On the symmetries of the synchronization problem in Cryo-EM: Multi-Frequency Vector Diffusion Maps on the Projective Plane
Authors: Gabriele Cesa
Arash Behboodi
Taco Cohen
Max Welling
Venue: NeurIPS 2022 Accept (2022)
Keywords: cryo-EM
synchronization
representation-theory
equivariance
spectral

Abstract: Cryo-Electron Microscopy (Cryo-EM) is an important imaging method which allows high-resolution reconstruction of the 3D structures of biomolecules. It produces highly noisy 2D images by projecting a molecule's 3D density from random viewing directions. Because the projection directions are unknown, estimating the images' poses is necessary to perform the reconstruction. We focus on this task and study it under the group synchronization framework: if the relative poses of pairs of images can be approximated from the data, an estimation of the images' poses is given by the assignment which is most consistent with the relative ones.
In particular, by studying the symmetries of cryo-EM, we show that relative poses in the group O(2) provide sufficient constraints to identify the images' poses, up to the molecule's chirality. With this in mind, we improve the existing multi-frequency vector diffusion maps (MFVDM) method: by using O(2) relative poses, our method not only predicts the similarity between the images' viewing directions but also recovers their poses. Hence, we can leverage all input images in a 3D reconstruction algorithm by initializing the poses with our estimation rather than just clustering and averaging the input images. We validate the recovery capabilities and robustness of our method on randomly generated synchronization graphs and a synthetic cryo-EM dataset.

On the symmetries of the synchronization problem in Cryo-EM: Multi-Frequency Vector Diffusion Maps on the Projective Plane
Gabriele Cesa
Arash Behboodi
Qualcomm AI Research, Amsterdam" AMLab, University of Amsterdam gcesauqti. acesa@@ti.qualcomme qumm. com
Qualcomm AI Research, Amsterdam" behboodi@qt: qualcomm. com
Taco Cohen
Max Welling
Qualcomm AI Research, Amsterdam" tacoseqti, qual com
AMLab, University of Amsterdam welling max@gmail com
Abstract
Cryo-Electron Microscopy (Cryo-EM) an important imaging method which allows high-resolution reconstruction of the 3D structures of biomolecules. It produces highly noisy 2D images by projecting molecule's 3D density from random viewing directions. Because the projection directions are unknown, esti- mating the images' poses is necessary to perform the reconstruction. We focus on this task and study it under the group synchronization framework: the relative poses of pairs of images can be approximated from the data, an estimation of the images' poses given by the assignment which most consistent with the relative ones. In particular, by studying the symmetries of cryo-EM, we show that rela- tive poses in the group ((2) provide sufficient constraints to identify the images' poses, up to the molecule's chirality. With this in mind, we improve the existing mulli-frequency vector diffusion maps (MFVDM) method: by using O(2) relative poses, method not only predicts the similarity between the images' viewing directions but also recovers their poses. Hence, we can leverage all input images 3D reconstruction algorithm by initializing the poses with our estimation rather than just clustering and averaging the input images. We validate the recovery capabilities and robustness of our method randomly generated synchronization graphs and synthetic cryo-EM dataset.
Introduction
Cryo-Electron Microscopy (Cryo-EM) revolutionized the field of structural biology, enabling the reconstruction protein structures at unprecedented resolutions. Indeed, the 2017 Nobel Prize in Chemistry was awarded to three scientists for their pioneering works it [ In single-particle cryo-EM, a purified solution containing the molecule of interest is frozen on a thin film and then bombarded with electrons to obtain 2D tomographic (integral) projection of it. The resulting image contains the projection of each copy of the molecule in the solution; in a particle picking phase, these projections randomly rotated in 3D, each image is projection the molccule's density random unknown pose. Moreover, the produced images are characterized by very low signal-to-noise ratios (SNR) [2].
cropped to obtain a dataset of 2D images. Since each copy in the solution is
"Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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e.g. see Fig. D The objective is reconstructing the molecule's 3D structure from these observations, but the high noise and the unknown poses make this inverse problem particularly challenging.
Common methods attempt to estimate the images' poses while performing the reconstruc- tion. This is typically done via iterative refinement with an Expecaaton-Maxxiiizatonn (EM) algorithm B. M.. Unfortunately, EM-like methods are known to suffer from convergence is- sues due local minima and generally require sufficiently good initializations and restarts [SI. The 3D density of known molecules can be used as initial guess but this often causes model bias [6. Ab-initio methods, which don't 't rely on an initial guess, are particularly useful to initialize the iterative methods above and, therefore, are a fundamental el- ement of the reconstruction pipeline, Moreover, a good initialization of the
dia
01
poses can speed up iterative refinement by providing a non-flat prior over poses, and so, reducing the search space.
Figure 1: Example of Cryo-EM images at different SNR values.
Due to the high level of noise, first phase of 2D classification and averaging is performed, where smaller denoised set of images is generated by clustering images with a similar viewing direction and averaging images within each cluster. Because this blurs out the fine details, it can reduce the final reconstruction's 's resolution and, therefore, it is desirable to minimize the averaging performed [7 [ Vector Diffusion Maps (VDM), and their variants, B IQ are powerful methods estimate the similarity between images' viewing directions and, therefore, provide a robust metric for clustering. In short, these methods leverage the equivariance of Cryo-EM to planar rotations (SO(2) and interpret the planar rotations aligning pairs of images as an approximation of the local parallel transport over sphere of viewing directions. The closer the viewing directions two images are, the more consistent is the transport along different paths between them. This notion of consistency is encoded in the top eigenvectors of an Hermitian matrix representing the discretized parallel transport operator.
Contributions and Outline In this work, we consider the problem of estimating Cryo-EM images' poses under the group synchronization framework, i.e. using the relative poses of pairs of images to recover an estimation of the absolute poses. First, in Sec. P we relate the symmetries of Cryo-EM with those of the synchronization task: we show that (2) relative poses not sufficient to solve the multi-view synchronization problem, but O(2) relative poses are. Second, in Sec. we use this insight to construct improved VDM method, which is able recover only similarity metric between the images, but also an estimation of each image's pose. In particular, we extend VDM by i) using vector diffusion over a projective plane (rather than sphere) reconstruct denoised similarity and parallel transport and by ii) including a final synchronization step to recover the absolute poses. This minor modification of VDM allows one to replace the 2D classification phase with pose estimation phase and, therefore, leverage all the raw cryo-EM images in the final 3D reconstruction without the loss of resolution caused by averaging. Additionally, our improved VDM method shares the same spectral properties of the original VDM method and, hence, the same noise stability and recovery guarantees. We discuss the complexity of our method in Sec. 3.3 and its limitations in Finally, we validate experimentally our theoretical results in Sec.
Sec. 3.4
1.1 The mathematics of Cryo-EM
Following [II] we consider the problem of single-particle homogeneous reconstruction and assume no image translations for simplicity. Then, a molccule's s 3D density is a function V : R - R e I.2 (R²) with compact support around the origin of TR3. An observation is a gray-scale image generated by the integral projection along the Z axis II (RR3 + (R²), which is defined as:
(T))) = / Ve. 2)2 E
Let [o be the collection of images observed. The image o, == I[ (V)) E L²(R²) is the projection of a copy of the molecule & rotated by random 9; E SO(3), where the action of SO(3) on L (RR3 is the standard action:
[g. (x)) = V(g -1.x).)) Vg SO(3).

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fr
Figure 2: Generation process Cryo-EM images. Projec- tions from similar or opposite directions are related by ele- ments of O(2), also Fig.
Figure 3: First row: images sharing similar viewing direction with the top-left image. Second row: images with opposite viewing direction with respect to the top-left image. The colored group elements are the relative transformation with respect the top-left image; is the identity, r is rotation by 2 around the viewing axis Z and f mirrors along the X axis.
An element 9, E S3(3) is identified with a real orthonormal matrix (æ,, yo, 2;) E 33x3 with positive determinant. The three columns 2; ys, 2, R3 form orthonormal basis for R3. We can write:
o,(2,,) [1[g M) (x, v) (( + yy: zzz)dz.
In particular, if g; (æ;, Vi, z.), then E 3& is the viewing direction along which v is projected, while (æ;; yi) defines the camera rotation around 2,. We define the projection map т (gg) = 2; R
2 On the symmetries of the Cryo-EM synchronization problem
One of the main challenges in cryo-EM the fact that the pose 9i E SO(3) of each observation o, is unknown. This prevents the set of observations {o:}: can be used to estimate the relative poses (ūi R gij 9j 9. A synchronization problem is defined by "synchronization graph" EV,E), i.. graph with node in V per image and an edge (i,j) E & associated with 9.j its estimation available. Given a synchronization graph G, the goal is now to "synchronize" them, i.e. find global assignment of the poses (ō: h consistent with the estimated relative poses:
glue the observations together reconstruct the density V. However,
9j
9:9.
v(i,j)
Global SO(3) ambiguity If ( Ji is solution, so is {9タ:}: hi for any g E SO(3). This ambiguity not unexpected and corresponds to the fact that any rotated molecule - an equally valid reconstruction, i.e. cryo-EM can not recover the orientation of molecule.
2.1 Relative poses in SO(2): the impossibility of synchronization
Before solving the synchronization problem, is necessary estimate the relative poses {gi 9j hi Unfortunately, since the projection loses information about the 3D structure, is not possible to estimate the relative pose of any pair of images. However, when the viewing directions (g.)) = 2; and (g)) = 2j of two images are sufficiently close, they generally differ only by planar rotation ray E SO(2) which can be directly estimated as
Ti arg min [lo, rr, if m ((x)) R * r2))(2)
where r acts image o by rotating it. The closer 2; and øj are, the better ray approximates 9yy- Indeed, if 2; zj. then gij 9 9: has form Loo1 1
where the top-left 2 x 2 block is an SO(2)
"Tii shown using the orthogonality of 9: and أj 's columns.

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matrix. We identify all the rotations of this form as the subgroug (O()) = R. < SO((3 of planar rotations around the Z axis. The projection operator is -quivvaria i.e.:
[) = (V) Vr
R2 SO(2).
(4)
Hence, if g, = rg, it follows that 0g r.Og. For example, all projections generated by the top view in Fig 2 are related by a rotations as shown in the top row of Fig. E
Global (2( ambiguity While this is a common approach in the literature, e.g. in VDM R 12
10 13 it does not provide sufficient information to estimate the poses (9:)- Indeed, the estimated relative rotations {ri E SO(2)) yield the following constraints:
9j - 9
v(i,j) E E
Since (2) abelian, if ] is solution this set of constraints, then (ārr is an equally valid solution for any E R₂ = (2))):
9 ~ J
Or O = Irr (rrrr
This implies that any method which estimates the poses {g from the estimated relative ones (rij E S0(2)); will only able to recover them up to a global rotation r Rg. This global ambiguity prevents using these estimations to directly invert the linear projection II and recover the original molecule V, since would result in averaging randomly rotated versions of the images.
Nevertheless, the ambiguous poses {grr still contain information about the viewing direction of the projections. Indeed, note that ((grr) = (gi), i.c. the viewing directions are invariant this global ambiguity and, therefore, can be recovered from this method. Previous works such as VDM 8 IC exploit this invariance to estimate the relative distance between different images. This distance is generally used for class averaging i.e. to cluster images and average images within each cluster to generate smaller, de-noised, images dataset.
2.2 Relative poses in O(2): sufficient condition for synchronization
In this work, we solve this global SO(2) ambiguity and directly estimate the absolute poses {9: ) of the images. To do so, we will exploit another symmetry of Cryo-EM. Indeed, the tomographic
8 E 0((3) i.e.:
projection IT is also invariant to mirroring along the 7. axis m = 0 0
Mmm .V) V(r,,, -2)2 / V v,
IT(V)
This additional symmetry turns out useful to break the global SO(2) symmetry explained earlier.
-1 SO(3) be rotation along the Y axis; note that Ty m, my, where my is mirroring along the X axis. Then, let f [-] ]] O(2) be the flip of planar image along the X axis. Then:
First, let Ty
0 -1
[1 v I1(m, my V) 11 .V) = f f.l(
(5)
ie. the projection operator II flip equivariant: projections along a direction (Tg 10--1-1 -v)) are related to projections from the opposite direction (f [TT((r,,,) V)) by a planar reflection f. For example, all projections generated by the bottom view in Fig are related by a rotation followed by the reflection with the images generated from the top view, as shown in the bottom row of Fig To simplify the notation, we will also let Ty act on image; this action should be intended as that of f.
That means that we can not only estimate the relative pose gif hij E R, when the viewing direc- tions (9:) and πt (9;) are sufficiently close, but also his E H _ O(2) when ((g)) ~ (9;),, (0))..e... i.e.:
hi = arg min |lo5 - h..0:l2 1 if (g:) = (g;) he h))#O(2)
(6)
where H = 0(2) is the subgrou F of (3) containing R2 (planar rotations along the Z axis) and ry.
SS)(3) has many subgroups isomorphic SO(2) so use R. to indicate this particular one.
4 'For simplicity, the action R2 on 3D densities or 2D images is not distinguished.
MMore precisely, H := Rs (e, rv) a O(2).

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Table 1: Summary the Cryo-EM symmetries.
description of the transformation
symmetty name
0_ 11(9 V)
9i (3)3)
generative process SO(2) equivariance
r.O: 1[(g v)
r E (2)2) any rotation around axis E ((3) mirroring along Z axis O(3), [ 0(2) mirrringg along X axis Vy My m, E S3(3) rotation by around axis
= 11 (m. 4)
Mirroring (Z) invariance
Co [1 9; v)
Mirroring (XY) equivariance
To-- [r W)
Flip equivariance
Global chirality ambiguity The relative poses {hij E O(2) still don't fully constrain the
absolute ones; if (ū: Ja is solution, so is (āeT') Ji. with r. E R2 being t-rotation around the Z axis Iyr= Geligrr
Ø3 gehiy
Jj"== R (āerz) beg
This ambiguity related to the well known problem that Cryo-EM can not recover the chirality of
0
molecule. To see why, let = 0 -1 be the inversion, i.e. mirroring along all axes; note that
o 0
m = rr i and that commutes with any element 9 SO(3). Then:
O, = -I[( ) = II(m: 9, .V) = II(r; ig. 1- . = II(r:9; .(..V) (((((.)))'..1119")
ie.. o, is equally likely to be generated by the molecule v with pose Ir or its mirrored version i V with pose gx Together with the global SO(3) symmetry introduced earlier, this means Cryo-EM intrinsically suffers from a global 0(3) ambiguity which we can not hope to resolve. Nevertheless, since is the only element of SO(3) commuting with all elements in H ~ O(2), we have that relative poses in H constrain the synchronization problem precisely up to this 0(3) symmetry and, thus provide sufficient constraints to identify the images' poses. We provide theoretical argument based on Vector Diffusion Maps (VDM) supporting this informal claim in Apx.
3 Multi-Frequency Vector Diffusion Maps on the Projective Plane
In this section, we use the new insights on the symmetries of cryo-EM to enable the Vector Diffusion Map (VDM) method to recover the poses b. The key component is the Graph Connection Lapla- cian (GCL) of the synchronization graph G with the estimated relative poses {hij E 0(2) on its edges. Under certain conditions, its eigenvectors converge to the ligen-vector-filld of the connection Laplacian operator on the projective plane, which contain information about the absolute poses. Our method involves two steps: a denoising step (like the original VDM) and a synchronization step; Alg. provides an overview of the proposed method, We study its computational complexity in Sec. 3.3
3.1 Quotient Spaces and Projective Plane
Each image o; is uniquely identified by element 9. SO(3) via Eq. assuming the molecule has no symmetries. Two images O; and oy related by a rotation rij E R₂ contain the same information and are considered equivalent. We can define similar equivalence relation their respective poses, i.e. 9j 9:Tiy- This defines the quotient space SO(3)/Ra which is isomorphic to the sphere S?. An element 9: E (3) is interpreted as point π (g.) = z; in S and a choice of frame on it, i.c. the choice of basig (w,, y1) for the tangent space at z; cquivalently, the rotation of the camera around the axis z;. Then, the relative rotation Tiş E R, estimated in Eq. 3] is interpreted as the unique rotation such that gy is the parallel transport of g, along the shortest geodesic connecting т (gg)) to ((g)) (this is unique unless (g:) = T (gg)). )) Here, we want to leverage the larger symmetry H. We define the new equivalence relation g gh for all / H on S3(3). which determines the quotient space SO(3) H. isomorphic to the projective plane PIR?. While it is harder to visualize. elements of SO(3) are now interpreted as points in 2g² together with a choice of frame for their tangent spaces") Finally, the relative pose hig E H from Eq. 6 is interpreted as the element such that gj h is the parallel transport of gi along the shortest geodesic between the projections of gi and gj on PRR
'One can verify that rz commutes with any r E R& and rz- Indeed, (ax, )1) span 2D plane through the origin parallel to the tangent plane at 'The projective plane can be thought as sphere where antipodal points are identified together. By moving from S to pig? change the structure group and consider frames 0(2) rather than SO(2).
5

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3.2 Vector Diffusion Maps on PR²
An initial synchronization graph & can be generated by comparing all pairs of images using Eq. and preserving only the edges with smallest distances. Ideally, this graph is a discretization of PR = SO(3) H with local connectivity, but the noise on the images negatively affects the estimated distances and relative poses, introducing "shortcut" edges in this graph. Therefore, the first step of our method is re-estimating the geodesic distances and the relative poses of the points in the graph.
Like previous works, we leverage the consistency of the local alignments to do this. Indeed, since manifold is locally Euclidean, parallel transport along sufficiently short paths approximately path-independent in the local neighborhood of a point. On the other hand, the further away two points are, the more inconsistent the cycles through them will be. When summing the transport over multiple paths between two points, inconsistent paths tend to average out.
We use the synchronization graph to build discretization of this local parallel transport operator. Let the graph connection Laplacian W be the 2N x 2.N matrix with its 2, 2 x 2 block defined as:
W (i, = pp(i)) if (1,5) E 9 otherwise
E =
0
where p(haz) is the standard representation of hu E (2) as a 2 orthogonal matrix. Let D be the 2N x 2N diagonal matrix containing the degree of each node in G (repeated twice) in its diagonal. The matrix W acts by parallel-transportin
tangent vectors along each edge of the graph while A = D D-'W W averages all vectors transported to node. Then, W transports vectors along all length- t paths in G and the block (i. is the average of the transport over all length-t paths from i to j. If the paths mostly inconsistent, A' tends to O, being the average of uncorrelated orthogonal matrices. Hence, A' (i.j) provides information about the geodesic distance between i and j.
approach to improve robustness to noise. In practice, we build a different graph connection Laplacian Ag for different choice 10 of irreducible representation (irrep) Pk of O(2). For k 0, Pk is 2 x orthogonal matrix with rotational frequency k;; see Apx. A
Like 12). we take multi-
In the limit & + co, the power iteration of A converges its top eigenspace and A (i, will be the product of the top eigenvectors of Ag. It is therefore important to discuss the spectral properties of Ag. 14] proved that the normalized graph connection Laplacian a discretization of the connection Laplacian operator and that, under certain conditions, its eigenvectors and eigenvalues converge to those of the connection Laplacian operator. In Apx. B we study the spectral properties of the connection Laplacian operator over PR2 and relate it with the operator over S studied in [K] In particular, we show that the eigenspaces of our Laplacian operator over PIR. are subspaces of the eigenspaces of the spherical Laplacian and they share the same eigenvalues. This also implies our graph connection Laplacians enjoy the same spectral properties studied in 13 including the spectral gap and the robustness to noise.
For frequency > 0, the top eigen- sigen-space of Ag is 2k + dimensional. Additionally, denote /2
PKk V RR (2k- x) the stack of the top 2k + 1 eigen-vectors. Let (x)
11 F (x( and
define ( (i) E R2x2, Then:
Theorem 3.1. Let Sij = (zj, z1) = (T(()), (gi)) [-1,, be the cosine similarity of the viewing directions π (g:), (gz;) E S².
The two following identities hold:
(A))))))
1 8ij
2k
+
88g
2k
det (c((t)))))) ~
1 + 8ij
2k
-
8ij
2k
2
The proof is in Apx. D Since we need a similarity over PR rather than S², we are interested in Wiy := | In other words, the top eigenspace of As allows the estimation of the geodesic distance between points. In practice, dę > 2k + top-eigenvectors can be used, as in g
Denoised similarity and parallel transport The information obtained from the different A matrices can be combined in multiple ways to obtain an estimation of stj. Define
sty - + det det(AM(),
1 + 8ij
2k
(8)
2
The standard representation of 0(2) rotation matrices correspond to le

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Algorithm 1 MFVDM based synchronization over PR
Require: {ox
I.....N .N V) the input raw images, max frequency L > 1: g(V. { < synchronization- graph ((hr) = arg min, |oo, h.ovl12)s {di = minA ||o, - h..0112)05))
2: for < k < I, do
3: Aş (px(he)) if (t. E & clse O1g3
Froquency-k graph connection Laplacian: Eq.
4: Pk f top-eigenvectors( AR. 2k +1
5:
Ag
( [Pabbil. PD (x)) (Tentenle (
v3
(1.))
del (Aa(t.))) Vijev
Eq.
7: 5 - exp
T. 2
By - sign(3))
81z) max 1g:
b Denoised similarity over PRR:; Eq.
9: Rn arg m Eoceel V dimpp
(Au(t, (ln(hh))))))
Denoised parallel transport; Eq.
10: G'(VE") - -synchoonizationgaaa
Build denoised Synchronization Graph
A' [
(hts) (i.j) E Ex else
b Build denoised graph connection Laplacian: Eq.
y + t top-eigenvectons(A", 33)
Final Synchronization
13: 3: + Projea(SVD((()) v(). x v(i)
Then, if is the largest frequency considered, the similarity sij can be estimated
L
1k log
with = exp 1
Σ
ii =
(9)
2k
In practice, we found the following estimator more robust for the nearest neighbors search:
Sig sign(S 8it) max (5ig) Sty) and
Wij
Bayl max Bijs 8ig)
(10)
although it approximates 2d rather than Sij. In Apx. E discuss alternative estimator which can be expressed as a dot product, enabling the use of a fast K-nearest neighbors (K-NN) search rather than computing the similarity between all pairs. Additionally, note that, due to the decaying spectrum of the Laplacian operators, the noise has a stronger effect on the lowest eigenvalues; discarding the lower eigenvectors also helps denoising the GCL matrices. That means that the top cigenvectors of A₁ can be used to partially denoise the frequency-k parallel transport. Next, we combine all denoised blocks (i, he by interpreting them the Fourier coefficients of function on H; we estimate the relative pose with the element h E maximizing this function:
1tsu
Te = arg max Σ
dimp, Tr (Ax(1.5)p&(hh)))
(11)
P
Final Synchronization So far, we have used the MFVDM method on PR2 to estimate the similarity
eij ر and parallel transport hTj over 2R With respect to 12 our method includes a final synchronization step, where these quantities are used recover 9 We construct a new graph connection Laplacian matrix A' as in Eq. 7 using the new estimations and the standard representation (k: = of (2)). The top eigen-space of A' is 3 dimensional and its top eigenvectors define tangent frame at each node in the following way. Let yp V 3332 be the stack of the top 3 3 be its two columns. Then,
eigen-vectors, (p(i) [T
(() and x + 3 and y
(1) and v(i) R define basis for the tangent space ((g)) E S², see Apx. C Recall that (z; yr 2x) < (O))), then xr)) and y(i) approximates respectively æ;; and yi. while z; can v((),=(()))) 2(1)) present the global O(3) symmetry discussed in Sec. 2 Indeed, eigenvalue decomposition is unique up to an orthogonal change of basis in each eigenspace. Since the top eigenspace is 3 dimensional, the solution is unique up to global E 0(3) transformation, i.e. if so is set of orthogonal eigenvectors,
be recovered as 2(i) x(i) v(). Note that recovered poses (Ør (z(i), v(i),
((x)), =(0) (=(i) v()) x(1) x v(()) det (E6)) x(0) zs sty using Theorem 3.1 with & 1.
7

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so is gi, defined as [g) = gy((t.. Hence, all tangent frames {x((), y(a)) can be simultaneously rotated by any g E Ó(3). Finally, because xti) and y(i) are not perfectly unitary and orthogonal to each other, the matrix (ar()), (()), =(i) will not be orthogonal; therefore, we project it to the closest SO(3) element via SVD.
3.3 Cryo-EM Pipeline and Computational Complexity
We study the complexity of our method in a pipeline similar to the one implemented in ASPIRE and described in [13]_ Assume a dataset of N images of resolution D x D; consider the MFVDM algorithm with frequencies up to I. and that at most M top-eigenvectors of Ag are computed for each frequency k. The pipeline consists of the following three stages, with relative complexity:
Preprocessing O( + Nk log N): a number of invariant features are built using (fast) steer- able PCA O(N
I and the bispectrum, and are used for a K-nearest neighbors (K-NN) search ( K log N) as in [[5]. Then, the ((2) relative alignments of each pair are estimated in O(N log D + NKD²) by leveraging Polar and Fast Fourier Transforms (FFT).
MFVDM denoising O(NLM + NKLM² log N): the eigenvalue decomposition of the matrices ~ (M)) as in [12.. The denoised similarities Sij cost ((N2). By aggregating multiple frequencies without log as in 12 however, a faster K-NN
{AA t can be accelerated to ((NL(M*+MK)
search in D(NKLM log N) can be used, see Apx. E. Finally, the denoised parallel transports h:j
between neighbors are computed in O log L) with an FFT.
Synchronization ((N since only the top 3 eigenvectors of A' are required, the eigenvalue decomposition only costs ((N
3.4 Limitations
The main limitation of our method is that VDM assumes uniformly distributed poses, which is not generally the case for real cryo-EM data. Renormalization techniques such as [ir] [ can help relaxing this requirement. Our analysis of the synchronization symmetries, however, is not limited to VDM but applies to any cryo-EM synchronization method. Moreover, 2D classification or pose estimation methods which do not explicitly account for the image formation model might fail at the lowest SNR regimes [5]. Indeed, if the noise is too high, the relative poses estimation itself can be impossible and increasing the number of observations can not solve this problem 19 20), although methods like steerable PCA [I6] can alleviate that. Nevertheless, in practice 2D classification with VDM is leveraged as a denoising step in reconstruction pipelines like ASPIRE: our method can replace this step, providing pose estimation and reducing the need for averaging. Finally, while assumed non-symmetric molecules, our method can handle symmetries, provided they are known a priori, since they enforce precise sparsity patterns in the spectrum of the diffusion operators; we leave this as future work.
4 Related Works
Vector-Diffusion Maps (VDM) were initially proposed in 9 3 for solving the cryo-EM synchro- nization problem with SO()) constraints. The O(2) symmetry was noted also in [8] which suggests augmenting a dataset with mirrored images; still, it only uses SO(2) relative alignments and, therefore, recovers only a similarity metric over the sphere. VDMs were later extended to a multi-frequencc 2 describes strategy based on the classical method of moments solve the generic problem of recovering orbits from invariants. Additionally, the authors provide theoreti- cal analysis of sample complexity of such problems; in particular, they show that in the cryo-EM synchronization problem with uniform viewing distribution, the sample complexity scales as SNRS Some works exploit the property that the Fourier transforms of any pair of cryo-EM images must agree on line passing through the origin (common lines approach) to define set of constraints on each pair of absolute poses. [22] use Semi-Definite Programming (SDP) relaxation find the solution; however, SDP is known for its high computational cost. Instead, 7 23
setting in 12 TC [3
rely on faster
spectral relaxation of the problem. Still, the estimation of common lines itself is expensive- since requires comparing each pair of images- and very sensitive to noise; for this reason, it usually follows a first 2D classification and averaging phase 24 Popular reconstruction methods include RELION 2 and cryoSPARC (26]. which are based EM-like algorithm.
27 suggested using stochastic
8

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rather than batch optimization for iterative refinement, thereby improving the computational cost. Recent advances employ neural networks optimized by SGD represent the 3D density 128 29 Note that these methods still need to periodically perform an expensive search over the images' poses and found resetting the model's weights throughout training essential for convergence.
Experiments
We first evaluate our method on a synchronization task over some synthetic datasets of vectors which share the same symmetries of cryo-EM. Then, we include our method in simple 3D reconstruction pipeline which we test on synthetic cryo-EM data. We include additional experiments in Apx.
5.1 Synthetic Vector dataset
We generate number of synthetic datasets which im- itate the symmetry of the cryo-EM generative process. To do so, we first construct a matrix [ M Rdxb which has the symmetries of II described in Table where H acts on the left with dddiimensional rep- resentation of the group ((2) while 0(3) acts on its right with b-dimensional representation vb. Both representations contain irreps of different frequencies of the two groups. See Apx. Apx.AIIt for for more details about the representations of these groups. A dataset is gen- erated by sampling a random vector E R², a set of random poses {9. SO(3)}, and then evaluating fo, (9, V + f& RR), where e entries normally distributed and X² is the SNR.
Poses Cor
0.8
SNR=A
0.4
1.00
0:44
0.2
0.16
1000
2000 3000
4000
5000
#samples
Figure 4: Correlation with the SO(3) poses in different variations of the vector dataset.
D121 MFVDM L-1
1.0 012) MEVDM, L-5
1.0
Estimated 5,
0.0
evaluate variants of our method on this data and compare it with MFVDM with SO(2) poses. We use the SO(3) correlation 13
and the correlation between the real and estimated co- sine similarities {s as metries. Fig. 4 shows
nV3 Ex
55
Real 5g
80
03
1.0
05
Real 5g
20
03
1.0
SO(2) MFVDM, L=1
S012) MFVDM, L
Estimated Log
10
Estimated 1+4
1.0
as
"es
the SO(3) correlation achieved by method in different settings; in the low-noise regime, perfect reconstruction achieved. As expected, in the high- est noise regime, reconstruction is almost impossi- ble as increasing the number of samples helps only marginally (SNR=0.100 Fig. 5 compares our method with the original MFVDM-SCC (2), for two choices of maximum frequency L. While the original estimator
0%
-05
5.0 0.5
1.0
-03
0.0
25
3.0
Real 5g
Real 5g
Figure 5: Estimated {&t by vsreal (stg hi cosine similarities in the synthetic compare our method with O(2) relative poses with the original MFVDM with SO(2) poses.
dataset. We
from MFVDM is stable when sig
> 0, our estimator
from Eq. 10 is stable when sigl >> 0. We also ob-
serve the effect of the fulti-fre approach: leveraging higher frequencies provide a more robust
estimation.
5.2 3D reconstruction of synthetic cryo-EM data
For a more realistic evaluation, we integrate our method in the pipeline described in Sec. 3.3 and test on the 70S Ribosome B0 structure from the Protein Data Bank database (structure 5060). See Apx. F2 for more details. In summary, a first synchronization graph is built via two K-NN searched by using invariant features generated with Steerable PCA and bispectrum [1]] (we pick K rotated and K mirrored neighbors). Then, the estimated relative poses can, optionally, be denoised with MFVDM, either using SO(2) O(2) and with different values of the maximum frequency MFVDM-S0(2) is used, information about reflections is lost, hence our VDM-based synchronization
M contains subset of the matrix decomposing Ind 0(3)
{ P into O(3) irreps.
Note that this metric is invariant the global ((3) symmetry.

---

Table 2: Pose correlation (and emerr in radians) with different methods.
Synchronization Demoising
VDM (our)
VDM (our)
VDM O L = 10
CL None
CL
None
0 L =
S0(2), L =
0 L. = 0 L = 10
N
SNR
3000
0.35
99.6 (0.4)
99.3 (0.15)
99.3 (0.15)
99.6 (0.00)
99.7 (0.08)
99.7 (0.06)
5000
0.12
82.0 (0.00)
97.5 (0.23)
97.5 (0.21)
98.0 (0:20)
97.9 (0.18)
99.0 (0:14)
99.0
5000
0.05
60.1 (1.11)
63.4 (1.05)
63.5 (1.05)
61.6 (1.03)
71.4 (0.87)
75.9 (0.37)
76.6 (0.75)
can not be used later. Next, the dataset can be clustered n < NN images and denoised by averaging nearest neighbors.
As synchronization baseline, consider the common line method (CL) from the estimation of common lines estimation is more sensitive to noise. Moreover, the method requires O(N2) comparisons find common lines between each pair of images; in comparison, MFVDM only needs to estimate the relative poses of ((N nearest neighbors. For these reasons, the CL method is usually applied on the smaller subset of < N averaged images.
22 Note that
Once the poses are estimated, we perform a first 3D reconstruction using an algorithm inspired by [3I] which optimizes via SGD Gaussian posterior over the 3D density leveraging the Fourier Slice Theorem. See also Apx. S for more details about it and other experiments. Finally, this low-resolution reconstruction is used to initialize the 3D Refine method in RELION 25 which performs the final refinement. RELION also re-estimates the images' poses and the final resolution using the gold-standard Fourier Shell Coefficient. Since RELION is based on EM method, it can show large variance in the final results and, sometimes, fail to converge. To reduce the impact of this on the variance of the results, for each method, we perform the initial reconstruction three times using the method above and, for each reconstruction, we consider the best run of RELION out of three trials. In the tables, we report the mean and the standard deviation over the three initial reconstructions of the corresponding best RELION trials.
In our experiments, and n = 1000 averaged images for the CL methods. In Tab. 2 we report the pose correlation (and the average error in radians) after synchronization on three synthetic datasets, with different levels of noise. Tab. 3 reports the correlation and error of the poses after the refinement performed by RELION, as well as the corresponding resolutions (lower is better). We observe that the CL method achieves better initial pose estimation in Tab. 2 although only by a small margin in the low noise regime. However, this does not reflect on the final resolution achieved by RELION in Tab. B which is generally lower for CL methods since fewer images are used and averaging loses details. Note also that our MFVDM-O(2) denoising improves the performance of the CL method with respect to (IPVDM-SO()) since it also recovers reflected neighbors and, therefore, allows averaging over a larger set of images with similar viewing direction.
images of resolution D 97, K = 20 for the nearest neighbors search
Table 3: Pose correlation, pose error (radians) and estimated resolution after refinement with RELION.
Synchromization
VDM (our)
VDM (our)
VDM (our)
CL
CL
CL 0(3), 4. 10
Denoising
None
O12), J.
0(2). I. 10
None
80(2), 1. 5
0(2),
1
N
SNR
5000
0.12
99 20.900..
99.8
99.920.0
76.6
85.7
84.220.6
78, 78.5
0.01+0.0
0.06+2.02
0.01=0.0
0.52+0.27
0.50=0.24
0.67=0.a6
2. 2.42
3.31
2.4=0.03
8.0
7.65:1345
5.17=1.47
9.02+:..0
5000
0.05
98.7.00
99.01o.1
99.01o
67.12182
59.4 59.41115
76.4gm
69. 69.21aa
10:0.0
0.09
0.09
0.93:0.32
1.14:0.22
0.70=0.08
0.91
3.51=0.00
3.47+0.05
3.56=0.0
6.65+1149
8.21+4.41
5.11-0.88
6.71+0.22
Conclusions and Discussion
In this work, we studied the symmetries of cryo-EM and found that the 0(2) relative poses estimated from the images are sufficient to solve the SO(3) poses synchronization task. With this insight, we proposed new method which can replace current VDM-based 2D classification methods with limited changes to directly estimate poses rather than clustering and averaging images, while enjoying the same noise robustness properties.
10

---

Acknowledgments and Disclosure of Funding
Funding in direct support of this work: Qualcomm Technologies, Inc.
References
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1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
12

---

(b) Did you describe the limitations of your work? [Yes] See last paragraph of Sec.
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes]
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either the supplemental material or as URL)? [No]
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (c.g., with respect to the random seed after running experi- ments multiple times)? [No]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you eite the creators? [Yes]
(b) Did you mention the license of the assets? [No]
(c) Did you include any new assets either in the supplemental material or as URL? [No]
(d) Did you discuss whether and how consent was obtained from people whose data you're re
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(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "Yss_3vlAAJw", "reviewer": "Reviewer_PHyx", "summary": "This papers formally describes symmetries present in SPA cryoEM and proposes a method to enable the Vector Diffusion Map to recover particle poses using the insights from symmetries. The method can potentially replace current 2D classification methods and allow to skip clustering and averaging steps of cryoEM SPA pipeline altogether. The authors validate their approach on one synthetic and one experimental datasets, and show accurate pose estimations.", "questions": "1. One of the advantages of the proposed approach is that it would allow to skip 2D classification step, use all images for the final 3D reconstruction and subsequently increase the final resolution. Did you explore whether this holds true? Moreover, are there any estimations performance wise?", "limitations": "The authors adequately addressed the limitations and potential negative societal impact of their work. The authors also thoroughly list the limitations of their method in the last paragraph of section 3.2.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The paper establishes formal description of the symmetries present in cryoEM.\n2. The validation suggest that the proposed method can accurately recover poses. \n\nWeaknesses:\nThe paper is very careful in its formal description, which simultaneously is a strength and a weakness depending on readers' math well-read-ness. As a cryoEM practitioner, I found it very hard to follow and felt out of my depth (and reflected in my confidence score).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "JsqBA8BQBAd", "reviewer": "Reviewer_XAUC", "summary": "This paper studies the symmetries in group synchronization for pose estimation in cryo-EM. The authors find that relative poses between images in O(2), but not SO(2), provide sufficient constraints to identify image poses in SO(3). This insight is used to improve the multi-frequency vector diffusion map algorithm (MFVDM), extending its use from 2D classification of images to absolute pose estimation that can be used for ab initio 3D reconstruction. The authors show results on a synthetic cryo-EM dataset with varying noise levels.", "questions": "See above, what is the computational complexity (time and memory) of the algorithm?\nCan the authors report the error between the estimated and ground truth poses (e.g. mean/median error in degrees) instead of correlation?\nPlease include more detail on dataset generation and the experiments. In the supplement, the authors mention that poses are also updated from their initial values by backprop. What are the details here (e.g. the balance between pose updates vs. model updates) and how much are poses refined?\n", "limitations": "The authors discuss the method's limitations (assumption of uniformly distributed poses, which typically does not hold in practice and testing on synthetic projection images without CTF). The authors could further discuss the computational limitations of MFVDM. ", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well-written and motivated. I did not look deeply into the prior work on vector diffusion maps, but the insight is interesting, and the additional capability of absolute pose estimation is significant. \n\nSome details on the synthetic cryo-EM dataset generation and experiments are missing. More discussion of the algorithm's computational complexity would be interesting to include as well. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "8BjI3Q6tJO", "reviewer": "Reviewer_SWAQ", "summary": "The motivation of this work is to improve the 3D reconstruction quality from cryo-EM generated 2D images and builds on a method called MFVDM. \nFirst, the presented method estimates relative poses between pairs of 2D-images using a group synchronization framework. Here, relative pose can be computed when the viewing angles of the pair of images are either very similar or they differ by 180 degrees. \nThe authors propose to do a robust computation of similarity of viewing angles and the relative pose (parallel transport) with respect to noise using a multi-frequency approach. \nFinally, the authors explicitly estimate the pose of individual 2D images which can then be used to estimate the 3D density of the object. Presented ideas have been tested on a synthetic dataset of vectors created by the authors, which has similar symmetries as cryo-EM data would have. Additional testing was performed on another synthetic cryo-EM dataset.", "questions": "In the appendix C, sufficiency for the O(2) relative poses is shown. At what line will the proof fail if relative poses belong to SO(2), ie, O(2) minus the flips?", "limitations": "The authors have mentioned the limitation of the work: \n(1) This work assumes uniform distribution of poses. \n(2)This work does not handle symmetries present in the molecule to be scanned.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\nIn other ab initio methods, relative pose between a pair of 2D images can only be estimated if their viewing angles are very similar. \nIn the presented work, in contrast, the authors are able to also estimate the relative pose when the viewing angles differ by 180 degrees. \nThe positive effect are that (i) the quality of estimated related poses is improved, and that (ii) the number of connected components in the synchronization graph is reduced. \nSecondly, in literature, images belonging to similar viewing angles are aggregated. The aggregated images are then used during 3D reconstruction. However, this aggregation leads to a loss in the obtainable resolution. Hence, in the presented work, the authors estimate the pose of each 2D image and can thereby avoid the aggregation step and the raw images can directly be used in the 3D reconstruction task.\nFinally, I believe that the presented analysis of the synchronization symmetries could be helpful for any independent approach to improve 3D reconstruction in cryo-EM.\n\nWeaknesses:\n\nThe absence on a proof-of-concept on real-world data is regrettable. Showing the validity and effectiveness of the presented ideas would have made a strong case for the utility of the new approach. \nAccording to the submitted work, VDM assumes the viewing angles to be uniformly distributed. However, this assumption does not hold true for real world cryo-EM datasets. Hence, it would be illuminating to see how the effectiveness of the presented approach deteriorates as the viewing angle distribution is diverging from being uniformly distributed (and maybe getting closer and closer to to the configuration commonly used in cryo-EM).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept