Orientation Assignment in Cryo-EM - University of Chicago · 2014-07-24 · Orientation Assignment...

Orientation Assignment in Cryo-EM

Amit Singer

Princeton University

Department of Mathematics and Program in Applied and Computational Mathematics

July 24, 2014

Amit Singer (Princeton University) July 2014 1 / 30

Single Particle Reconstruction using cryo-EM

Schematic drawing of the imaging process:

The cryo-EM problem:


Orientation Estimation: Fourier projection-slice theorem

Projection Ii

Molecule φ

Electronsource

Ri ∈ SO(3)

Projection Ii

Projection Ij

Ii

Ij

3D Fourier space

3D Fourier space

(xij , yij )

(xji , yji )

Ri cij cij = (xij , yij , 0)T

Ri cij = Rj cji

Cryo-EM inverse problem: Find φ (and R1, . . . ,Rn) given I1, . . . , In.

n = 3: Vainshtein and Goncharov 1986, van Heel 1987

n > 3: S, Shkolnisky SIAM Imaging 2011

n > 3: Bandeira, Charikar, Chen, S, in preparation


Assumptions for today’s talk

Trivial point-group symmetry

Homogeneity


Experiments with simulated noisy projections

Each projection is 129x129 pixels.

SNR =Var(Signal)

Var(Noise),

(a) Clean (b) SNR=20 (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


Fraction of correctly identified common lines and the SNR

Define common line as being correctly identified if both radial linesdeviate by no more than 10 from true directions.

Fraction p of correctly identified common lines increases by PCA

log2(SNR) p

20 0.9970 0.980-1 0.956-2 0.890-3 0.764-4 0.575-5 0.345-6 0.157-7 0.064-8 0.028-9 0.019


Least Squares Approach

Projection Ii

Projection Ij

Ii

Ij

3D Fourier space

3D Fourier space

(xij , yij)

(xji , yji )

Ricij cij = (xij , yij , 0)T

Ricij = Rjcji

Least squares:

minR1,R2,...,Rn∈SO(3)

∑

i 6=j

‖Ricij − Rjcji‖2

Search space is exponentially large and non-convex.


Quadratic Optimization Under Orthogonality Constraints

Quadratic cost:∑

i 6=j ‖Ricij − Rjcji‖2

Quadratic constraints: RTi Ri = I3×3

(det(Ri ) = +1 constraint is ignored)

We approximate the solution using SDP and rounding(S, Shkolnisky 2011) Related to:

MAX-CUT (Goemans and Williamson 1995)

Generalized Orthogonal Procrustes Problem (Nemirovski 2007)

PhaseLift (Candes, Strohmer, Voroninski 2013)

Non-commutative Grothendick Problem (Naor, Regev, Vidick 2013)

“Robust” version – Least Unsquared Deviations (Wang, S, Wen 2013)


∑

i 6=j

‖Ricij − Rjcji‖


Detour: MAX-CUT

MAX-CUT: Given a weighted undirected graph with n vertices andnon-negative weights wij ≥ 0, split the vertices into two sets suchthat the total weight of edges across the cut is maximized.

1

5

4

3

2

w12

w34

w35

w25

w

45

w14

w24

CUT(1, 2, 3, 4, 5) = w14 + w24 + w25 + w34 + w35


The MAX-CUT Approximation of Goemans & Williamson

OPT = maxy1,y2,...,yn∈−1,+1

1

4

∑

i 6=j

wij(1− yiyj)

Combinatorial problem that is known to be NP-complete

SDP relaxation of Goemans-Williamson (1995): The n× n Grammatrix G = yyT (Gij = yiyj) is PSD (G 0), Gii = 1, rank(G ) = 1.

maxG∈Rn×n

1

4

∑

i ,j

wij −1

4Tr(GW )

s.t. G 0, Gii = 1, i = 1, 2, . . . , n.

Randomize a random vector z with i.i.d standard Gaussian entrieszi ∼ N (0, 1), and compute Yz , where G = YY T is the Choleskydecomposition of G .

Round yi = sign[(Yz)i ].

Theorem (G-W): E[CUTGW ] > 0.87 · OPT.Amit Singer (Princeton University) July 2014 10 / 30

SDP Relaxation for the Common-Lines Problem

Least squares is equivalent to maximizing the sum of inner products:


∑

i 6=j

‖Ricij−Rjcji‖2 ⇐⇒ maxR1,R2,...,Rn∈SO(3)

∑

i 6=j

〈Ricij ,Rjcji 〉

⇐⇒ maxR1,R2,...,Rn∈SO(3)

∑

i 6=j

Tr(cjicTij R

Ti Rj) ⇐⇒ max

R1,R2,...,Rn∈SO(3)Tr(CG )

C is the 2n × 2n matrix (“the common lines matrix”) with

Cij = cji cTij =

[

xjiyji

]

[

xij yij]

=

[

xjixij xjiyijyjixij yjiyij

]

, Cii =

[

0 00 0

]

G is the 2n × 2n Gram matrix G = RT R with Gij = RTi Rj :

G =

RT1

RT2...

RTn

[

R1 R2 · · · Rn

]

=

I2×2 RT1 R2 · · · RT

1 Rn

RT2 R1 I2×2 · · · RT

2 Rn

......

. . ....

RTn R1 RT

n R2 · · · I2×2


SDP Relaxation and Rounding

maxR1,R2,...,Rn∈SO(3)

Tr(CG )

SDP Relaxation:

maxG∈R2n×2n

Tr(CG )

s.t. G 0, Gii = I2×2, i = 1, 2, . . . , n.

Missing is the non-convex constraint rank(G ) = 3.

Randomize a 2n × 3 orthogonal matrix Q using (careful) QRfactorization of a 2n × 3 matrix with i.i.d standard Gaussian entries

Compute Cholesky decomposition G = YY T

Round using SVD: (YQ)i = UiΣiVTi =⇒ RT

i = UiVTi .

Use the cross product to find RTi .

Loss of handedness.


Spectral Relaxation for Uniformly Distributed Rotations

[ | |R1i R2

i

| |

]

=

[

x1i x2iy1i y2

i

z1i z2i

]

, i = 1, . . . , n.

Define 3 vectors of length 2n

x =[

x11 x21 x12 x22 · · · x1n x2n]T

y =[

y11 y21 y12 y22 · · · y1n y2n]T

z =[

z11 z21 z12 z22 · · · z1n z2n]T

Rewrite the least squares objective function as

maxR1,...,Rn∈SO(3)

∑

i 6=j

〈Ricij ,Rjcji 〉 = maxR1,...,Rn∈SO(3)

xTCx + yTCy + zTCz

By symmetry, if rotations are uniformly distributed over SO(3), thenthe top eigenvalue of C has multiplicity 3 and correspondingeigenvectors are x , y , z from which we recover R1,R2, . . . ,Rn!


Spectrum of C

Numerical simulation with n = 1000 rotations sampled from the Haarmeasure; no noise.

Bar plot of positive (left) and negative (right) eigenvalues of C :

0 10 20 30 40 50 600

100

200

300

400

500

600

λ

0 10 20 30 40 50 600

20

40

60

80

100

120

140

160

180

−λ

Eigenvalues: λl ≈ n(−1)l+1

l(l+1) , l = 1, 2, 3, . . .. (12 ,−16 ,

112 , . . .)

Multiplicities: 2l + 1.

Two basic questions:

1 Why this spectrum? Answer: Representation Theory of SO(3)(Hadani, S, 2011)

2 Is it stable to noise? Answer: Yes, due to random matrix theory.


Common Lines Kernel

Common line equation: Ricij = Rjcji = ±R3i × R3

j

‖R3i × R3

j ‖

cij = ±R−1i

R3i × R3

j

‖R3i × R3

j ‖, cji = ±R−1

j

R3i × R3

j

‖R3i × R3

j ‖

Basic properties of the cross product:Cij is only a function of the ratio Uij = R−1

i Rj

Cij =

(

xijxji xijyjiyijxji yijyji

)

=

(

xijyij

)

(

xji yji)

= (Pcij )(Pcji )T

= P[I 3 × U3

ij ][(U−1ij )3 × I 3]T

‖I 3 × U3ij‖2

PT , where P =

(

1 0 00 1 0

)

.

Common lines kernel: Cij = k(Ri ,Rj) = k(R−1i Rj) : R

2 7→ R2.


Common Lines Operator

Common lines kernel: Cij = k(Ri ,Rj) = k(R−1i Rj) : R

2 7→ R2.

Suppose f : SO(3) 7→ R2 then

1

n(Cf )(Ri ) =

1

n

n∑

j=1

Cij f (Rj) ≈∫

SO(3)k(Ri ,Rj)f (Rj) dµ(Rj) = (Cf )(Ri ).

Common lines integral operator C is a convolution over SO(3):

(Cf )(Ri ) =

∫

SO(3)k(R−1

i Rj)f (Rj) dµ(Rj) = λf (Ri)

Representation theory of SO(3) explains eigenvalues andeigenfunctions.

Eigenfunctions are tangent vector fields to the 2-sphere(sections of the tangent bundle).

They are the projected gradient of the scalar spherical harmonics.


Probabilistic Model and Wigner’s Semi-Circle Law

Simplistic Model: every common line is detected correctly withprobability p, independently of all other common-lines, and withprobability 1− p the common lines are falsely detected and areuniformly distributed over the unit circle.Let C clean be the matrix C when all common-lines are detectedcorrectly (p = 1).The expected value of the noisy matrix C is

E[C ] = pC clean,

as the contribution of the falsely detected common lines to theexpected value vanishes.Decompose C as

C = pC clean +W ,

where W is a 2n × 2n zero-mean random matrix.The eigenvalues of W are distributed according to Wigner’ssemi-circle law whose support, up to O(p) and finite samplefluctuations, is [−

√2n,√2n].


Threshold probability

Sufficient condition for top three eigenvalues to be pushed away fromthe semi-circle and no other eigenvalue crossings:(rank-1 and finite rank deformed Wigner matrices,Furedi and Komlos 1981, Feral and Peche 2007, ...)

p∆(C clean) >1

2λ1(W )

Spectral gap ∆(C clean) and spectral norm λ1(W ) are given by

∆(C clean) ≈ (1

2− 1

12)n

andλ1(W ) ≈

√2n.

Threshold probability

pc =5√2

6√n.


Numerical Spectra of C , n = 1000

−200 0 200 400 6000

200

400

600

800

λ

(a) SNR=20

−200 0 200 400 6000

200

400

600

800

λ

(b) SNR=2−1

−200 0 200 400 6000

100

200

300

400

λ

(c) SNR=2−2

−200 0 200 400 6000

50

100

150

200

250

λ

(d) SNR=2−3

−200 0 200 400 6000

50

100

150

200

λ

(e) SNR=2−4

−100 0 100 200 300 4000

20

40

60

80

100

λ

(f) SNR=2−5

−100 0 100 200 3000

20

40

60

λ

(g) SNR=2−6

−50 0 50 100 150 2000

10

20

30

40

λ

(h) SNR=2−7

−50 0 50 100 1500

5

10

15

20

25

λ

(i) SNR=2−8


Estimation Error

True rotations: R1, . . . ,Rn.

Estimated rotations: R1, . . . , Rn.

Registration:

O = argminO∈SO(3)

n∑

i=1

‖Ri − ORi‖2F

Mean squared error:

MSE =1

n

n∑

i=1

‖Ri − ORi‖2F


MSE for n = 1000

SNR p MSE

2−1 0.951 0.0182

2−2 0.890 0.0224

2−3 0.761 0.0361

2−4 0.564 0.0737

2−5 0.342 0.2169

2−6 0.168 1.8011

2−7 0.072 2.5244

2−8 0.032 3.5196

Model fails at low SNR. Why?

Wigner model is too simplistic – cannot have n2 independent randomvariables from just n images.

Cij = K (Ii , Ij ), “random kernel matrix”(Koltchinskii and Gine 2000, El-Karoui 2010).

Kernel is discontinuous (Cheng, S, 2013)


From Common Lines to Maximum Likelihood

The images contain more information than that expressed by thecommon lines.

Common line based algorithms can succeed only at “high” SNR.

(Quasi) Maximum Likelihood: We would like to try all possiblerotations R1, . . . ,Rn and choose the combination for which theagreement on the common lines as observed in the images is maximal.

Computationally intractable: exponentially large search space.


Maximum Likelihood Solution using SDP

Main idea: Lift SO(3) to Sym(S2)Suppose ω1, ω2, . . . , ωL ∈ S

2 are “evenly” distributed points over thesphere (e.g., a spherical t-design).

g ∈ SO(3)←→ π ∈ SL (Assignment→ ,

Procrustes← )(in practice no explicit construction is needed).


Convex Relaxation of Permutations arising from Rotations

The convex hull of permutation matrices are the doubly stochasticmatrices (Birkhoff-von Neumann polytope):

Π ∈ RL×L, Π ≥ 0, 1TΠ = 1T , Π1 = 1

Rotation by an element of SO(3) should “map nearby-points tonearby-points”. More precisely, SO(3) preserves inner products:

Ωij = 〈ωi , ωj〉 ǫ= 〈ωπ(i), ωπ(j)〉 = Ωπ(i),π(j)

ΠΩΠT ǫ= Ω =⇒ ΠΩ

ǫ= ΩΠ

Axis of rotation (Euler’s rotation theorem):

Tr(Π)ǫ

≥ 2

Notice: We are discretizing S2, not SO(3)

(substantial gain in computational complexity)


Convex Relaxation of Cycle Consistency: SDP

Let G be a block-matrix of size n × n with Gij ∈ RL×L.

We want Gij = ΠiΠTj .

G is PSD, Gii = IL×L, and rank(G ) = L

(the rank constraint is dropped).

Convex relaxation of search space (putting it all together):

G ∈ RnL×nL G 0 Gii = IL×L (cycle consistency)

Gij ≥ 0 Gij1 = 1 1TGij = 1T (doubly stochastic)

GijΩǫ= ΩGij Tr(Gij)

ǫ

≥ 2 (SO(3))

Related work: Charikar, Makarychev, Makarychev,“Near optimal algorithms for Unique Games” (STOC 2006)


Maximum Likelihood: Linear Objective Function

The common line depends on RiRTj .

Log-Likelihood function is of the form

∑

i 6=j

fij(RiRTj )

Nonlinear in RiRTj , but linear in G .

Proof by picture.


Linear Objective: Proof by picture


Tightness of the SDP

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

σ

p

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

σ

p

Figure : Fraction of trials (among 100) 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 ofnoise (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 andalternating minimization.

The SDP gives a certificate whenever it finds the MLE.


Are we there yet?

The SDP approach can be used in a variety of problems where theobjective function is a sum of pairwise interactions

Need better theoretical understanding for the phase transitionbehavior and conditions for tightness of the SDP.

Need better computational tools for solving large scale SDPs.(we use ADMM).


References

A. Singer, Y. Shkolnisky, “Three-Dimensional Structure Determination fromCommon Lines in Cryo-EM by Eigenvectors and Semidefinite Programming”,SIAM Journal on Imaging Sciences, 4 (2), pp. 543–572 (2011).

R. Hadani, A. Singer, “Representation theoretic patterns in threedimensional Cryo-Electron Microscopy I – The intrinsic reconstitutionalgorithm”, Annals of Mathematics, 174 (2), pp. 1219–1241 (2011).

L. Wang, A. Singer, Z. Wen, “Orientation Determination of Cryo-EMimages using Least Unsquared Deviations”, SIAM Journal on Imaging

Sciences, 6(4), pp. 2450–2483 (2013).

A. S. Bandeira, M. Charikar, A. Singer, A. Zhu, “Multireference Alignmentusing Semidefinite Programming”, 5th Innovations in Theoretical ComputerScience (ITCS 2014).

A. S. Bandeira, Y. Khoo, A. Singer, “Open problem: Tightness of maximumlikelihood semidenite relaxations”, COLT 2014. Also available athttp://arxiv.org/abs/1404.2655.


http://arxiv.org/abs/1404.2655

Orientation Assignment in Cryo-EM - University of Chicago · 2014-07-24 · Orientation Assignment...

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