Original: On the symmetries of the synchronization problem in Cryo-EM: Multi-Frequency Vector Diffusion Maps on the Projective Plane
Revised: On the symmetries of the synchronization problem in Cryo-EM: Multi-Frequency Vector Diffusion Maps on the Projective Plane

Original: Anonymous Author
Revised: Anonymous Author(s)

Original: AfﬁliationAddress email
Revised: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Cryo-Electron Microscopy (Cryo-EM) is an important imaging method whichallows high-resolution reconstruction of the 3D structures of biomolecules.
Revised: Cryo-Electron Microscopy (Cryo-EM) is an important imaging method whichallows high-resolution reconstruction of the 3D structures of biomolecules.

Original: Itproduces highly noisy 2D images by projecting a molecule’s 3D density fromrandom viewing directions.
Revised: Itproduces highly noisy 2D images by projecting a molecule’s 3D density fromrandom viewing directions.

Original: Because the projection directions are unknown , estimating the images’ poses is necessary to perform the reconstruction.
Revised: Because the projection directions are unknown , estimating the images’ poses is necessary to perform the reconstruction.

Original: We focus onthis task and study it under the group synchronization framework: if the relativeposes of pairs of images can be approximated from the data, an estimation of theimages’ poses is given by the assignment which is most consistent with the relativeones.
Revised: We focus onthis task and study it under the group synchronization framework: if the relativeposes of pairs of images can be approximated from the data, an estimation of theimages’ poses is given by the assignment which is most consistent with the relativeones.

Original: 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.
Revised: 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.

Original: With this in mind, we improve the existingmulti-frequency vector diffusion maps (MFVDM) method: by using O(2) relativeposes, our method not only predicts the similarity between the images’ viewingdirections but also recovers their poses.
Revised: With this in mind, we improve the existingmulti-frequency vector diffusion maps (MFVDM) method: by using O(2) relativeposes, our method not only predicts the similarity between the images’ viewingdirections but also recovers their poses.

Original: Hence, we can leverage all input images ina 3D reconstruction algorithm by initializing the poses with our estimation ratherthan just clustering and averaging the input images.
Revised: Hence, we can leverage all input images ina 3D reconstruction algorithm by initializing the poses with our estimation ratherthan just clustering and averaging the input images.

Original: We validate the recoverycapabilities and robustness of our method on randomly generated synchronizationgraphs and a synthetic cryo-EM dataset.
Revised: We validate the recoverycapabilities and robustness of our method on randomly generated synchronizationgraphs and a synthetic cryo-EM dataset.

Original:  Introduction
Revised:  Introduction

Original: Cryo-Electron Microscopy (Cryo-EM) revolutionized the ﬁeld of structural biology, enabling thereconstruction of protein structures at unprecedented resolutions.
Revised: Cryo-Electron Microscopy (Cryo-EM) revolutionized the ﬁeld of structural biology, enabling the 20 reconstruction of protein structures at unprecedented resolutions.

Original: Indeed, the 2017 Nobel Prize in
Revised: Indeed, the 2017 Nobel Prize in

Original: Chemistry was awarded to three scientists for their pioneering works on it [1].
Revised: Chemistry was awarded to three scientists for their pioneering works on it [1].

Original: In single-particlecryo-EM, a puriﬁed solution containing the molecule of interest is frozen on a thin ﬁlm and thenbombarded with electrons to obtain a 2D tomographic (integral) projection of it.
Revised: In single-particle 22 cryo-EM, a puriﬁed solution containing the molecule of interest is frozen on a thin ﬁlm and thenbombarded with electrons to obtain a 2D tomographic (integral) projection of it.

Original: The resulting imagecontains the projection of each copy of the molecule in the solution; in a particle picking phase,these projections are cropped to obtain a dataset of 2D images.
Revised: The resulting image 24 contains the projection of each copy of the molecule in the solution; in a particle picking phase,these projections are cropped to obtain a dataset of 2D images.

Original: Since each copy in the solution israndomly rotated in 3D, each image is a projection of the molecule’s density in a random unknownpose.
Revised: Since each copy in the solution is 26 randomly rotated in 3D, each image is a projection of the molecule’s density in a random unknownpose.

Original: The objective is reconstructing the molecule’s 3D structure from these observations.
Revised: The objective is reconstructing the molecule’s 3D structure from these observations.

Original: However,the produced images are characterized by very low signal-to-noise ratios (SNR) [2], e.g. see Fig.
Revised: However, 28 the produced images are characterized by very low signal-to-noise ratios (SNR) [2], e.g. see Fig.

Original: 1;the high noise and the unknown poses make this inverse problem particularly challenging.
Revised: 1;the high noise and the unknown poses make this inverse problem particularly challenging.

Original: Common methods attempt to estimate the images’ poses while performing the reconstruction.
Revised: Common methods attempt to estimate the images’ poses while performing the reconstruc- 31 tion.

Original: This is typically done via iterative reﬁnement with an Expectation-Maximization (EM)algorithm [3, 4].
Revised: This is typically done via iterative reﬁnement with an Expectation-Maximization (EM)algorithm [3, 4].

Original: Unfortunately, EM-like methods are known to suffer from convergence
Revised: Unfortunately, EM-like methods are known to suffer from convergence

Original: is-  sues due to local minima and generally require a sufﬁciently good initialization and restarts
Revised: is- 33 sues due to local minima and generally require a sufﬁciently good initialization and restarts

Original: The 3D density of known molecules can be used as initial guess but this often causesmodel bias [6].
Revised: The 3D density of known molecules can be used as initial guess but this often causes 35 model bias [6].

Original: Ab-initio methods, which don’t rely on an initial guess, are particularly useful to initialize the iterative methods aboveand, therefore, are a fundamental element of the reconstruction pipeline.
Revised: Ab-initio methods, which don’t rely on an initial guess, are particularly useful to initialize the iterative methods above 37 and, therefore, are a fundamental element of the reconstruction pipeline.

Original: Moreover, a good initialization of theposes can speed up iterative reﬁnementby providing a non-ﬂat prior over poses,and so, reducing the search space.
Revised: Moreover, a good initialization of theposes can speed up iterative reﬁnement 41 by providing a non-ﬂat prior over poses,and so, reducing the search space.

Original: Due to the high level of noise, a ﬁrst phase of 2D classiﬁcation and averaging is performed, where asmaller denoised set of images is generated by clustering images with a similar viewing direction andaveraging images within each cluster.
Revised: Due to the high level of noise, a ﬁrst phase of 2D classiﬁcation and averaging is performed, where a 44 smaller denoised set of images is generated by clustering images with a similar viewing direction andaveraging images within each cluster.

Original: Because this blurs out the ﬁne details, it can reduce the ﬁnalreconstruction’s resolution and, therefore, it is desirable to minimize the averaging performed [7, 5].
Revised: Because this blurs out the ﬁne details, it can reduce the ﬁnal 46 reconstruction’s resolution and, therefore, it is desirable to minimize the averaging performed [7, 5].

Original: Vector Diffusion Maps (VDM), and their variants, [8, 9, 10] are powerful methods to estimate thesimilarity between images’ viewing directions and, therefore, provide a robust metric for clustering.
Revised: Vector Diffusion Maps (VDM), and their variants, [8, 9, 10] are powerful methods to estimate the 48 similarity between images’ viewing directions and, therefore, provide a robust metric for clustering.

Original: Inshort, these methods leverage the equivariance of Cryo-EM to planar rotations ( SO(2) ) and interpretthe planar rotations aligning pairs of images as an approximation of the local parallel transport over asphere of viewing directions.
Revised: Inshort, these methods leverage the equivariance of Cryo-EM to planar rotations ( SO(2) ) and interpretthe planar rotations aligning pairs of images as an approximation of the local parallel transport over a 51 sphere of viewing directions.

Original: The closer the viewing directions of two images are, the more consistentis the transport along different paths between them.
Revised: The closer the viewing directions of two images are, the more consistentis the transport along different paths between them.

Original: This notion of consistency is encoded in the topeigenvectors of an Hermitian matrix representing the discretized parallel transport operator.
Revised: This notion of consistency is encoded in the top 53 eigenvectors of an Hermitian matrix representing the discretized parallel transport operator.

Original: Contributions and Outline
Revised: Contributions and Outline

Original: 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 torecover an estimation of the absolute poses.
Revised: In this work, we consider the problem of estimating Cryo-EM images’ 55 poses under the group synchronization framework, i.e. using the relative poses of pairs of imagesto recover an estimation of the absolute poses.

Original: First, in Sec. 2, we relate the symmetries of Cryo-EMwith those of the synchronization task: we show that SO(2) relative poses are not sufﬁcient to solvethe multi-view synchronization problem, but O(2) relative poses are .
Revised: First, in Section 2, we relate the symmetries of 57 Cryo-EM with those of the synchronization task: we show that SO(2) relative poses are not sufﬁcientto solve the multi-view synchronization problem, but O(2) relative poses are .

Original: Second, in Sec. 3, we use thisinsight to construct an improved VDM method, which is able to recover not only a  similarity metricbetween the images, but also an estimation of each image’s pose.
Revised: Second, in Section 3,we use this insight to construct an improved VDM method, which is able to recover not only a 60 similarity metric between the images, but also an estimation of each image’s pose.

Original: In particular, we extend VDM by i)using vector diffusion over a projective plane (rather than a sphere) to reconstruct  denoised similarityand parallel transport and by ii) including a ﬁnal synchronization step to recover the absolute poses.
Revised: In particular, weextend VDM by i) using vector diffusion over a projective plane (rather than a sphere) to reconstruct 62 denoised similarity and parallel transport and by ii) including a ﬁnal synchronization step to recoverthe absolute poses.

Original: This minor modiﬁcation of VDM allows one to replace the 2D classiﬁcation  phase with a poseestimation phase and, therefore, leverage all the raw cryo-EM images in the ﬁnal 3D reconstructionwithout the loss of resolution caused by averaging.
Revised: This minor modiﬁcation of VDM allows one to replace the 2D classiﬁcation 64 phase with a pose estimation phase and, therefore, leverage all the raw cryo-EM images in the ﬁnal3D reconstruction without the loss of resolution caused by averaging.

Original: Additionally, our improved  VDM method sharesthe same spectral properties of the original VDM method and, hence, the same noise stability andrecovery guarantees.
Revised: Additionally, our improved 66 VDM method shares the same spectral properties of the original VDM method and, hence, the samenoise stability and recovery guarantees.

Original: Finally, we validate experimentally our theoretical results in Sec. 5.
Revised: Finally, we validate experimentally our theoretical results in 68 Section 5.

Original: 1.1 The mathematics of Cryo-EM
Revised: 1.1 The mathematics of Cryo-EM

Original: Following [11], we consider the problem of single-particle homogeneous reconstruction and assumeno image translations for simplicity.
Revised: Following [11], we consider the problem of single-particle homogeneous reconstruction and assume 71 no image translations for simplicity.

Original: Then, a molecule’s 3D density is a function : R 3 !
Revised: Then, a molecule’s 3D density is a function : R!

Original: R 2 71 L 2 ( R 3 ) with compact support around the origin of R 3 .
Revised: R 2 72 L 2 ( R 3 ) with compact support around the origin of R 3 .

Original: An observation is a gray-scale image generatedby the integral projection along the Z axis ⇧ : L 2 ( R 3 ) !
Revised: An observation is a gray-scale image generatedby the integral projection along the Z axis ⇧ : L 2 ( R 3 ) !

Original: L 2 ( R 2 ) , which is deﬁned as:
Revised: L 2 ( R 2 ) , which is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: Let { o i } Ni =1 be the collection of images observed.
Revised: Let { o i } Ni =1 be the collection of images observed.

Original: The image o i := ⇧ ( g  1 i . )
Revised: The image o i := ⇧ ( g  1 i . )

Original: 2 L 2 ( R 2 ) is theprojection of a copy of the molecule rotated by a random g  1 i 2 SO(3) , where the action of SO(3) 75 on L 2 ( R 3 ) is the standard action:
Revised: 2 L 2 ( R 2 ) is theprojection of a copy of the molecule rotated by a random g  1 i 2 SO(3) , where the action of SO(3) 76 on L 2 ( R 3 ) is the standard action:

Original: [Equation]
Revised: [Equation]

Original: An element g i 2 SO(3) is identiﬁed with a real orthonormal matrix ( x i , y i , z i ) 2 R 3 ⇥ 3 with positive 77 determinant.
Revised: An element g i 2 SO(3) is identiﬁed with a real orthonormal matrix ( x i , y i , z i ) 2 R 3 ⇥ 3 with positive 78 determinant.

Original: The three columns x i , y i , z i 2 R 3 form an orthonormal basis for R 3 .
Revised: The three columns x i , y i , z i 2 R 3 form an orthonormal basis for R 3 .

Original: We can write:
Revised: We can write:

Original: [Equation]
Revised: [Equation]

Original: In particular, if g i =
Revised: In particular, if g i =

Original: ( x i , y i , z i ) , then z i 2 R 3 is the viewing direction along which is projected, 79 while ( x i , y i ) deﬁnes the camera rotation around z i .
Revised: ( x i , y i , z i ) , then z i 2 R 3 is the viewing direction along which is projected, 80 while ( x i , y i ) deﬁnes the camera rotation around z i .

Original: We deﬁne the projection map ⇡ ( g i ) =
Revised: We deﬁne the projection map ⇡ ( g i ) =

Original: z i 2 R 3 .
Revised: z i 2 R 3 .

Revised: r similar or opposite directions are related by elements of O(2) , see also Fig.

Original:  On the symmetries of the Cryo-EM synchronization problem
Revised:  On the symmetries of the Cryo-EM synchronization problem

Original: One of the main challenges in cryo-EM is the fact that the pose g i 2 SO(3) of each observation o i is 82 unknown.
Revised: One of the main challenges in cryo-EM is the fact that the pose g i 2 SO(3) of each observation o i is 83 unknown.

Original: This prevents one to glue the observations together to reconstruct the density .
Revised: This prevents one to glue the observations together to reconstruct the density .

Original: However,the set of observations { o i } i can be used to estimate the relative poses {b
Revised: However,the set of observations { o i } i can be used to estimate the relative poses {

Original: [Equation]
Revised: [Equation]

Original: Asynchronization problem is deﬁned by a “synchronization graph” G =
Revised: Asynchronization problem is deﬁned by a “synchronization graph” G =

Original: ( V , E ) , i.e. a graph with a 85 node in
Revised: ( V , E ) , i.e. a graph with a 86 node in

Original: V per image and an edge ( i, j ) 2 E associated with b g ij if its estimation is available.
Revised: V per image and an edge ( i, j ) 2 E associated with b g ij if its estimation is available.

Original: Given a 86 synchronization graph
Revised: Given a 87 synchronization graph

Original: G , the goal is now to “synchronize” them, i.e. ﬁnd a global assignment of the 87 poses { b g i } i consistent with the estimated relative poses:
Revised: G , the goal is now to “synchronize” them, i.e. ﬁnd a global assignment of the 88 poses { b g i } i consistent with the estimated relative poses:

Original: [Equation]
Revised: [Equation]

Original: Global SO(3) ambiguity If { b g i } i is a solution, so is { g b g i } i for any g 2 SO(3) .
Revised: Global SO(3) ambiguity If { b g i } i is a solution, so is { g b g i } i for any g 2 SO(3) .

Original: This ambiguity 89 is not unexpected and corresponds to the fact that any rotated molecule g  1 . is an equally validreconstruction, i.e. cryo-EM can not recover the orientation of a molecule.
Revised: This ambiguity 90 is not unexpected and corresponds to the fact that any rotated molecule g  1 . is an equally validreconstruction, i.e. cryo-EM can not recover the orientation of a molecule.

Original: .1 Relative 
Revised: 2.1 Relative poses in

Original: poses in SO(2): the impossibility of synchronization
Revised: SO(2) the impossibility of synchronization

Original: Before solving the synchronization problem, it is necessary to estimate the relative poses { g ij =g  

Revised: Before solving the synchronization problem, it is necessary to estimate the relative poses { g ij =g

Original: j g i } ij .
Revised: j g i } ij .

Original: Unfortunately, since the projection ⇧ loses information about the 3D structure, it is not 94 possible to estimate the relative pose of any pair of images.
Revised: Unfortunately, since the projection ⇧ loses information about the 3D structure, it is not 95 possible to estimate the relative pose of any pair of images.

Original: However, when the viewing directions⇡ ( g i ) =
Revised: However, when the viewing directions⇡ ( g i ) =

Original: z i and ⇡ ( g j ) =
Revised: z i and ⇡ ( g j ) =

Original: z j of two images are sufﬁciently close, they generally differ only by aplanar rotation r ij 2 SO(2) which can be directly estimated as
Revised: z j of two images are sufﬁciently close, they generally differ only by a 97 planar rotation r ij 2 SO(2) which can be directly estimated as

Original: [Equation]
Revised: [Equation]

Original:  where r acts on image o i by rotating it.
Revised: ) where r acts on image o i by rotating it.

Original: The closer z i and z j are, the better r ij approximates g ij .
Revised: The closer z i and z j are, the better r ij approximates g ij .

Original: Indeed, if z i =
Revised: Indeed, if z i =

Original: z j , then g ij = g  1 j g i has form
Revised: z j , then g ij = g  1 j g i has form

Original: [Equation]
Revised: [Equation]

Original: i , where the top-left 2 ⇥ 2 block is an SO(2)matrix.
Revised: i , where the top-left 2 ⇥ 2 block is an SO(2)matrix.

Original: We identify all the rotations of this form as the subgroup 2 SO(2) ⇠
Revised: We identify all the rotations of this form as the subgroup 2 SO(2) ⇠

Original: = R z < SO(3) of planarrotations around the Z axis.
Revised: = R z < SO(3) of planar 101 rotations around the Z axis.

Original: The projection operator ⇧ is SO(2) - equivariant 3 , i.e.:
Revised: The projection operator ⇧ is SO(2) - equivariant 3 , i.e.:

Original: [Equation]
Revised: [Equation]

Original: Hence, if g  1 j = rg  1 i , it follows that o j = r.o i .
Revised: Hence, if g  1 j = rg  1 i , it follows that o j = r.o i .

Original: For example, all projections generated by the topview in Fig 2 are related by a rotations r as shown in the top row of Fig.
Revised: For example, all projections generated by the top 103 view in Fig 2 are related by a rotations r as shown in the top row of Fig.

Original: Global SO(2) ambiguity While this is a common approach in the literature, e.g. in VDM [8, 12,10, 13], it does not provide sufﬁcient information to estimate the poses { g i } i .
Revised: Global SO(2) ambiguity While this is a common approach in the literature, e.g. in VDM [8, 12, 105 10, 13], it does not provide sufﬁcient information to estimate the poses { g i } i .

Original: Indeed, the estimated 105 relative rotations { r ij 
Revised: Indeed, the estimated 106 relative rotations { r ij 2 SO

Original: [Equation]
Revised: [Equation]

Original: SO

Original: [Equation]

Original: [Equation]

Original: ij

Original: [Equation]
Revised: [Equation]

Original:  2 G yield the following constraints:
Revised: ij ) 2 G yield the following constraints:

Original: [Equation]
Revised: [Equation]

Original: Since SO(2) is abelian , if { b g i } i is a solution to this set of constraints, then { b g i r } i is an equally valid 107 solution for any r
Revised: Since SO(2) is abelian , if { b g i } i is a solution to this set of constraints, then { b g i r } i is an equally valid 108 solution for any r

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: This implies that any method which estimates the poses { g i } i from the estimated relative ones{ r ij 2 SO(2) } ij will only be able to recover them up to a global rotation r 2 R z .
Revised: This implies that any method which estimates the poses { g i } i from the estimated relative ones{ r ij 2 SO(2) } ij will only be able to recover them up to a global rotation r 2 R z .

Original: This global 110 ambiguity prevents using these estimations to directly invert the linear projection ⇧ and recover theoriginal molecule , since it would result in averaging randomly rotated versions of the images.
Revised: This global 111 ambiguity prevents using these estimations to directly invert the linear projection ⇧ and recover theoriginal molecule , since it would result in averaging randomly rotated versions of the images.

Original: Nevertheless, the ambiguous poses { g i r } i still contain information about the viewing direction of 113 the projections.
Revised: Nevertheless, the ambiguous poses { g i r } i still contain information about the viewing direction of 114 the projections.

Original: Indeed, note that ⇡ ( g i r ) =
Revised: Indeed, note that ⇡ ( g i r ) =

Original: ⇡ ( g i ) , i.e. the viewing directions are invariant to thisglobal ambiguity and, therefore, can be recovered from this method.
Revised: ⇡ ( g i ) , i.e. the viewing directions are invariant to thisglobal ambiguity and, therefore, can be recovered from this method.

Original: Previous works such as VDM[8, 12, 10] exploit this invariance to estimate the relative distance between different images.
Revised: Previous works such as VDM 116 [8, 12, 10] exploit this invariance to estimate the relative distance between different images.

Original: Thisdistance is generally used for class averaging i.e. to cluster images and average images within eachcluster to generate a smaller, de-noised, images dataset.
Revised: Thisdistance is generally used for class averaging i.e. to cluster images and average images within each 118 cluster to generate a smaller, de-noised, images dataset.

Original: 2.2 Relative poses in O(2) : a sufﬁcient condition for synchronization
Revised: 2.2 Relative poses in O(2) : a sufﬁcient condition for synchronization

Original: In this work, we solve this global SO(2) ambiguity and directly estimate the absolute poses { g i } i 120 of the images.
Revised: In this work, we solve this global SO(2) ambiguity and directly estimate the absolute poses { g i } i 121 of the images.

Original: To do so, we will exploit another symmetry of Cryo-EM.
Revised: To do so, we will exploit another symmetry of Cryo-EM.

Original: Indeed, the tomographicprojection ⇧ is also invariant to mirroring along the Z axis m
Revised: Indeed, the tomographicprojection ⇧ is also invariant to mirroring along the Z axis m

Original: z = h {+1 0 0 0 1 0 
+}
Revised: z = h {+1 0 0 0 1 0 
+}

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: O

Original: [Equation]
Revised: [Equation]

Revised: i.e.:

Original: [Equation]
Revised: [Equation]

Original: This additional symmetry turns out useful to break the global SO(2) symmetry explained earlier.
Revised: This additional symmetry turns out useful to break the global SO(2) symmetry explained earlier.

Original: First, let r y = 
Revised: First, let r y = h

Original: [Equation]
Revised: [Equation]

Original:  1 i 2 SO(3) be a ⇡ rotation along the Y axis; note that r y =
Revised:  1 i 2 SO(3) be a ⇡ rotation along the Y axis; note that r y =

Original: m
Revised: m

Original: z m x , wherem x is a mirroring along the X axis.
Revised: z m x , wherem x is a mirroring along the X axis.

Original: Then, let f = ⇥ [Equation]{+ 1 0 
+}
Revised: Then, let f = ⇥ {+ 1 0 
+}

Original:  2 O(2) be the ﬂip of a planar image along 125 the X axis.
Revised: ⇤ 2 O(2) be the ﬂip of a planar image along 126 the X axis.

Original: Then:
Revised: Then:

Original: [Equation]
Revised: [Equation]

Original: i.e. the projection operator ⇧ is ﬂip equivariant : projections along a direction ( ⇧ ( g  1 i . ) )
Revised: i.e. the projection operator ⇧ is ﬂip equivariant : projections along a direction ( ⇧ ( g  1 i . ) )

Original: arerelated to projections from the opposite direction ( f .
Revised: arerelated to projections from the opposite direction ( f .

Original: ⇧ ( r y g  1 i . ) )
Revised: ⇧ ( r y g  1 i . ) )

Original: by a planar reﬂection f .
Revised: by a planar reﬂection f .

Original: Forexample, all projections generated by the bottom view in Fig 2 are related by a rotation r followed bythe reﬂection f with the images generated from the top view, as shown in the bottom row of Fig.
Revised: For 129 example, all projections generated by the bottom view in Fig 2 are related by a rotation r followed bythe reﬂection f with the images generated from the top view, as shown in the bottom row of Fig.

Original: Tosimplify the notation, we will also let r y act on a image; this action should be intended as that of f .
Revised: To 131 simplify the notation, we will also let r y act on a image; this action should be intended as that of f .

Original: That means that we can not only estimate the relative pose g ij ⇡ h ij 2 R z when the viewing direc- 132 tions ⇡ ( g i ) and ⇡ ( g j ) are sufﬁciently close, but also
Revised: That means that we can not only estimate the relative pose g ij ⇡ h ij 2 R z when the viewing direc- 133 tions ⇡ ( g i ) and ⇡ ( g j ) are sufﬁciently close, but also

Original: h ij 2 H ⇠ = O(2) when ⇡ ( g i ) ⇡  ⇡ ( g j ) , i.e.:
Revised: h ij 2 H ⇠ = O(2) when ⇡ ( g i ) ⇡  ⇡ ( g j ) , i.e.:

Original: [Equation]
Revised: [Equation]

Original: 6) where H ⇠ = O(2) is the subgroup 4 of SO(3) containing R z (planar rotations along the Z axis) and r y .
Revised: if [Equation] where H ⇠ = O(2) is the subgroup 4 of SO(3) containing R z (planar rotations along the Z axis) and r y .

Original: Global chirality ambiguity The relative poses { h ij 2 O(2) } ij still don’t fully constrain theabsolute ones; if { b g i } i is a solution, so is { b g
Revised: Global chirality ambiguity The relative poses { h ij 2 O(2) } ij still don’t fully constrain theabsolute ones; if { b g i } i is a solution, so is { b g

Original: i r z } i , with r z 2 R z being a ⇡ -rotation around the Z axis 5 :
Revised: i r z } i , with r z 2 R z being a ⇡ -rotation around the Z axis 5 :

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: This ambiguity is related to the well known problem that Cryo-EM can not recover the chirality of amolecule.
Revised: This ambiguity is related to the well known problem that Cryo-EM can not recover the chirality of amolecule.

Original: To see why, let i =
Revised: To see why, let i =

Original: h

Original: [Equation]

Original: [Table]

Original: 0

Original: [Equation]

Original: [Table]
Revised: [Table]

Original: 0 1 i be the inversion , i.e. a mirroring along all axes; note thatm z
Revised:  1 i be the inversion , i.e. a mirroring along all axes; note thatm z

Original: = r z i and that i commutes with any element g 2 SO(3) .
Revised: = r z i and that i commutes with any element g 2 SO(3) .

Original: Then:
Revised: Then:

Original: [Equation]
Revised: [Equation]

Original:  i.e., o i is equally likely to be generated by the molecule with pose g i or its mirrored version i .with
Revised: ) i.e., o i is equally likely to be generated by the molecule with pose g i or its mirrored version i .

Original:  pose g
Revised: 141 with pose g

Original: i r z .
Revised: i r z .

Original: Together with the global SO(3) symmetry introduced earlier, this means Cryo-EMintrinsically suffers from a global O(3) ambiguity which we can not hope to resolve.
Revised: Together with the global SO(3) symmetry introduced earlier, this means Cryo-EMintrinsically suffers from a global O(3) ambiguity which we can not hope to resolve.

Original: Nevertheless,since r z is the only element of SO(3) commuting with all elements in H ⇠ = O(2) , we have thatrelative poses in H constrain the synchronization problem precisely up to this O(3) symmetry and,thus provide sufﬁcient constraints to identify the images’ poses.
Revised: Nevertheless, 143 since r z is the only element of SO(3) commuting with all elements in H ⇠ = O(2) , we have that 144 relative poses in H constrain the synchronization problem precisely up to this O(3) symmetry and,thus provide sufﬁcient constraints to identify the images’ poses.

Original: We provide a theoretical argumentbased on Vector Diffusion Maps (VDM) [9] supporting this informal claim in Apx. C.
Revised: We provide a theoretical argument 146 based on Vector Diffusion Maps (VDM) [9] supporting this informal claim in Appendix C.

Original:  Multi-Frequency Vector Diffusion Maps on the Projective Plane
Revised:  Multi-Frequency Vector Diffusion Maps on the Projective Plane

Original: 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 { g i } i .
Revised: In this section, we use the new insights on the symmetries of cryo-EM to enable the Vector Diffusion 149 Map (VDM) method to recover the poses { g i } i .

Original: The key component  is the Graph  Connection Lapla- 149 cian (GCL) of the synchronization graph
Revised: The key component of the method is the Graph 150 Connection Laplacian (GCL) of the synchronization graph

Original: G with the estimated relative poses { h ij 2  O(2) } ij on its150 edges.
Revised: G with the estimated relative poses{ h ij 2 H ⇠ = O(2) } ij on its edges.

Original: Under certain conditions, its eigenvectors converge to the eigen-vector-ﬁelds of the connection Laplacian operator on the projective plane, which contain information about the absolute poses.
Revised: Under certain conditions, its eigenvectors converge to the eigen- 152 vector-ﬁelds of the connection Laplacian operator on the projective plane, which contain informationabout the absolute poses.

Original: Ourmethod involves two steps : a  denoising step (like the original  VDM) and a  synchronization step ; Alg. 1provides an overview of the proposed method.We study its computational complexity in Apx. F.
Revised: Our method involves two steps: a ﬁrst denoising step (like the original 154 VDM) and a ﬁnal synchronization step; Algorithm 1 provides an overview of the proposed method.

Original: 3.1 Quotient Spaces and Projective Plane
Revised: 3.1 Quotient Spaces and Projective Plane

Original: Each image o i is uniquely identiﬁed by an element g i 2 SO(3) via Eq.
Revised: Each image o i is uniquely identiﬁed by an element g i 2 SO(3) via Eq.

Original: 1, assuming the molecule has 156 no symmetries.
Revised: 1, assuming the molecule has 157 no symmetries.

Original: Two images o i and o j related by a rotation r ij 2 R z contain the same information 157 and are considered equivalent.
Revised: Two images o i and o j related by a rotation r ij 2 R z contain the same information 158 and are considered equivalent.

Original: We can deﬁne a similar equivalence relation on their respective poses,i.e. g i ⇠ g j = g i r ij .
Revised: We can deﬁne a similar equivalence relation on their respective poses,i.e. g i ⇠ g j = g i r ij .

Original: This deﬁnes the quotient space SO(3) /R
Revised: This deﬁnes the quotient space SO(3) /R

Original: z which is isomorphic to the sphere
Revised: z which is isomorphic to the sphere

Original: S 2 .
Revised: S 2 .

Original: An element g i 2 SO(3) is interpreted as a point ⇡ ( g i ) =
Revised: An element g i 2 SO(3) is interpreted as a point ⇡ ( g i ) =

Original: z i in S 2 and a choice of frame on it, i.e. 160 the choice of basis 6 ( x i , y i ) for the tangent space at z
Revised: z i in S 2 and a choice of frame on it, i.e. 161 the choice of basis 6 ( x i , y i ) for the tangent space at z

Original: i or, equivalently, the rotation of the camera 161 around the axis z i .
Revised: i or, equivalently, the rotation of the camera 162 around the axis z i .

Original: Then, the relative rotation r ij 2 R z estimated in Eq.
Revised: Then, the relative rotation r ij 2 R z estimated in Eq.

Original: 3 is interpreted as the uniquerotation such that g j r  1 ij is the parallel transport of g i along the shortest geodesic connecting ⇡
Revised: 3 is interpreted as the uniquerotation such that g j r  1 ij is the parallel transport of g i along the shortest geodesic connecting ⇡

Original: ( g i ) to⇡ ( g j ) (this is unique unless ⇡ ( g i )
Revised: ( g i ) to⇡ ( g j ) (this is unique unless ⇡ ( g i )

Original: =
Revised: =

Original:  ⇡ ( g j ) ).
Revised:  ⇡ ( g j ) ).

Original: Here, we want to leverage the larger symmetry H .
Revised: Here, we want to leverage the larger symmetry H .

Original: We 164 deﬁne the new equivalence relation g ⇠ gh for all h 2 H on SO(3) , which determines the quotient 165 space SO(3) /H , isomorphic to the projective plane 7 P R 2 .
Revised: We 165 deﬁne the new equivalence relation g ⇠ gh for all h 2 H on SO(3) , which determines the quotient 166 space SO(3) /H , isomorphic to the projective plane 7 P R 2 .

Original: While it is harder to visualize, elements of
Revised: While it is harder to visualize, elements of

Original: SO(3) are now interpreted as points in P R 2 together with a choice of frame for their tangent spaces 8 .
Revised: SO(3) are now interpreted as points in P R 2 together with a choice of frame for their tangent spaces 8 .

Original: Finally, the relative pose h ij 2 H from Eq.
Revised: Finally, the relative pose h ij 2 H from Eq.

Original: 6 is interpreted as the element such that g j h  1 ij is theparallel transport of g i along the shortest geodesic between the projections of g i and g j on P R 2 .
Revised: 6 is interpreted as the element such that g j h  1 ij is theparallel transport of g i along the shortest geodesic between the projections of g i and g j on P R 2 .

Original: 3.2 Vector Diffusion Maps on P R 2
Revised: 3.2 Vector Diffusion Maps on P R 2

Original: An initial synchronization graph G can be generated by comparing all pairs of images using Eq.
Revised: An initial synchronization graph G can be generated by comparing all pairs of images using Eq.

Original: 6 171 and preserving only the edges with smallest distances.
Revised: 6 172 and preserving only the edges with smallest distances.

Original: Ideally, this graph is a discretization of
Revised: Ideally, this graph is a discretization of

Original: P R 2 ⇠ = SO(3) /H with local connectivity, but the noise on the images negatively affects the estimateddistances and relative poses, introducing “shortcut” edges in this graph.
Revised: P R 2 ⇠ = SO [Equation] 3) /H with local connectivity, but the noise on the images negatively affects the estimated 174 distances and relative poses, introducing “shortcut” edges in this graph.

Original: Therefore, the ﬁrst step ofour method is re-estimating the geodesic distances and the relative poses of the points in the graph.
Revised: Therefore, the ﬁrst step ofour method is re-estimating the geodesic distances and the relative poses of the points in the graph.

Original: Like previous works, we leverage the consistency of the local alignments to do this.
Revised: Like previous works, we leverage the consistency of the local alignments to do this.

Original: Indeed, sincea manifold is locally Euclidean, parallel transport along sufﬁciently short paths is approximatelypath-independent in the local neighborhood of a point.
Revised: Indeed, since 177 a manifold is locally Euclidean, parallel transport along sufﬁciently short paths is approximatelypath-independent in the local neighborhood of a point.

Original: On the other hand, the further away twopoints are, the more inconsistent the cycles through them will be.
Revised: On the other hand, the further away two 179 points are, the more inconsistent the cycles through them will be.

Original: When summing the transport overmultiple paths between two points, inconsistent paths tend to average out.
Revised: When summing the transport overmultiple paths between two points, inconsistent paths tend to average out.

Original: We use the synchronization graph to build a discretization of this local parallel transport operator.
Revised: We use the synchronization graph to build a discretization of this local parallel transport operator.

Original: Letthe graph connection Laplacian W be the 2 N ⇥ 2 N matrix with its i, j -th 2 ⇥ 2 block deﬁned as:
Revised: Let 182 the graph connection Laplacian W be the 2 N ⇥ 2 N matrix with its i, j -th 2 ⇥ 2 block deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where ⇢ ( h ij ) is the standard representation of h ij 2 O(2) as a 2 ⇥ 2 orthogonal matrix.
Revised: where ⇢ ( h ij ) is the standard representation of h ij 2 O(2) as a 2 ⇥ 2 orthogonal matrix.

Original: Let D be the 183 2 N ⇥ 2 N diagonal matrix containing the degree of each node in G (repeated twice) in its diagonal.
Revised: Let D be the 184 2 N ⇥ 2 N diagonal matrix containing the degree of each node in G (repeated twice) in its diagonal.

Original: The matrix W acts by parallel-transporting tangent vectors along each edge of the graph while
Revised: The matrix W acts by parallel-transporting tangent vectors along each edge of the graph while

Original: A := D  1 W averages all vectors transported to a node.
Revised: A := D  1 W averages all vectors transported to a node.

Original: Then, W t transports vectors along all lengtht paths in G and the block
Revised: Then, W t transports vectors along all length- 187 t paths in G and the block

Original: A t ( i, j ) is the average of the transport over all length- t paths from i to j . {+
+}
Revised: A t ( i, j ) is the average of the transport over all length- t paths from i to j . {+
+}

Original: If the paths are mostly inconsistent, A t ( i, j ) tends to 0 , being the average of uncorrelated orthogonalmatrices.
Revised: If the paths are mostly inconsistent, A t ( i, j ) tends to 0 , being the average of uncorrelated orthogonalmatrices.

Original: Hence, A t ( i, j ) provides information about the geodesic distance between i and j .
Revised: Hence, A t ( i, j ) provides information about the geodesic distance between i and j .

Original: Like [12], we take a multi-frequency approach to improve robustness to noise.
Revised: Like [12], we take a multi-frequency approach to improve robustness to noise.

Original: In practice, we builda different graph connection Laplacian A k for different choices 9 of irreducible representation (irrep) 191 ⇢ k of O(2) .
Revised: In practice, we build 191 a different graph connection Laplacian A k for different choices 9 of irreducible representation (irrep)⇢ k of O(2) .

Original: For k > 0 , ⇢ k is a 2 ⇥ 2 orthogonal matrix with rotational frequency k ; see Apx. A.
Revised: For k > 0 , ⇢ k is a 2 ⇥ 2 orthogonal matrix with rotational frequency k ; see Appendix A.

Original: In the limit t 7! 1 , the power iteration of A tk converges to its top eigenspace and A tk ( i, j ) will be the 193 product of the top eigenvectors of A k .
Revised: In the limit t 7! 1 , the power iteration of A tk converges to its top eigenspace and A tk ( i, j ) will be the 194 product of the top eigenvectors of A k .

Original: It is therefore important to discuss the spectral properties of
Revised: It is therefore important to discuss the spectral properties of

Original: A k .
Revised: A k .

Original: [14] proved that the normalized graph connection Laplacian is a discretization of the connection  Laplacian operator and that, under certain conditions, its eigenvectors and eigenvalues convergeto those of the connection Laplacian operator.
Revised: [14] proved that the normalized graph connection Laplacian is a discretization of the connection 196 Laplacian operator and that, under certain conditions, its eigenvectors and eigenvalues converge tothose of the connection Laplacian operator.

Original: In Apx. B, we study the spectral properties of theconnection Laplacian operator over P R 2 and relate it with the operator over S 2 studied in [13].
Revised: In Appendix B, we study the spectral properties of the 198 connection Laplacian operator over P R 2 and relate it with the operator over S 2 studied in [13].

Original: Inparticular, we show that the eigenspaces of our Laplacian operator over P R 2 are subspaces of theeigenspaces of the spherical Laplacian and they share the same eigenvalues.
Revised: In 199 particular, we show that the eigenspaces of our Laplacian operator over P R 2 are subspaces of theeigenspaces of the spherical Laplacian and they share the same eigenvalues.

Original: This also implies ourgraph connection Laplacians enjoy the same spectral properties studied in [13], including the spectralgap and the robustness to noise.
Revised: This also implies our 201 graph connection Laplacians enjoy the same spectral properties studied in [13], including the spectralgap and the robustness to noise.

Original: For frequency k > 0 , the top eigen-space of A k is 2 k + 1 dimensional.
Revised: For frequency k > 0 , the top eigen-space of A k is 2 k + 1 dimensional.

Original: Additionally, denote' k : V !
Revised: Additionally, denote' k : V !

Original: R (2 k +1) ⇥ 2 the stack of the top 2 k + 1 eigen-vectors.
Revised: R (2 k +1) ⇥ 2 the stack of the top 2 k + 1 eigen-vectors.

Original: Let ' k
Revised: Let ' k

Original: ( i ) =
Revised: ( i ) =

Original: pk ' k ( i ) k F ' k ( i ) anddeﬁne c
Revised: pk ' k ( i ) k F ' k ( i ) anddeﬁne c

Original: A k ( i, j ) = ' k ( j ) T ' k ( i ) 2 R 2 ⇥ 2 .
Revised: [Equation]

Original: Then:
Revised: Then:

Original: Theorem 3.1.
Revised: Theorem 3.1.

Original: Let s ij =
Revised: Let s ij =

Original: h
Revised: h

Original: z j , z i
Revised: z j , z i

Original: i =
Revised: i =

Original: h ⇡ ( g i ) , ⇡ ( g j ) i
Revised: h ⇡ ( g i ) , ⇡ ( g j ) i

Original: [  1 , 1] be the cosine similarity of the viewingdirections ⇡ ( g i ) , ⇡ ( g j ) 2 S 2 .
Revised: [  1 , 1] be the cosine similarity of the viewing 207 directions ⇡ ( g i ) , ⇡ ( g j ) 2 S 2 .

Original: The two following identities hold:
Revised: The two following identities hold:

Original: [Equation]
Revised: [Equation]

Original: The proof is in Apx. D. Since we need a similarity over P R 2 rather than S 2 , we use b w ij := | s ij | .
Revised: The proof is in Appendix D. Since we need a similarity over P R 2 rather than S 2 , we use b w ij := | s ij | .

Original: In208 other words, the top eigenspace of A k allows the estimation of the geodesic distance between points.
Revised: In other words, the top eigenspace of A k allows the estimation of the geodesic distance between points.

Original: Denoised similarity and parallel transport
Revised: Denoised similarity and parallel transport

Original: The information obtained from the different A k 210 matrices can be combined in multiple ways to obtain an estimation of s ij .
Revised: The information obtained from the different A k 211 matrices can be combined in multiple ways to obtain an estimation of s ij .

Original: Deﬁne
Revised: Deﬁne

Original: [Equation]
Revised: [Equation]

Original: For numerical stability, we estimate s ij as
Revised: For numerical stability, we estimate s ij as

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: ,j ) 2 E 0 else 0{+
+} i ij.
Revised: j ) 2 E 0 else {+
+} i ij

Original: Build denoised graph connection Laplacian; Eq.11: x,y ' top-eigenvectors ( A 0 , 3).
Revised: [Table] [Table] .11: x,y ' top-eigenvectors ( A 0 , 3)

Original: Final Synchronization 12:
Revised: [Table]

Original:  b g i projectSVD
Revised: 12: b g i projectSVD

Original: ( x ( i ) ,y ( i ) ,x ( i ) ⇥
Revised: ( x ( [Equation] ,y ( [Equation]

Revised: [Equation]

Original: y ( i )) where L is the largest frequency considered.
Revised: ⇥ [Equation] i )) where L is the largest frequency considered.

Original: Additionally, note that, due to the decaying spectrumof the Laplacian operators, the noise has a stronger effect on the lowest eigenvalues; discarding thelower eigenvectors also helps denoising the GCL matrices.
Revised: Additionally, note that, due to the decaying spectrum 214 of the Laplacian operators, the noise has a stronger effect on the lowest eigenvalues; discarding thelower eigenvectors also helps denoising the GCL matrices.

Original: That means that the top eigenvectors of 215 A k can be used to partially denoise the frequency- k parallel transport.
Revised: That means that the top eigenvectors of 216 A k can be used to partially denoise the frequency- k parallel transport.

Original: Next, we combine all denoisedblocks { c
Revised: Next, we combine all denoisedblocks { c

Original: A k ( i, j ) } k by interpreting them as the Fourier coefﬁcients of a function on H ; we estimate 217 the relative pose with the element h
Revised: A k ( i, j ) } k by interpreting them as the Fourier coefﬁcients of a function on H ; we estimate 218 the relative pose with the element h

Original: [Equation]
Revised: [Equation]

Original: maximizing this function:
Revised: maximizing this function:

Original: [Equation]
Revised: [Equation]

Original: Final Synchronization So far, we have used the MFVDM method on P R 2 to estimate the similarityb w ij =
Revised: Final Synchronization So far, we have used the MFVDM method on P R 2 to estimate the similarityb [Equation] ij =

Original: |
Revised: |

Original: b s ij | and parallel transport b h ij over P R 2 .
Revised: b s ij | and parallel transport b h ij over P R 2 .

Original: With respect to [12], our method includes a ﬁnalsynchronization step, where these quantities are used to recover { b g i } i .
Revised: With respect to [12], our method includes a ﬁnal 221 synchronization step, where these quantities are used to recover { b g i } i .

Original: We construct a new graph 221 connection Laplacian matrix A 0 as in Eq.
Revised: We construct a new graph 222 connection Laplacian matrix A 0 as in Eq.

Original: 7 using the new estimations and the standard representation( k = 1 ) of O(2) .
Revised: 7 using the new estimations and the standard representation( k = 1 ) of O(2) .

Original: The top eigen-space of A 0 is 3 dimensional and its 3 top eigenvectors deﬁne atangent frame at each node in the following way.
Revised: The top eigen-space of A 0 is 3 dimensional and its 3 top eigenvectors deﬁne atangent frame at each node in the following way.

Original: Let ' :
Revised: Let ' :

Original: V !
Revised: V !

Original: R 3 ⇥ 2 be the stack of the top 3eigen-vectors, ' ( i ) =
Revised: R 3 ⇥ 2 be the stack of the top 3eigen-vectors, ' ( i ) =

Original: pk ' ( i ) k F ' ( i ) and x : V !
Revised: pk ' ( i ) k F ' ( i ) and x : V !

Original: R 3 and y : V !
Revised: R 3 and y : V !

Original: R 3 be its two columns.
Revised: R 3 be its two columns.

Original: Then,x ( i ) and y ( i ) 2 R 3 deﬁne a basis for the tangent space at ⇡ ( g i ) 2 S 2 , see Apx.
Revised: Then,x ( i ) and y ( i ) 2 R 3 deﬁne a basis for the tangent space at ⇡ ( g i ) 2 S 2 , see Appendix C. Recall that 227 g

Original: C. Recall that 226 g i =
Revised: i =

Original: ( x i , y i , z i ) 2 SO(3) , then x ( i ) and y ( i ) approximates respectively x i and y i , while z i can227 be  recovered 10 as z ( i ) =
Revised: ( x i , y i , z i ) 2 SO(3) , then x ( i ) and y ( i ) approximates respectively x i and y i , while z i can be 228 recovered 10 as z ( i ) =

Original: x ( i ) ⇥
Revised: x ( i ) ⇥

Original: y ( i ) .
Revised: y ( i ) .

Original: Note that recovered poses { b g i =
Revised: Note that recovered poses { b g i =

Original: ( x ( i ) , y ( i ) , z ( i ))
Revised: ( x ( i ) , y ( i ) , z ( i ))

Original: } i present228 the  global O(3) symmetry discussed in Sec. 2.
Revised: } i present the 229 global O(3) symmetry discussed in Section 2.

Original: Indeed, eigenvalue decomposition is unique up toan orthogonal change of basis in each eigenspace.
Revised: Indeed, eigenvalue decomposition is unique up toan orthogonal change of basis in each eigenspace.

Original: Since the top eigenspace is 3 dimensional, thesolution is unique up to a global g 2 O(3) transformation, i.e. if ' is a set of orthogonal eigenvectors, 231 so is g ' , deﬁned as [ g ' ]
Revised: Since the top eigenspace is 3 dimensional, the 231 solution is unique up to a global g 2 O(3) transformation, i.e. if ' is a set of orthogonal eigenvectors, 232 so is g ' , deﬁned as [ g ' ]

Original: ( i ) =
Revised: ( i ) =

Original: g ' ( i ) .
Revised: g ' ( i ) .

Original: Hence, all tangent frames { ( x ( i ) , y ( i ))
Revised: Hence, all tangent frames { ( x ( i ) , y ( i ))

Original: } i can be simultaneously 232 rotated by any g 2 O(3) .
Revised: } i can be simultaneously 233 rotated by any g 2 O(3) .

Original: Finally, because x ( i ) and y ( i ) are not perfectly unitary and orthogonal toeach other, the matrix ( x ( i ) , y ( i ) , z ( i )) will not be orthogonal; therefore, we project it to the closest234 SO(3) element via SVD.
Revised: Finally, because x ( i ) and y ( i ) are not perfectly unitary and orthogonal to 234 each other, the matrix ( x ( i ) , y ( i ) , z ( i )) will not be orthogonal; therefore, we project it to the closest SO(3) element via SVD.

Original: Limitations  VDM  assumes  uniformly distributed poses, which might not  always be true; normalization methods such as [15, 16] can help relaxing this requirement.
Revised: Limitations The VDM method assumes the poses to be uniformly distributed, which might not 237 always be true in practice.

Original: Our analysis of the synchronizationsymmetries, however, is not limited to VDM but applies to any cryo-EM synchronization method.
Revised: Our analysis of the synchronization symmetries, however, is not limitedto VDM but applies to any cryo-EM synchronization method.

Original: 2Dclassiﬁcation or pose  estimation methods which do not  account for the image formation model mightfail at the lowest SNR regimes [5].
Revised: Moreover, 2D classiﬁcation or pose 239 estimation methods which do not explicitly account for the image formation model might fail atthe lowest SNR regimes [5].

Original: Indeed, if the noise  is too high, the  relative  poses estimation itselfcan be impossible and increasing the number of observations can not solve this problem [17, 18], although methods like steerable PCA [19] can alleviate that.
Revised: Indeed, if the noise level is too high, the estimation of the relative 241 poses itself can be impossible and increasing the number of observations can not solve this issue[15, 16].

Original: Nevertheless, in practice, 2D classiﬁcationwith VDM is leveraged as a denoising step  in reconstruction pipelines (e.g. in ASPIRE); our methodcan replace this step, providing pose estimation and reducing the need for averaging.
Revised: Nevertheless, in practice, 2D classiﬁcation with VDM is leveraged as a denoising step 243 in reconstruction pipelines (e.g. in ASPIRE); our method can replace this step, providing poseestimation and reducing the need for averaging.

Original: Finally, whilewe assumed non-symmetric molecules,  symmetries directly affect the spectral properties of the GCLmatrices and can theoretically be accounted for.
Revised: Finally, while we assumed non-symmetric molecules, 245 symmetries directly affect the spectral properties of the GCL matrices and can theoretically beaccounted for.

Original: We leave this as future work.
Revised: We leave this as future work.

Revised: [Equation]

Revised: [Equation]

Original:  Related Works
Revised:  Related Works

Original: [Figure]
Revised: [Figure]

Original: Vector-Diffusion Maps (VDM) were initially proposed in [9, 8] for solving the cryo-EM synchronization problem with SO(2) constraints.
Revised: Vector-Diffusion Maps (VDM) were initially proposed in [9, 8] for solving the cryo-EM synchro- 249 nization problem with SO(2) constraints.

Original: The O(2) symmetry was noted also in [8] which suggestsaugmenting a dataset with mirrored images; still, it only uses SO(2) relative alignments and, therefore,recovers only a similarity metric over the sphere.
Revised: The O(2) symmetry was noted also in [8] which suggestsaugmenting a dataset with mirrored images; still, it only uses SO(2) relative alignments and, therefore,recovers only a similarity metric over the sphere.

Original: VDMs were later extended to a multi-frequencysetting in [12, 10, 13].
Revised: VDMs were later extended to a multi-frequency 252 setting in [12, 10, 13].

Original: [20] describes a strategy based on the classical method of moments to solvethe generic problem of recovering orbits from invariants.
Revised: [17] describes a strategy based on the classical method of moments to solvethe generic problem of recovering orbits from invariants.

Original: Additionally, the authors provide theoreticalanalysis 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 1 SNR 3 . {+
+}
Revised: Additionally, the authors provide theoreti- 254 cal analysis of sample complexity of such problems; in particular, they show that in the cryo-EMsynchronization problem with uniform viewing distribution, the sample complexity scales as 1 SNR 3 . {+
+}

Original: Some 255 works exploit the property that the Fourier transforms of any pair of cryo-EM images must agree on aline passing through the origin ( common lines approach ) to deﬁne a set of constraints on each pair ofabsolute poses.
Revised: [18] exploits the property that the Fourier transforms of any pair of cryo-EM images must agree on aline passing through the origin ( common lines approach ) to deﬁne a set of constraints on each pair of 258 absolute poses.

Original: [21] use a Semi-Deﬁnite Programming (SDP) relaxation to ﬁnd the solution; however,
Revised: Then, they use a Semi-Deﬁnite Programming (SDP) relaxation to ﬁnd the solution.

Original:  SDP is known for its high computational cost. Instead, [7, 22] rely on a spectral relaxation of theproblem; still, common lines estimation is very  sensitive to noise and is generally performed only aftera ﬁrst 2D classiﬁcation and averaging phase [23].
Revised: However, SDP is known for its high computational cost; moreover, common lines estimation is very 260 sensitive to noise and is generally performed only after a ﬁrst 2D classiﬁcation and averaging phase.

Original: Popular reconstruction methods include RELION[24] and cryoSPARC [25], which are based on  an EM-like algorithm.
Revised: Popular reconstruction methods include RELION [19] and cryoSPARC [20], which are based on 262 an EM-like algorithm.

Original: [26] suggested using stochasticrather than batch optimization for iterative reﬁnement, thereby improving the computational cost.
Revised: [21] suggested using stochastic rather than batch optimization for iterativereﬁnement, thereby improving the computational cost.

Original: Recent advances employ neural networks optimized by SGD to represent the 3D density [27, 28].
Revised: Recent advances employ neural networks opti- 264 mized by SGD to represent the 3D density [22, 23].

Original: Note that these methods still need to periodically perform an expensive search over the images’ posesand [28] found resetting the model’s weights  throughout training essential for convergence.
Revised: Note that these methods still need to periodicallyperform an expensive search over the images’ poses and [23] found resetting the model’s weights 266 throughout training essential for convergence.

Original:  Experiments
Revised:  Experiments

Original: [Figure]
Revised: [Figure]

Original: [Figure] the synthetic vector dataset. We com- pare our method with O(2) relative poses with the original MFVDM with SO(2) poses.
Revised: the original MFVDM with SO(2) poses.

Original: We ﬁrst evaluate our method on a synchronizationtask over some synthetic datasets of vectors whichshare the same symmetries of cryo-EM.
Revised: We ﬁrst evaluate our method on a synchronization 269 task over some synthetic datasets of vectors whichshare the same symmetries of cryo-EM.

Original: Then, weinclude our method in a simple 3D reconstructionpipeline which we test on synthetic cryo-EM data.
Revised: Then, we 271 include our method in a simple 3D reconstructionpipeline which we test on synthetic cryo-EM data.

Original: We include additional experiments in Appendix E.
Revised: We include additional experiments in Appendix E.

Original: 5.1 Synthetic Vector dataset
Revised: 5.1 Synthetic Vector dataset

Original: We generate a number of synthetic datasets which imitate the symmetry of the cryo-EM generative process.
Revised: We generate a number of synthetic datasets which im- 276 itate the symmetry of the cryo-EM generative process.

Original: To do so, we ﬁrst construct a matrix 11 M 2 R d ⇥ bwhich has the symmetries of ⇧ described in Table 1,where H acts on the left with a d -dimensional representation ⇢ of the group O(2) while O(3) acts onits right with a b -dimensional representation .
Revised: To do so, we ﬁrst construct a matrix 11 M 2 R d ⇥ b 278 which has the symmetries of ⇧ described in Table 1,where H acts on the left with a d -dimensional rep- 280 resentation ⇢ of the group O(2) while O(3) acts onits right with a b -dimensional representation .

Original: Bothrepresentation contain multiple irreps of different frequencies of the two groups.
Revised: Both 282 representation contain multiple irreps of different frequencies of the two groups.

Original: A dataset is generatedby sampling a random vector 2 R b and a set of 284 random poses { g i 2 SO(3) } ni and then evaluating{ o i =
Revised: A dataset is generated 284 by sampling a random vector 2 R b and a set of 285 random poses { g i 2 SO(3) } ni and then evaluating{ o i =

Original: M ( g  1 i ) + 1  ✏ i 2 R d } i , where ✏ i ’s entriesare normally distributed and  2 is the SNR.
Revised: M ( g  1 i ) + 1  ✏ i 2 R d } i , where ✏ i ’s entriesare normally distributed and  2 is the SNR.

Original: We evaluate variants of our method on this data andcompare it with MFVDM with SO(2) poses.
Revised: We evaluate variants of our method on this data and 289 compare it with MFVDM with SO(2) poses.

Original: We usethe SO(3) correlation 12 1 n pP
Revised: Weuse the SO(3) correlation 12 1 n pP

Original: i g i
Revised: i g i

Original: b
Revised: b

Original: g  1 i 
Revised: g  1 i

Original: F and thecorrelation between the real and estimated cosine  similarities { s ij } ij as metrics.
Revised: F andthe correlation between the real and estimated cosine 292 similarities { s ij } ij as metrics.

Original: Fig. 4 shows the  SO(3)292 correlation achieved by our method in different settings; in the low-noise regime, perfect reconstruction is achieved.
Revised: Figure 4 shows the 293 SO(3) correlation achieved by our method in different settings; in the low-noise regime, perfectreconstruction is achieved.

Original: As expected, in the highest noise regime, reconstruction is almost  impossible asincreasing the number of samples helps only marginally (SNR= 0 . 16 ).
Revised: As expected, in the highest noise regime, reconstruction is almost 295 impossible as increasing the number of samples helps only marginally (SNR= 0 . 16 ).

Original: Fig. 5 compares our method11 M contains a subset of with the original MFVDM- SO(2) , for two choices of maximum frequency  L .
Revised: Figure 5 compares our method with the original MFVDM- SO(2) , for two choices of maximum frequency 297 L .

Original: While the originalestimator from MFVDM is stable when s ij  0 , our estimator from Eq.
Revised: While the original estimator from MFVDM is stable when s ij  0 , our estimator from Eq.

Original: 9 is  stable when | s ij |  0 .
Revised: 9 is 298 stable when | s ij |  0 .

Original: We also observe the effect of the multi-frequency approach: leveraging higher  frequencies provide amore robust estimation.
Revised: We also observe the effect of the multi-frequency approach: leveraging higher 299 frequencies provide a more robust estimation.

Revised: [Figure]

Original: 5.2 3D reconstruction of synthetic cryo-EM data
Revised: 5.2 3D reconstruction of synthetic cryo-EM data

Original: [Figure]
Revised: [Figure]

Original: For a more realistic evaluation, we use the 70S Ribosome[29] structure from the Protein Data Bank database (structure 5O60).
Revised: For a more realistic evaluation, we use the 70S Ribosome 302 [24] structure from the Protein Data Bank database (structure 5O60).

Original: We adopt a simple 3D reconstruction pipelinefrom [30] which optimizes via SGD a Gaussian posteriorover the 3D density in Fourier space leveraging the Fourier  Slice Theorem.
Revised: We adopt a simple 3D reconstruction pipeline 304 from [25] which optimizes via SGD a Gaussian posteriorover the 3D density in Fourier space leveraging the Fourier 306 Slice Theorem.

Original: Like [30], we ignore the contrast transferfunction (CTF) in the rendering step for simplicity.
Revised: Like [25], we ignore the contrast transferfunction (CTF) in the rendering step for simplicity.

Original: Thedifferentiability of the render allows us to also optimize theposes in the quaternion space.
Revised: The 308 differentiability of the render allows us to also optimize theposes in the quaternion space.

Original: We compare three differentinitializations of the poses - with the ground truth, with theestimation produced by our method and with estimationsproduced via SO(2) alignment - and a version where welearn full posteriors over the poses using a discretization of 313 SO(3) and the Gumbel-Softmax trick [31].
Revised: We compare three different 310 initializations of the poses - with the ground truth, with theestimation produced by our method and with estimations 312 produced via SO(2) alignment - and a version where welearn full posteriors over the poses using a discretization of 314 SO(3) and the Gumbel-Softmax trick [26].

Original: In the SO(2)alignment case, we use the similarity metric recovered by MFVDM to build a diffusion operator on
Revised: In the SO(2)alignment case, we use the similarity metric recovered by MFVDM to build a diffusion operator on

Original: S 2 , whose top eigenvectors estimate z i ; we initialize ( x i , y i ) to random orthogonal frames.
Revised: S 2 , whose top eigenvectors estimate z i ; we initialize ( x i , y i ) to random orthogonal frames.

Original: To evaluate the reconstruction, we compute the likelihood of a number of new clean images.
Revised: To evaluate the reconstruction, we compute the likelihood of a number of new clean images.

Original: Weestimate the test images’ poses by, ﬁrst, estimating the optimal alignment g 2 O(3) between the 318 training images’ ground truth and estimated poses and, then, correcting the test images’ poses by g .
Revised: We 318 estimate the test images’ poses by, ﬁrst, estimating the optimal alignment g 2 O(3) between the 319 training images’ ground truth and estimated poses and, then, correcting the test images’ poses by g .

Original: We also ﬁne-tune these estimate poses before computing their ﬁnal likelihood.
Revised: We also ﬁne-tune these estimate poses before computing their ﬁnal likelihood.

Original: Before synchronization,we perform a ﬁrst dimensionality reduction and denoising step of the images using steerable PCA[32, 19] with a Fourier-Bessel basis.
Revised: Before synchronization, 321 we perform a ﬁrst dimensionality reduction and denoising step of the images using steerable PCA[27, 28] with a Fourier-Bessel basis.

Original: Since this basis is O(2) -steerable, we use PCA’s embeddings toestimate the relative poses rather than the raw images.
Revised: Since this basis is O(2) -steerable, we use PCA’s embeddings toestimate the relative poses rather than the raw images.

Original: We estimate the initial synchronization graphby picking for each node its nearest neighbors according to the H -invariant distance.
Revised: We estimate the initial synchronization graph 324 by picking for each node its nearest neighbors according to the H -invariant distance.

Original: Table 2 reports the correlation of the training images’ poses and the negative log-likelihood (NLL)of 500 clean test images in different training settings.
Revised: Table 2 reports the correlation of the training images’ poses and the negative log-likelihood (NLL) of 326 500 clean test images in different training settings.

Original: Our MFVDM- O(2) method can accuratelyrecover the poses, yielding reconstruction quality close to the one obtained using ground truth poses(GT).
Revised: Our MFVDM- O(2) method can accurately recoverthe poses, yielding reconstruction quality close to the one obtained using ground truth poses (GT).

Original:  Conversely, we do not ﬁnd initializing the poses with the spherical embeddings recovered by MFVDM- SO(2) useful.
Revised: 328 Conversely, we do not ﬁnd initializing the poses with the spherical embeddings recovered by MFVDMSO(2) useful.

Original: Unfortunately, we ﬁnd that the NLL - essentially, a mean-squared error - doesnot reﬂect the reconstruction quality in the lowest SNR regimes (e.g., the baselines have losses lowerthan the GT model).
Revised: Unfortunately, we ﬁnd that the NLL - essentially, a mean-squared error - does notreﬂect the reconstruction quality in the lowest SNR regimes (e.g., the baselines have losses lower than 331 the GT model).

Original: Still, the sample projections of the produced densities in Fig. 6 demonstrate thatour method (MFVDM- O(2) )
Revised: Still, the sample projections of the produced densities in Figure 6 demonstrate thatour method (MFVDM- O(2) )

Original: achieves similar visual quality to the ground-truth.
Revised: achieves similar visual quality to the ground-truth.

Original: Instead, the baselinemodels did not converge to meaningful structures, emphasizing the importance of pose initialization.
Revised: Instead, the baselinemodels did not converge to meaningful structures, emphasizing the importance of pose initialization.

Original:  Conclusions and Discussion
Revised:  Conclusions and Discussion

Original: In this work, we studied the symmetries of cryo-EM and found that the O(2) relative poses estimatedfrom the images are sufﬁcient to solve the SO(3) poses synchronization task.
Revised: In this work, we studied the symmetries of cryo-EM and found that the O(2) relative poses estimatedfrom the images are sufﬁcient to solve the SO(3) poses synchronization task.

Original: With this insight,we proposed a new method which can replace current VDM-based 2D classiﬁcation methods withlimited changes to directly estimate poses rather than clustering and averaging images, while enjoyingthe same noise robustness properties.
Revised: With this insight,we proposed a new method which can replace current VDM-based 2D classiﬁcation methods with 338 limited changes to directly estimate poses rather than clustering and averaging images, while enjoyingthe same noise robustness properties.

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: Checklist
Revised: Checklist

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Revised: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’scontributions and scope?

Original: [Yes](b) Did you describe the limitations of your work?
Revised: [Yes](b) Did you describe the limitations of your work?

Original: [Yes] See last paragraph of Sec. 3(c)
Revised: [Yes] See last paragraph of Sec. [Table]

Original:  Did you discuss any potential negative societal impacts of your work?
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Original: [No](b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
Revised: [No](b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?

Original: [No] (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?[No]
Revised: [No](c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Table] (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
Revised: [No](d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?

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Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re [Table] using/curating?
Revised: [No](d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating?

Original: [N/A]
Revised: [N/A](e)

Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?
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Original: [N/A] [Table]
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Original: If you used crowdsourcing or conducted research with human subjects...
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