)(8) Kam's method also enjoys other remarkable properties:…”

Algebraic Theory of Phase Retrieval

“…A wide variety of MRA models have been studied in recent years (see for example [41,10,16,4]), mainly motivated by the single-particle cryo-electron microscopy (cryo-EM) technology, as discussed in Section 6.1. In [2,41], an intimate connection between the method of moments and the sample complexity of the MRA problem was uncovered.…”