## Applied Algebra Seminar: Estimating the covariance of a protein

Seminar | March 23 | 5:15-6:15 p.m. | 891 Evans Hall

David Dynerman, University of California, Berkeley

Department of Mathematics

In biological electron microscopy, experimentalists produce 2D images of unknown 3D objects (proteins) in unknown orientations, and process these images to recover the unknown 3D object. If we consider these unknown objects and orientations as random variables, the images produced by an experiment can be modeled by$$I = PX,$$

where $X$ is a random variable representing the 3D object and $P$ is a random linear operator producing an image of $X$.

In this situation, a natural question is: what statistics of $X$ (the object biologists would like to find) can we determine from samples of the random variable $I$ (the data available from experiment)?

This question also has biological motivation - for example, in certain situations the covariance matrix of $X$ can reveal biologically relevant information about the different shapes of a protein.

In this talk we will survey three recent papers by Amit Singer's (Princeton University, Applied Mathematics) group on this problem. We will present work of Katsevich, Katsevich and Singer [1] on estimating the mean and covariance of $X$, and briefly discuss two other related papers [2], [3].

[1] Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem E. Katsevich, A. Katsevich, A. Singer http://www.math.princeton.edu/~amits/publications/CovarianceSIIMS.pdf

[2] Covariance estimation using conjugate gradient for 3D classification in Cryo-EM J. Andén, E. Katsevich, A. Singer

[3] Denoising and Covariance Estimation of Single Particle Cryo-EM Images T. Bhamre, T. Zhang, A. Singer http://www.math.princeton.edu/~amits/publications/JSB2016-Published.pdf

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