Unbiasing Procedures for Scale-Invariant Multi-Reference Alignment - IPAM at UCLA

Explore unbiasing procedures for scale-invariant multi-reference alignment in this 51-minute lecture by Anna Little from the University of Utah. Delve into the mathematical analysis of multi-reference alignment problems, focusing on recovering hidden signals from noisy observations. Examine the generalization of the classic problem by incorporating random dilations alongside random translations and additive noise. Discover multiple approaches to solving this challenging model based on translation invariant representations. Learn about wavelet-based unbiasing procedures for unknown dilation distributions and more accurate methods for known distributions. Investigate the convergence rates of estimators and their dependence on sample size and noise levels. Explore the application of signal processing tools in distribution learning from biased, sparse batches. Gain insights into cryoelectron microscopy challenges, wavelet transforms, and signal recovery techniques through theoretical discussions and numerical experiments.

Introduction
Outline
Introduction to cryoelectron microscopy
Challenges of cryoelectron microscopy
Orientation of frozen molecules
Deformations of molecules
Challenges of multireference alignment
Onedimensional multireference alignment model
Other sources of noise
Scaling the translation
Rotations
Why consider this model
Classic MRA
General MRA
Wavelet Transform
Wavelet Invariant
Translation Invariant
Results
Observations
Order K Wavelet Variant estimator
Air Decay
Pros and Cons
Infinite Sample Limit
Key Point
Signal Recovery
Error Decay
Full Signal Recovery
Power Spectrum
Key Observation
Inversion
Future questions
Learning the dilation distribution