Three-dimensional structure determination of molecules without crystallization

Transcription

1 Three-dimensional structure determination of molecules without crystallization Amit Singer Princeton University, Department of Mathematics and PACM June 4, 214 Amit Singer (Princeton University) June / 34

2 Multi-Reference Alignment Let G be a group of transformations acting on a vector space X. Estimate x X and g 1,...,g n G from n measurements of the form y i = P g i x +ǫ i, i = 1,...,n where ǫ i are independent noise terms. P : X Y is a linear operator, where Y is the measurement space and X is the object space. Amit Singer (Princeton University) June / 34

3 Multi-Reference Alignment: Single Particle Reconstruction (Ex. 1) G = SO(3) X = L 2 (R 3 ) g φ(r) = φ(g 1 r) for φ X,g G,r R 3. Y = R N N P is the discrete X-ray transform from computerized tomography Image formation model: I i = P g i φ+ǫ i Amit Singer (Princeton University) June / 34

4 Multi-Reference Alignment: Registration (Ex. 2) G = SO(d) X = harmonic polynomials of degree t over S d 1 R d (band-limited functions on the unit sphere) g h(ω) = h(g 1 ω) for h X,g G,ω S d 1. Y = R L P is a sampling operator h (h(ω 1 ),h(ω 2 ),...,h(ω L )) where ω 1,...,ω L S d 1. Signal/Image formation model: y i = P g i f +ǫ i Amit Singer (Princeton University) June / 34

5 Registration (d = 1, periodic signals) Clean Template Clean Template Noisy Reference Very Noisy Reference Shifted Template Noisy Reference Noisy Reference Very Noisy Reference High SNR: Matched filtering, phase-correlation Low SNR: Failure of pairwise comparisons. How to use information in all signals? Invariants: Power spectrum, higher order spectra (bispectrum). Is maximum likelihood estimation (MLE) computationally tractable? Amit Singer (Princeton University) June / 34

6 Multi-Reference Alignment: Shape Matching (Ex. 3) Basic se ng Generalized se ng Figure : Huang and Guibas, Consistent Shape Maps via Semidefinite Programming, Eurographics 213 Current approach: Use as input a set of pair-wise permutations (between any pair of shapes). Ignore origin of permutations as smooth deformations. Can we... Find all permutations (correspondences) simultaneously as the minimizer of one global cost function? Impose smoothness on permutations? Amit Singer (Princeton University) June / 34

7 Canonical optimization problem min g 1,g 2,...,g n G n i,j=1 f ij (g 1 i g j ) where G is some group of transformations, and f ij are non-linear functions over G. Application: finding ground states of interacting particle systems Amit Singer (Princeton University) June / 34

8 Single Particle Reconstruction using Cryo-Electron Microscopy Drawing of the imaging process: Amit Singer (Princeton University) June / 34

9 Single Particle Cryo-Electron Microscopy: Model Projection I i Molecule φ R i = Ri 1 Ri 2 Ri 3 T T T SO(3) Electron source Projection images I i (x,y) = φ(xr1 i +yri 2 +zri 3 )dz + noise. φ : R 3 R is the electric potential of the molecule. Cryo-EM problem: Find φ and R 1,...,R n given I 1,...,I n. Amit Singer (Princeton University) June / 34

10 Toy Example Amit Singer (Princeton University) June / 34

11 E. coli 5S ribosomal subunit: sample images Fred Sigworth, Yale Medical School Movie by Lanhui Wang and Zhizhen (Jane) Zhao Amit Singer (Princeton University) June / 34

12 Algorithmic Pipeline Particle Picking: manual, automatic or experimental image segmentation. Class Averaging: classify images with similar viewing directions, register and average to improve their signal-to-noise ratio (SNR). Orientation Estimation: common lines Three-dimensional Reconstruction: a 3D volume is generated by a tomographic inversion algorithm. Iterative Refinement Assumptions for today s talk: Trivial point-group symmetry Homogeneity (no structural variability) = Heterogeneity Amit Singer (Princeton University) June / 34

13 What mathematics do we use to solve the problem? Tomography Convex optimization and semidefinite programming Representation theory of SO(3) Random matrix theory Non-linear dimensionality reduction, (vector) diffusion maps Fast (randomized) algorithms... Amit Singer (Princeton University) June / 34

14 Orientation Estimation: Fourier projection-slice theorem (x ij,y ij ) R i c ij c ij = (x ij,y ij,) T Projection I i Î i (x ji,y ji ) 3D Fourier space R i c ij = R j c ji Projection I j Î j 3D Fourier space Amit Singer (Princeton University) June / 34

15 Angular Reconstitution (Vainshtein and Goncharov 1986, Van Heel 1987) Amit Singer (Princeton University) June / 34

16 Experiments with simulated noisy projections Each projection is 129x129 pixels. SNR = Var(Signal) Var(Noise), (a) Clean (b) SNR=2 (c) SNR=2 1 (d) SNR=2 2 (e) SNR=2 3 (f) SNR=2 4 (g) SNR=2 5 (h) SNR=2 6 (i) SNR=2 7 (j) SNR=2 8 Amit Singer (Princeton University) June / 34

17 Fraction of correctly identified common lines and the SNR Define common line as being correctly identified if both radial lines deviate by no more than 1 from true directions. log 2 (SNR) p Amit Singer (Princeton University) June / 34

18 Least Squares Approach Consider the unit directional vectors as three-dimensional vectors: c ij = (x ij,y ij,) T, c ji = (x ji,y ji,) T. Being the common-line of intersection, the mapping of c ij by R i must coincide with the mapping of c ji by R j : (R i,r j SO(3)) R i c ij = R j c ji, for 1 i < j n. Least squares: min R 1,R 2,...,R n SO(3) R i c ij R j c ji 2 i j Search space is exponentially large and non-convex. Amit Singer (Princeton University) June / 34

19 Quadratic Optimization Under Orthogonality Constraints Quadratic cost: i j R ic ij R j c ji 2 Quadratic constraints: R T i R i = I 3 3 (det(r i ) = +1 constraint is ignored) We approximate the solution using SDP and rounding (S, Shkolnisky 211) Related to: MAX-CUT (Goemans and Williamson 1995) Generalized Orthogonal Procrustes Problem (Nemirovski 27) PhaseLift (Candes, Strohmer, Voroninski 213) Non-commutative Grothendick Problem (Naor, Regev, Vidick 213) Robust version Least Unsquared Deviations (Wang, S, Wen 213) min R 1,R 2,...,R n SO(3) R i c ij R j c ji i j Amit Singer (Princeton University) June / 34

20 Spectral Relaxation for Uniformly Distributed Rotations C is the 2n 2n matrix ( the common lines matrix ) with [ ] [ ] [ C ij = c ji c ij T xji [ ] xji x = xij y ij = ij x ji y ij, C y ji x ij y ji y ii = ij y ji Rewrite the least squares objective function as max R 1,...,R n SO(3) i j R i c ij,r j c ji = max R 1,...,R n SO(3) rt 1 Cr 1 +r T 2 Cr 2 +r T 3 Cr 3 R i = r 1 (i) T r 2 (i) T r 3 (i) T By symmetry, if rotations are uniformly distributed over SO(3), then the top eigenvalue of C has multiplicity 3 and corresponding eigenvectors are r 1,r 2,r 3 (first order spherical harmonics) from which we recover R 1,R 2,...,R n. Amit Singer (Princeton University) June / 34 ]

21 Spectrum of C Numerical simulation with n = 1 rotations sampled from the Haar measure; no noise. Bar plot of positive (left) and negative (right) eigenvalues of C: λ λ Eigenvalues: λ l n ( 1)l+1 l(l+1), l = 1,2,3,... (1 2, 1 6, 1 12,...) Multiplicities: 2l + 1. Two basic questions: 1 Why this spectrum? Answer: Representation Theory of SO(3) (Hadani, S, 211) 2 Is it stable to noise? Answer: Yes, due to random matrix theory. Amit Singer (Princeton University) June / 34

22 Numerical Spectra of C, n = λ (a) SNR= λ (b) SNR= λ (c) SNR= λ (d) SNR= λ (e) SNR= λ (f) SNR= λ (g) SNR= λ (h) SNR= λ (i) SNR=2 8 Amit Singer (Princeton University) June / 34

23 Estimation Error True rotations: R 1,...,R n. Estimated rotations: ˆR1,..., ˆR n. Registration: Ô = argmin O SO(3) n R i OˆR i 2 F i=1 Mean squared error: MSE = 1 n n R i ÔˆR i 2 F i=1 Amit Singer (Princeton University) June / 34

24 MSE for n = 1 SNR p MSE Model fails at low SNR. Why? Wigner model is too simplistic cannot have n 2 independent random variables from just n images. C ij = K(P i,p j ), random kernel matrix (Koltchinskii and Giné 2, El-Karoui 21). Kernel is discontinuous (Cheng, S, 213) Amit Singer (Princeton University) June / 34

25 From Common Lines to Maximum Likelihood The images contain more information than that expressed by the common lines. Common line based algorithms can succeed only at high SNR similar to other methods that are based on pairwise comparison (recall registration). (Quasi) Maximum Likelihood: We would like to try all possible rotations R 1,...,R n and choose the combination for which the agreement on the common lines as observed in the images is maximal. Computationally intractable: exponentially large search space. Amit Singer (Princeton University) June / 34

26 Maximum Likelihood Solution using SDP Main idea: Lift SO(3) to Sym(S 2 ) Suppose ω 1,ω 2,...,ω L S 2 are evenly distributed points over the sphere (e.g., a spherical t-design). g SO(3) π S L ( Assignment, Procrustes ) (in practice no explicit construction is needed). Amit Singer (Princeton University) June / 34

27 Convex Relaxation of Permutations arising from Rotations The convex hull of permutation matrices are the doubly stochastic matrices (Birkhoff-von Neumann polytope): Π R L L, Π, 1 T Π = 1 T, Π1 = 1 Rotation by an element of SO(3) should map nearby-points to nearby-points. More precisely, SO(3) preserves inner products: Ω ij = ω i,ω j ǫ = ω π(i),ω π(j) = Ω π(i),π(j) ΠΩΠ T ǫ = Ω = ΠΩ ǫ = ΩΠ Axis of rotation (Euler s rotation theorem): Tr(Π) ǫ 2 Notice: We are discretizing S 2, not SO(3) (substantial gain in computational complexity) Amit Singer (Princeton University) June / 34

28 Convex Relaxation of Cycle Consistency: SDP Let G be a block-matrix of size n n with G ij R L L. We want G ij = Π i Π T j. G is PSD, G ii = I L L, and rank(g) = L (the rank constraint is dropped). Convex relaxation of search space (putting it all together): G R nl nl G G ii = I L L (cycle consistency) G ij G ij 1 = 1 1 T G ij = 1 T (doubly stochastic) G ij Ω = ǫ ΩG ij Tr(G ij ) ǫ 2 (SO(3)) Related work: Charikar, Makarychev, Makarychev, Near optimal algorithms for Unique Games (STOC 26) Amit Singer (Princeton University) June / 34

29 Maximum Likelihood: Linear Objective Function The common line depends on R i R T j. Log-Likelihood function is of the form f ij (R i Rj T ) i j Nonlinear in R i R T j, but linear in G. Proof by picture. Amit Singer (Princeton University) June / 34

30 Linear Objective: Proof by picture Amit Singer (Princeton University) June / 34

31 Tightness of the SDP p σ p σ Figure : Fraction of trials (among 1) on which rank recovery was observed. Left: Generalized Orthogonal Procrustes. Right: multi-reference alignment (registration). The solution of the SDP has the desired rank up to a certain level of noise (w.h.p). In other words, even though the search-space is exponentially large, we typically find the MLE in polynomial time. This is a viable alternative to heuristic methods such as EM and alternating minimization. The SDP gives a certificate whenever it finds the MLE. Amit Singer (Princeton University) June / 34

32 Are we there yet? The SDP approach can be used in a variety of problems where the objective function is a sum of pairwise interactions Need better theoretical understanding for the phase transition behavior and conditions for tightness of the SDP. Need better computational tools for solving large scale SDPs. (we use ADMM). Amit Singer (Princeton University) June / 34

33 References A. Singer, Y. Shkolnisky, Three-Dimensional Structure Determination from Common Lines in Cryo-EM by Eigenvectors and Semidefinite Programming, SIAM Journal on Imaging Sciences, 4 (2), pp (211). R. Hadani, A. Singer, Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I The intrinsic reconstitution algorithm, Annals of Mathematics, 174 (2), pp (211). L. Wang, A. Singer, Z. Wen, Orientation Determination of Cryo-EM images using Least Unsquared Deviations, SIAM Journal on Imaging Sciences, 6(4), pp (213). A. S. Bandeira, M. Charikar, A. Singer, A. Zhu, Multireference Alignment using Semidefinite Programming, 5th Innovations in Theoretical Computer Science (ITCS 214). A. S. Bandeira, Y. Khoo, A. Singer, Open problem: Tightness of maximum likelihood semidenite relaxations, COLT 214. Also available at Amit Singer (Princeton University) June / 34

34 Thank You! Funding: NIH/NIGMS R1GM92 AFOSR FA Simons Foundation LTR DTD Amit Singer (Princeton University) June / 34