CryoSPIN: Improving Ab-Initio Cryo-EM Reconstruction with Semi-Amortized Pose Inference

Authors: shayan  shekarforoush
david b lindell
marcus a brubaker
david j fleet
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 01:25:33

Abstract: Cryo-EM is an increasingly popular method for determining the atomic resolution 3D structure of macromolecular complexes (eg, proteins) from noisy 2D images captured by an electron microscope. The computational task is to reconstruct the 3D density of the particle, along with 3D pose of the particle in each 2D image, for which the posterior pose distribution is highly multi-modal. Recent developments in cryo-EM have focused on deep learning for which amortized inference has been used to predict pose. Here, we address key problems with this approach, and propose a new semi-amortized method, cryoSPIN, in which reconstruction begins with amortized inference and then switches to a form of auto-decoding to refine poses locally using stochastic gradient descent. Through evaluation on synthetic datasets, we demonstrate that cryoSPIN is able to handle multi-modal pose distributions during the amortized inference stage, while the later, more flexible stage of direct pose optimization yields faster and more accurate convergence of poses compared to baselines. On experimental data, we show that cryoSPIN outperforms the state-of-the-art cryoAI in speed and reconstruction quality.

Main paper: 
 introduction: 
 Single particle electron cryo-microscopy (cryo-EM) has gained popularity among structural biologists as a powerful experimental method for determining the 3D structure of macromolecular complexes, such as proteins and viruses, unlocking our understanding of biological function at the scale of the cell. Thanks to pioneering advances in hardware and data processing techniques, cryo-EM has enabled reconstruction of challenging structures at atomic or near-atomic resolution [1] .During a cryo-EM experiment, 10 4 -10 7 particle images, each containing an instance of the target bio-molecule, are acquired using a transmission electron microscope. From that particle stack, the goal is to reconstruct the unknown 3D density map [2] . This ab-initio reconstruction task presents some challenges. First, the 3D pose (orientation and position) of the particle in each image is unknown, and must be inferred along with the 3D structure. Second, low electron exposures are used to limit radiation damage to the particles (e.g., see Fig. 1 ). But this reduces the signal-to-noise ratio (SNR), obscuring high-resolution image detail and hindering pose and structure estimation. Third, bio-molecules are typically non-rigid and exhibit structural variations within a sample. Hence, for such heterogeneous datasets, it is crucial to account for the conformational variability in order to obtain high-resolution reconstruction [3] [4] [5] [6] . θ (2) , ω (2) , t (2) θ (3) , ω (3) , t (3) θ (1) , ω (1) , t (1) θ (N) , ω (N) , t (N) Fourier Volume Decoder Fourier Volume DecoderLook-up Table (learnable) H 1 H 2 H M ϕ (i) = (R (i) , t (i) )Figure 1 : CryoSPIN consists of two stages: (i) an auto-encoding stage where an image encoder equipped with multiple heads maps the input image to the pose candidate set {ϕ 1 , . . . ϕ M }, followed by computing projections by slicing through the volume decoder in Fourier space based on the pose set. These projections are compared with the input image and the one with the minimum error is used. (ii) An auto-decoding stage where pose parameters in axis-angle representation are stored for all images. The same volume decoder is used to obtain projections, and the reconstruction loss is computed for a single projection.The development of methods in cryo-EM has recently focused on deep learning. CryoDRGN [5] proposed an image encoder-volume decoder architecture to model continuous heterogeneity, but the poses were assumed to be known. CryoDRGNv2 [7] introduced hierarchical pose search, comprising grid search followed by a form of branch-and-bound (BnB) optimization, akin to cryoSPARC [8] .More recently, cryoPoseNet [9] and cryoAI [10] introduced amortized pose inference, to avoid the expense of orientation matching for each particle image. They employ a convolutional neural network (CNN) to map an input image to a pose estimate. Nevertheless, although efficient, amortized inference may not accurately represent multi-modal posterior pose distributions, which occur frequently early in reconstruction before the structure is estimated well. As such, compared to previous methods like cryoSPARC [8] and cryoDRGNv2 [7] that consider many pose candidates, amortized methods sometimes fail to identify the correct mode. Moreover, amortization confines the pose search to a global parametric function (the encoder) which can lead to slow convergence.Here, we introduce a new approach, called cryoSPIN, to ab-initio homogeneous reconstruction that is able to handle multi-modal pose distributions with a tailored encoder, and accelerates pose optimization with semi-amortization [11] . First, we perform amortized pose inference through a shared multi-choice encoder that maps each input image to multiple pose estimates (Fig. 1 , top).In contrast to cryoPoseNet [9] and cryoAI [10] , which use pose encoders that produce one or two estimates, we attach multiple pose predictor heads to a shared CNN feature extractor to predict multiple plausible poses for each image. By design, our multi-output encoder is able to account for pose uncertainty and encourage exploration of pose space during the initial stages of reconstruction.For the 3D decoder, unlike cryoDRGN and cryoAI that use MLPs with computationally expensive feed-forward implicit networks, we adopt an explicit parameterization to further accelerate the reconstruction. To train the encoder-decoder architecture, we apply a "winner-takes-all" loss in which the 3D decoder is queried to obtain a 3D-to-2D projection for each predicted pose. Inspired by the loss function in Multi-Choice Learning [12] [13] [14] , we select the projection with the lowest image reconstruction error to determine the loss.During the course of optimization, as higher resolution details are resolved, the pose posterior becomes predominantly uni-modal. Thus, the pose search can be narrowed down to a neighborhood containing the most likely mode. In this stage, we propose to transition from auto-encoding to auto-decoding (Fig. 1 , bottom). In particular, for each image, we choose the pose with the lowest reconstruction loss, iteratively refine it with stochastic gradient descent (SGD), and continue pose optimization and reconstruction in an alternating fashion. This direct, per-image optimization procedure achieves faster convergence to more accurate poses than amortized inference alone, which relies on potentially sub-optimal predictions of the encoder as a parametric function.We evaluate cryoSPIN in comparison to state-of-the-art methods, cryoSPARC [8] , cryoAI [10] ,and cryoDRGN [7] on both synthetic and real experimental datasets. Using synthetic data, we find that semi-amortized inference noticeably accelerates the convergence of poses, and yields 3D reconstructions at similar or higher resolutions than baseline methods. We also show that our multichoice encoder empirically accounts for multiple modes in the pose posterior. Finally, we apply cryoSPIN to an experimental dataset and obtain reconstructions with higher resolution than those from cryoAI, while matching the resolution of cryoSPARC reconstructions.To summarize, we make following contributions.• Building upon cryoAI, we develop a new encoder based on a multi-head architecture to return multiple plausible candidates to further mitigate pose ambiguity. • We train the encoder coupled with an explicit volume decoder using a winner-take-all loss, which penalizes the pose hypothesis with least reconstruction error. • Beyond the architecture and objective, we introduce semi-amortized pose inference, which begins with feed-forward amortized pose inference, followed by direct, per-particle pose optimization. • In ab-initio reconstruction, our semi-amortized method, cryoSPIN, is faster than amortized method of cryoAI and achieves higher-resolution on synthetic and experimental datasets. 
related work: 
 Cryo-EM reconstruction. Methods for cryo-EM reconstruction can be categorized as either homogeneous or heterogeneous. Homogeneous techniques [8] [9] [10] assume a rigid structure while heterogeneous ones [5, 7, 6, 15] allow for conformational variation. We focus on homogeneous reconstruction, but our optimization framework could be extended to heterogeneous data as well.Early reconstruction techniques rely on common-lines [16] [17] [18] or projection mapping [19, 20] to select optimal poses. Other works [21, 22] frame the reconstruction problem in the context of maximum a posteriori (MAP) estimation, and jointly reconstruct pose and structure via expectation maximization (EM). We compare our approach to cryoSPARC [8] , a state-of-the-art method that uses stochastic gradient descent and a branch-and-bound search [23] for ab initio reconstruction and pose estimation. Like these methods, our auto-decoding stage directly optimizes pose of every image.More recently, amortized inference techniques have been proposed for pose estimation [24, 9, 10, 15] . These techniques avoid explicit per-particle pose optimization; instead, they train an auto-encoder or variational one [25] to associate each particle image with a predicted pose [9] . One challenge is that the auto-encoders can become stuck in local optima during training [9] . To address this issue, cryoAI [10] produces two pose estimates per image coupled with a symmetrized loss function that penalizes the best one. We build on this concept by adopting a multi-head neural architecture as the encoder to output multiple plausible pose candidates and avoid local optima.Inspired by scenarios where a set of hypotheses needs to be generated to account for uncertainty in the prediction task, MCL [12] was introduced in a supervised setup to learn multiple structured-outputs with SSVMs [26] . Their motivating question was: can we learn to produce a set of plausible hypotheses? To address this, they define an"oracle" loss in which only the most accurate output pays the penalty. This loss is minimized even if there is only a single accurate prediction in the set. The early follow-up work [27, 13] uses the same loss to learn a deep CNN ensemble composed of M heads with a shared backbone network. Importantly, they show that the ensemble-mean loss hurts prediction diversity across different heads, while training with the "oracle" loss yields specialized heads. Variations have since been proposed to mitigate hypothesis collapse or overconfidence issues in MCL by modifying the loss or applying learnable probabilistic scoring schemes [28] [29] [30] 14] . MCL has been used to mitigate the ambiguity in several tasks including image segmentation [31] , optical-flow estimation [32] , trajectory forecasting [33] , human pose and shape estimation [14, 34] . In our work, we use the "oracle" loss in context of auto-encoder which supervises the pose encoder indirectly through projections provided by the decoder. 
problem definition: 
 Image formation in cryo-EM is well-approximated using the weak-phase object model [35] . Under this model, the structure of interest is an unknown 3D density map, V : R 3 → R ≥0 , represented in some canonical coordinate frame. Cryo-EM images, {I i } N i=1 , are approximated as orthographic projections of the 3D map under some unknown coordinate transformation. We denote the unknown orientation by a rotation R i ∈ SO(3) and the unknown position in terms of an in-plane translation t i = (t x , t y ) ∈ R 2 . Formally,EQUATIONwhere P Ri (•) is the linear operator computing the integral along the optical axis, z, over the input density map rotated by R i , and S is the shift operator. The projection is convolved with the imagespecific point-spread function (PSF), g i , and corrupted by additive noise n. It is common to assume that n follows a zero-mean white (or colored) Gaussian distribution.By the Fourier slice theorem [36] , the Fourier transform of a projection is equal to a central slice through the density map's 3D Fourier spectrum. That is,EQUATIONwhere Î and V denote the 2D and 3D Fourier transforms of the image and the density map. The slice perpendicular to the projection is computed by ( PRi V ). The translation by S ti becomes a phase shift operator Ŝti , and convolution with g i is equivalent to element-wise multiplication with, ĝi , the contrast transfer function (CTF). The noise n remains zero-mean Gaussian.Under this model, given the structure V , the negative log-likelihood of observing image Îi with noise variance σ 2 and poseEQUATIONAb-initio reconstruction methods [8] [9] [10] 7] solve jointly for the unknown structure V and per-particle poses (R i , t i ). They often follow an Expectation-Maximization (EM) [37, 21] procedure in which the E-step aligns images with the structure yielding pose estimates (R i , t i ), and then in the M-step the volume V is updated by minimizing the negative log-likelihood in Eq. 3. Since errors in pose estimates lead to blurry reconstructions, accurate pose estimates are crucial to finding high-resolution structures. As discussed above, poses are either optimized through search and projection matching [8, 7] or estimated by an encoder network [10, 9, 15] .Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [NA] Justification: We provide no theoretical results.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. • All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
methodology: 
 We introduce cryoSPIN, a two-staged approach to ab-initio cryo-EM reconstruction, combining auto-encoding and auto-decoding. We start with amortized inference (Fig. 1 , auto-encoding) where a shared encoder equipped with multiple heads provide a set of pose guesses. Multiple guesses enable multimodal inference in the initial stages of reconstruction when the 3D structure is poorly resolved. As reconstruction improves, the pose posterior becomes less uncertain, at which point we switch to direct pose optimization (Fig. 1 , auto-decoding). The former enables handling the pose uncertainty early in reconstruction while the latter circumvents sub-optimality of encoder network leading to arguably more accurate poses and faster convergence. We couple our pose estimation module with an explicit volumetric decoder representing the 3D structure in Hartley space [5] . Our explicit model enables faster evaluation of projections compared to multiple passes through implicit neural representations [38] [39] [40] . The following sections discuss these components in detail.With randomly initialized or a low-resolution reconstruction, the pose posterior contains multiple modes. Also, due to high levels of noise in images and near-symmetries in biological structures, pose estimation inherently involves high uncertainty. As a result, there exist several plausible poses for each image, and naive optimization or search is prone to local minima.To account for uncertainty in pose estimation, we build upon cryoAI [10] and extend the encoder to return multiple candidate poses ϕ. Formally, given the image I i ∈ R H×W , M pairs of rotations and translations are computed as,EQUATIONwhere R i,j ∈ SO(3) and t i,j ∈ R 2 . The image-specific intermediate features, F i = VGG(I i ) ∈ R C×H×W are extracted by a shared convolutional backbone based on the VGG16 architecture [41] , which are then supplied to M separate pose predictor heads, H θj , 1 ≤ j ≤ M , yielding poses ϕ i,j . The heads are composed of fully-connected layers with distinct learnable weights, while the backbone is shared across all heads. Using multiple heads facilitates pose exploration and reduces uncertainty in the pose estimation process. Moreover, as θ j are randomly initialized, different heads are free to specialize on subsets of pose space so that they collectively produce a set of likely poses {ϕ i,1 , . . . , ϕ i,M }.Inspired by multi-choice learning [12] [13] [14] , we optimize the encoder-decoder using a "winner-take-all" loss. For each image I i and predicted pose ϕ i,j , the negative log-likelihood, L i,j in (Eq. 3), is computed, and the minimum is selected as the final loss for the corresponding image, i.e.,EQUATIONMinimizing this loss requires only one of the predicted poses to be accurate. Interestingly, we find that this approach produces heads that specialize to somewhat disjoint regions of pose space (see Supp. C), consistent with previous work showing that it improves diversity in prediction tasks [13, 27] . This 'divide-and-conquer' approach facilitates pose space exploration, which is crucial in order to avoid local minima during the early stages of reconstruction where uncertainty in the 3D structure is significant. CryoAI [10] can be viewed a special case of this formulation; it assigns two poses to each image as a consequence of input augmentation, and selects the best one with a symmetrized loss. In contrast, our approach augments the output of the encoder network with multiple heads, each providing a pose estimate.As optimization progresses and fine-grained details of the 3D structure are resolved, the posterior pose variance tends to decrease, with the posterior approaching a unimodal distribution. At this point, the gap between the amortized and variational posterior distributions is mainly determined by the error in the pose estimate (predicted mean), prioritizing accuracy over exploration. However, a feed-forward network, as a globally parameterized function of the input image, may be limited in the accuracy of its predictions, and hence the approximations inherent in amortized inference become significant sources of error in the refinement of the 3D structure. In prior work [11, [42] [43] [44] , a related issue, called the amortization gap, is characterized by the KL-divergence between the true and predicted (amortized) variational posteriors.To address this issue, we adopt a semi-amortized inference scheme [11] comprising two stages. First, the encoder predicts a set of pose candidates using a multi-head architecture. In the second stage, rather than amortized inference, pose parameters are directly optimized for each image using stochastic gradient descent. To initialize poses for the i-th image, we choose the one with the lowest reconstruction loss from the set of candidatesEQUATIONSubsequently, the pose and structure are optimized by coordinate descent using the negative loglikelihood (Eq. 3) as the objective function. See Supp. A for more details on rotation optimization.Recent works [10, 5] use coordinate networks [39, 40, 38] to implicitly model the Fourier representation of the 3D structure. Instead, we couple the pose estimation module with an explicit parameterization (3D voxel array) of the structure in the Fourier domain. The explicit representation is less computationally expensive than an MLP to evaluate and update. Also, this choice is motivated by the fact the implicit decoder needs to be queried multiple times for each image with the multi-head encoder.We parameterize the volume using the Hartley representation [5] . The Fourier and Hartley transforms, respectively denoted as F (ω) and H(ω), are related asEQUATIONwhere ω denotes the frequency coordinate and R and I are the real and imaginary part, respectively. The Hartley representation is real-valued, and so more memory efficient to use than storing complexvalued Fourier coefficients. To account for high dynamic range of the Hartley coefficients, we assume the Hartley field is decomposed into mantissa, m(ω) and exponent e(ω) fields [10] as,EQUATIONThis decomposition restricts the range of values for m(ω) and e(ω) and makes the reconstruction less sensitive to the initialization of the field.Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: In a subsection called implementation details, we included all details needed to reproduce reconstruction results of our method, cryoAI and cryoSPARC. Steps to generate synthetic datasets and preprocessing steps to prepare experimental data are also provided. 
experiments: 
 In what follows, we consider both synthetic and real datasets to empirically compare our semiamortized ab-initio reconstruction method with state-of-the-art methods, cryoAI [10] and cryo-DRGN [7] which use amortized inference for reconstruction, and cryoSPARC [8] which performs pose search separately for each particle image.Synthetic data. We generate synthetic datasets by simulating the image formation process formalized in Sec. 3 using atomic models deposited in the Protein Data Bank (PDB). We compute ground-truth density maps of size 128 3 for a heat shock protein (HSP) [45] (1.5 Å), pre-catalytic spliceosome [46] (4.33 Å), and SARS-CoV-2 spike protein [47] (2.13 Å). Then, N = 50, 000 projections of size L = 128 are generated by randomly rotating and projecting the density map along the canonical z-axis. To simplify the analysis and interpretation of results (e.g. Figs. 4, 5 ), we consider synthetic particles are centered, and therefore omit the estimation of 2D translation and allowing us to visualize the pose posterior more easily. Finally, a random CTF is applied in the Fourier space and Gaussian noise is added yielding SNR = 0.1.Experimental data. As a widely adopted experimental benchmark, we use the 80S ribosome dataset (EMPIAR-10028 [48] ), containing 105,247 particle images of linear box size L = 360 with pixel size 1.34 Å. Following cryoAI and cryoDRGN, we downsample the images to L = 128 (3.76 Å) using cryoSPARC software [8] , and randomly split the data into two halves and run the reconstruction methods independently on each (to enable Fourier Shell Correlation as a quantitative measure of the resolution of the 3D reconstruction). Unlike synthetic data, particles are not well-centered so it is required to estimate translation parameters as well as rotations. This is achieved by augmenting each head to return translation parameters as well.Implementation details. During auto-encoding, we use encoders with M = 7 and M = 15 heads for reconstruction on synthetic and real datasets, respectively, with Adam [49] to optimize encoder and decoder with learning rates 0.0001 and 0.05. Once switched to direct optimization (after 7 epochs for synthetic and 15 epochs for real data), we reduce the decoder learning rate to 0.02 and allocate a new optimizer for pose parameters with learning rate 0.05. To be consistent with cryoAI, we use a batch size of 64 and train for the same number of epochs (20 for synthetic, 30 for real data). We use the public cryoAI codebase, run cryoSPARC v4.4.0 [8] with default settings, and run cryoDRGN homogenous ab-initio job. Methods are implemented in Pytorch [50] . Experiments are run on a single NVIDIA A40 GPU.We first qualitatively compare final reconstructions of cryoSPIN with cryoAI [10] , cryoDRGN [7] ,and cryoSPARC [8] on synthetic and real datasets (Fig. 2 , left). We observe that cryoAI gets stuck in local minima when evaluated on experimental data (eg, 80S ribosome). In particular, we found that in the presence of unknown planar shift, cryoAI fails to estimate effective translation parameters, but after centering all particles via pre-processing, we could reproduce the cryoAI results. Hence, the provided results for cryoAI in Fig. 2 are obtained after this pre-processing step, whereas our method and others are given off-centered experimental images. Both cryoSPIN and cryoSPARC capture high-frequency details of the 3D structure on all datasets, whereas reconstructions by amortized , cryoDRGN [7] and cryoSPARC [8] We note that cryoAI often becomes stuck in local minima when particles are not centered, so for cryoAI we translate the input images to correct for the spatial offset. (Left) Final 3D reconstructions on three synthetic datasets and one experimental dataset (EMPIAR-10028) are depicted using ChimeraX [51] . (Right) FSC curves are visualized for quantitative comparison. The red dashed lines show the standard threshold levels of 0.5 and 0.143 to report the resolution (in Angstrom) for synthetic and real data, respectively. CryoSPIN achieves higher resolution on the Spliceosome and HSP datasets, and it is competitive with the state of the art on the Spike and EMPIAR-10028 datasets.methods, cryoAI and cryoDRGN, are inferior on the HSP and 80S datasets. In particular, on HSP, cryoAI gets stuck in local minima as it fails to handle high uncertainty in poses caused by symmetries in this structure.For quantitative comparison, we visualize the Fourier Shell Correlation (FSC) [52] between the reconstruction and the ground truth (Fig. 2 , right). FSC is the gold-standard metric measuring the normalised cross-correlation coefficient between two 3D Fourier volumes along shells of increasing radius. Higher FSC implies more accurate reconstruction. On the Spliceosome, HSP and 80S experimental datasets, our reconstruction outperforms amortized methods of cryoAI and cryoDRGN. Also, our method outperforms cryoSPARC on Spliceosome, HSP and 80S, while achieving competitive FSC on Spike. We use the standard 0.5 and 0.143 cutoff thresholds to report the resolution for synthetic and real data, respectively. CryoSPIN achieves higher or competitive resolution compared to others on all datasets. To visualize pose inference, images depict the approximate log posterior for three particles (marginalized over in-plane rotations) as view-direction distribution over a HEALPix [53] uniform grid on a unit sphere S 2 . After Gnomonic projection to 2D, we show the neighborhood of the mode of interest. Black dots depict starting points.Blue and red dots are pose estimates from fully-and semi-amortized methods.In Fig. 3 , we visualize resolution-time plots, showing that cryoSPIN gets to high-resolution reconstructions significantly faster than cryoAI. In fact, semi-amortization in cryoSPIN accelerates the improvement in the resolution and with our explicit structure decoder, we circumvent expensive MLP evaluations and train faster without any degradation to the final reconstruction quality as shown in Fig. 2 . In Supp. B, through a detailed comparison with cryoAI, we show cryoSPIN is ∼ 6x faster and uses ∼ 5x less memory compared to cryoAI. To showcase the benefit of the auto-decoding stage, we compare our semi-amortized method with a fully-amortized baseline on spike and spliceosome datasets. Starting with auto-encoding, we branch the running reconstruction into two after 7 epochs: the first continues using the encoder while the second switches to direct pose optimization. We plot the average pose error with respect to the ground-truth at different epochs in Fig. 4 (left). When using the multi-head encoder, the pose with the least reconstruction error is selected among the candidates. Interestingly, as our method switches to direct pose optimization, the error in pose drops quickly, whereas the fully-amortized baseline exhibits slow convergence. This clearly shows the superiority of auto-decoding compared to auto-encoding during the later stages of optimization.Next, we analyze the evolution of pose estimates over the optimization landscape depicted as a heat map (Fig. 4 , right). Qualitatively, poses obtained by direct optimization (blue dots) show stable convergence to the optimal point (highlighted area) while those inferred by the encoder (red dots) frequently oscillate. Indeed, the encoder in amortized inference is a globally parameterized function which might be too restrictive, yielding sub-optimal pose parameters. Therefore, poses inferred in an amortized fashion might fail to consistently converge to the optimal point. On other hand, direct optimization is intuitively more flexible as it is performed separately and locally for each image, exhibiting more stable convergence. See Supp. D for more examples.Lastly, we examine how well cryoSPIN multi-head encoder performs vs. cryoAI encoder in terms of handling the uncertainty in the pose on the challenging dataset of HSP [45] . To simplify the visualization, we run our method with M = 4 heads in this experiment. We first inspect the behavior of encoders for two example cryo-EM images (Fig. 5 , left, see Supp. E for more examples). In each row, the approximate posterior distribution over view direction is visualized for both cryoAI and multi-head encoders given the input image and reconstruction. Our multi-head encoder returns a set of plausible candidates, while cryoAI obtains two pose estimates by data augmentation. In both examples, the multi-head encoder identifies the correct mode while cryoAI selects the incorrect one. Note that the pose set predicted by the multi-head encoder contains other posterior modes as well.Intuitively, cryoAI encoder with two pose predictions cannot explore the pose space as much as our multi-head encoder. Thus, the pose posterior in cryoAI remains uncertain and multi-modal during reconstruction. This also contributes to slower convergence to higher-resolution reconstructions as depicted in the top left of Fig. 5 . Finally, we visualize the percentage of images assigned to each head by our "winner-takes-all" loss. The relatively even spread of assignments to different heads shows that the multi-head encoder does not suffer hypotheses collapse [30] , namely all heads actively participate in the pose prediction. In Supp. C, we investigate how each head specializes in pose prediction for non-overlapping regions of SO(3).Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes] Justification:Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
conclusion, limitations and future work: 
 In this paper, we introduce cryoSPIN, a new semi-amortized approach to ab initio cryo-EM reconstruction. We develop a new multi-head encoder that estimates a set of plausible candidate pose to handle the uncertainty. As the uncertainty is reduced, we transition to an auto-decoding stage where poses are iteratively optimized using SGD for each image. Our results show that our multi-head encoder is able to capture multiple modes of the pose distribution, and our flexible direct optimization enables accelerated convergence of poses and reconstructions. CryoSPIN outperforms cryoAI on experimental data and achieves competitive results with cryoSPARC.Limitations and future work. In this work, we assume that the 3D structure is rigid while many biomolecules are flexible in practice, exhibiting different conformations. Our results show that, even in this relatively constrained homogeneous case, there is a significant gap between amortized inference and our semi-amortized approach. We expect that the benefits of semi-amortized inference will also apply to such heterogeneous cases. In particular, our proposed multi-head encoder should be applicable to the estimation of the multimodal posterior over heterogeneity latent parameters as well as pose. Yet, there are significant challenges in heterogeneous reconstruction, such as extra uncertainty due to heterogeneity and conflation of latent spaces, making it outside the scope of our analysis on semi-amortized inference.The development of principled criteria for switching from amortized to auto-decoding is also an interesting direction for future research. Our experiments with a range of values show that after a sufficient number of epochs, the reconstruction error becomes insensitive to the switching time. Our intuition is that when switching too early, the pose posterior may have significant uncertainty, and the pose estimates may be too far from the correct basin of attraction to afford robust convergence. Therefore, it is better to switch late than early. However, a well-defined heuristic for the optimal switch time is lacking.Societal impact. Structure determination with Cryo-EM for macromolecules is one of the most exciting, fundamental areas in structural biology, and is already having with vast social impact. Cryo-EM was used to determine the structure of the SARS-COV2 spike protein, the discovery of its pre-fusion conformation, and the evaluation of medical counter-measures. It compliments advances in methods like alpha-fold for structure prediction, transforming our understanding of the process of life at the scale of the cell, and the design and development of new therapeutics.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes]Justification: We have discussed all the necessary information about experimental settings in implementation details.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes]Justification: When reporting the resolution, we follow the standard to compute FSC by splitting data into two halves and make independent reconstructions and compare resulting half-maps.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
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 Appendix: CryoSPIN: Improving Ab-Initio Cryo-EM Reconstruction with Semi-Amortized Pose InferenceFor the auto-encoding stage, we follow cryoAI and design the convolutional backbone to output the six-dimensional representation commonly referred to as S2S2. To compute the rotation matrix from this representation, the 6D vector is split into two 3D vectors and normalized, denoted as v 1 , v 2 ∈ R 3 .We then compute the cross product between them, v 3 = v 1 × v 2 , yielding a new unit vector. Selecting v 1 and v 3 as the first and third columns of the target rotation matrix, we compute ṽ2 = v 3 × v 1 , which is another unit vector orthogonal to both v 1 and v3 . Then, R = [v 1 , ṽ2 , v 3 ].Once switched to direct optimization, we change parameterization to axis-angle representation.During auto-decoding, we alternate between five iterations of pose SGD updates and one iteration of volume update. To update poses, we keep the volume fixed and optimize for the negative loglikelihood (Eq. 3) with respect to the pose parameters. We define the new pose estimate based on the current one as follows:R t+1 = R δ R t (9)where R δ is an infinitesimal rotation matrix perturbing the current estimate. The perturbation matrix R δ is parameterized by axis-angle representation. By a single vector ω ∈ R 3 , one can represent both the axis ||ω|| and the angle 0 < ω ||ω|| < π for any given rotation. Using Rodrigues formula, the perturbation matrix R δ (ω) can be parameterized as a function of ω. To find the optimal ω, one can initialize it with zero vector, and then use automatic differentiation [54] in pytorch to compute the gradient with respect to ω and make updates using Adam [49] . However, a naive implementation of the function R δ (ω) would lead to numerically unstable calculations of the partial derivative ∂R δ ∂ω . In fact, there is a singularity at the zero vector, and the partial derivative involves terms that are unstable around the origin. Formally, the derivative of i-th column of the rotation matrix R δ with respect to the vector ω is [55, 56] ,∂R (i) δ ∂ω = -e (i) ⊗ ω + [e (i) ] × sin(∥ω∥) ∥ω∥ + [(ω • e (i) )I + ω ⊗ e (i) ] 1 -cos(∥ω∥) ∥ω∥ 2 + (ω ⊗ ω) (ω • e (i) ) 2 cos(∥ω∥) -2 + ∥ω∥ sin(∥ω∥) ∥ω∥ 4 + [(ω × e (i) ) ⊗ ω] ∥ω∥ cos(∥ω∥) -sin(∥ω∥) ∥ω∥ 3 .where ⊗ and × are tensor and cross products, respectively. e (i) is the i-th standard basis in 3D and [v] × denotes the cross product matrix for the vector v. In all four terms, there are scalars such assin(∥ω∥) ∥ω∥ or 1-cos(∥ω∥) ∥ω∥ 2that evaluate to 0 0 at zero angle ω = 0. Similar to [55] , for ||ω|| ≪ 1, we substitute these terms with their numerically robust Taylor expansion, for instance,sin(∥ω∥) ∥ω∥ = 1 - ||ω|| 2 6 + O(||ω|| 4 ) , 1 -cos(∥ω∥) ∥ω∥ 2 = 1 2 - ||ω|| 2 24 + O(||ω|| 4 ) .We implement a differentiable and numerically stable version of the function R δ (ω) in pytorch and use it in our pose estimation module.We perform an ablation study on the decoder, detailed in Table 2 , as well as a comparison with baselines in terms of reconstruction time per epoch, GPU memory usage, and number of parameters.As our method consists of two stages, we report numbers for each stage separately. In the first stage, the encoder has H=7 heads, while in the second stage, it is replaced with a pose module with size depending on the number of particles (N ). To facilitate comparison, we include another baseline, cryoAI-explicit, in which the implicit decoder is replaced by an explicit one as in our method. The explicit and implicit decoders have 4.29 and 0.33 million parameters, respectively. Despite a larger decoder, our method is 6x faster and uses 5x less memory compared to cryoAI. Importantly, swapping the implicit decoder with an explicit one (cryoAI-explicit) significantly drops time and memory, indicating that the implicit decoder is a major computational and memory bottleneck. Yet, cryoAI-explicit uses 2x more GPU memory for encoding and is 1.5x slower than our method as it performs early input augmentation and runs the entire encoder twice per image. Our method, by augmenting the encoder head, saves memory and time during pose encoding. Finally, the amortized baseline, which uses an explicit decoder coupled with a multi-head encoder (H=7), uses more memory in decoding (negligible vs implicit decoder) and runs slower than the direct optimization stage of the semi-amortized method.A natural question about the multi-head encoder is: how each head does take part in pose encoding process? To address this, using the synthetic datasets, we conduct an experiment with our multi-head architecture (M = 4) and visualize the performance of each head on different regions of SO(3) space. In particular, as before, we define a uniform grid on the unit sphere using HEALPix [53] and assign images to their corresponding cells based on the view-direction. Now, for all images end up in the same cell, we compute the average rotation error and visualize it separately for each head. As shown in Fig. 6 , all heads actively participate in pose estimation and they are able to specialize in prediction of poses for images with certain view-direction. A similar result has been provided in prior work on MCL [27, 13] , to show that minimizing the error made by the best prediction ("oracle" loss) encourages diversity in deep ensembles. In our problem, by optimizing a "winner-takes-all" loss, the whole burden of pose estimation is no longer on a single network but it gets divided between multiple heads as separate predictors.To validate the advantages of direct pose optimization in our semi-amortized method, we further show more qualitative examples of paths taken by pose estimates over the optimization landscape during reconstruction in Fig. 7 . For both methods, optimization start from the same point marked by black dot in the vicinity of the distribution mode. It will then continue in paths colored in blue and red for semi-amortized and full-amortized methods, respectively. We observe in all examples that iterative updates by stochastic gradient descent demonstrate a stable convergence toward the optima while poses obtained by amortized inference show unstable behavior around the mode. Figure 6 : Average rotation error is visualized over the sphere for different heads of the multi-head pose encoder (M = 4). The sphere is uniformly divided into cells using HEALPix [53] ; images are assigned to cells based on ground-truth view-direction. For each cell, the average rotation error is visualized, showing diverse behavior of different heads across the space. Blue and red colors show low and high error regions, respectively. Error ranges from zero to 180 degrees.Figure 7 : We compare the behavior of fully-amortized and semi-amortized pose inference on four examples per dataset. Two rows correspond to Spike and Spliceosome datasets, respectively. Each plot shows the approximate log pose posterior, marginalized over in-plane rotations represented as a heat map on a uniform grid over the unit sphere S 2 . Gnomonic projection to 2D is also applied, followed by zooming on the proximity of the mode of interest. Black cross is the starting point while blue dots and red dots show poses estimated by fully-and semi-amortized methods, respectively.Through more examples (Fig. 8 ), we demonstrate that cryoAI fails to handle ambiguity in pose estimation on HSP dataset. The visualization shows that pose estimates by cryoAI become stuck in incorrect modes whereas our pose encoder with multi-head architecture is able to return a pose candidate that captures the correct mode.Figure 8 : The approximate log posterior of view-direction visualized on the unit sphere with highlighted areas showing modes of the distribution. CryoAI [10] and our multi-head encoders provide two and four pose estimates, respectively, which are marked with colored dots on the sphere (the order of poses is arbitrary). The corresponding projections are also illustrated. CryoAI cannot identify the correct mode of pose distribution.Justification: We discuss limitations in the conclusion section. We acknowledge that ours assumptions about homogeneity and centered images could be limiting in some experimental setups. 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Decision: This paper introduces a method for 3D ab initio reconstruction in cryo-EM using amortized inference. The paper makes several important contributions to state-of-the-art deep learning approaches, and it is clear and well-written.

Reviews: review: paper_summary: paper_summary: The submission addresses ab-initio cryo-EM reconstruction, where both image poses and the 3D structure are estimated. The authors adopt a multihead architecture to estimate multiple poses for each image, encouraging the exploration of pose space early in the reconstruction process. They then refine poses in an auto-decoding fashion using SGD. Experiments on synthetic and experimental datasets demonstrate acceleration and resolution improvement over baseline approaches.
strength_weakness: strength_weakness: 1. The paper's technical contribution is primarily limited to a multi-head architecture for estimating multiple plausible poses.
2. The results presented for cryoSPARC are significantly worse than my experience suggests (i.e., on Splicesome or EMPIAR-10028). CryoSPARC's results should not be inferior to those obtained by the proposed approach.
3. The paper cannot handle heterogeneous reconstruction, which limits its general applicability in structural biology research.
4. Import baselines are missing: RELION, cryoFIRE, cryoDRGN-BNB and cryoDRGN2.
questions: questions: 1. The results presented for cryoSPARC are significantly worse than my experience. What is the setting for cryoSPARC? Are there any important assumptions?
2. Will the multi-head structure predict very similar poses, thereby undermining the assumption that this design encourages exploration of the pose space?
3. Can the authors provide a comparison of the training time and memory consumption between the proposed method and the baselines?
limitations: limitations: n/a
paper_summary: paper_summary: This work introduces a new method for 3D ab initio estimation in cryo-EM using amortized inference. It relies on an existing technique which predicts the 3D rotation of a cryo-EM image using a convolutional neural network. That estimated rotation is then used together with a neural representation of the 3D volume to regenerate the projected image. The network weights (both of the pose estimation network and the 3D neural representation) are then optimized to reduce this auto-encoding loss.

The first important difference between the proposed framework and previous work is that this method produces several estimates using multiple heads and computes the loss with respect to the rotation estimates that yields the closest matching image. The second important difference is that after a few epochs of training, the pose estimation network is discarded, and the best pose estimates are refined using alternating minimization together with the 3D neural representation. The resulting algorithm is shown to perform well on both simulated and experimental datasets.
strength_weakness: strength_weakness: The exposition of the pose inference network and the 3D neural representation are quite superficial and deserve a more detailed explanation. For example, how are the 3D rotations obtained from the convolutional backbone output? Are they encoded in axis–angle parametrization? If so, how are the outputs properly constrained? Similarly, the 3D neural representation is described as leveraging the Hartley transform and somehow uses a decomposition of this transform into its mantissa and exponent. How is this relevant to the actual neural network architecture?

The work would also benefit from more extensive testing on experimental datasets beside EMPIAR-10028. This is a relatively high-SNR dataset depicting a very large molecule (an 80S ribosome) and is not representative of experimental datasets in general.
questions: questions: – On like 54, what is meant by “we adopt an explicit paremeterization to further accelerate the reconstruction”? How does the parametrization affect convergence here?
– The idea of “semi-amortized inference” is never actually explicitly described. Presumably this means switching from amortized estimation to pose refinement at some point.
– How is L_{i,j} defined in eq. (6)?
– Why can the proposed method not handle shifts? Should this not be just a matter of adding another output to the pose estimation heads?
– For the synthetic and real datasets, the method switches from amortized inference to pose refinement after 7 and 15 epochs, respectively. How was this switching point determined and is there some way to automate this?
– The labels (Left) and (Right) appear to be switched in the caption for Figure 4.
– On line 227, “experiment” should be “experiments”.
limitations: limitations: The authors have largely addressed the limitations of the work in the manuscript.
paper_summary: paper_summary: This paper introduces an approach to ab-initio homogeneous reconstruction that handles multi-modal pose distributions with a tailored encoder and accelerates pose optimization with semi-amortization. The approach uses a shared CNN feature extractor with multiple pose predictor heads, predicting several plausible poses for each image to account for pose uncertainty. Unlike the computationally expensive implicit networks used by cryoDRGN and cryoAI, this method employs an explicit 3D decoder, speeding up reconstruction. The encoder-decoder architecture is trained with a "winner-takes-all" loss, where the 3D decoder generates multiple 3D-to-2D projections, and the one with the lowest reconstruction error determines the loss. This method achieves faster convergence and more accurate poses compared to relying solely on the encoder's predictions. Evaluations show that the approach outperforms cryoAI and is competitive with cryoSPARC on both synthetic and experimental datasets.
strength_weakness: strength_weakness: Despite its strengths, I see some major weaknesses in this paper:
1. It is unfair to claim "our semi-amortized method is faster than the amortized method of cryoAI" (Line 79) as there are some minor modifications, like changing the decoder from an implicit representation to an explicit volume to reduce the computational cost. An explicit density volume or a feature volume can easily replace all coordinate-based neural networks. In this paper, there is no ablation study about the representation of the decoder or report on how much training and inference time has been reduced by this replacement. Explicit volume with multi-head predictions leads to higher memory costs during training. Advanced representations that can achieve a better trade-off between computational and memory costs, like TensoRF, triplanes, and hash encodings, are not well discussed or explored.

2. The improvement of results is incremental. In Figure 2, the performance improvements on synthetic datasets are very subtle. On the experimental results, it seems cryoSPARC, which uses a traditional ab-initio method, achieves the best performance. Additionally, why do the resolutions of EMPIAR-10028's FSC curve all meet the resolution limit?

3. Translation estimation is important for a complete ab-initio model. Why does this method not support translation estimation while cryoAI is capable?
questions: questions: Overall, I am not convinced that the proposed semi-amortized approach can fundamentally improve pose prediction results without any significant improvements on the image encoder side. All experiments conducted in this paper show only minor performance improvements or even no improvements compared to cryoSPARC, which has no deep learning but only traditional optimization, does it imply that the actual bottleneck is the weak image encoder? The image encoder (VGG in this paper) is not pre-trained on a diverse cryo-EM dataset but trained from scratch on a per-scene dataset in a self-supervised manner. How can it be ensured that the extracted features can be applied for accurate pose prediction? I believe the image encoder is also a performance bottleneck. A PCA visualization of the feature space of the encoder (or calculating the cosine similarity) would be very helpful to validate if images with similar poses can be clustered and images with different poses can be far from each other. It is important for this work to conduct a performance analysis for both the encoder and decoder to identify the actual problem. Unfortunately, I see no analysis in the current version of this paper. I am happy to improve my ratings if my main concerns (including weaknesses) can be addressed by the authors.
limitations: limitations: Yes, there is a discussion of the limitations in this paper, but I believe the limitation of the image encoder used in this work should be also discussed.

Comments: comment: Thanks for the response. I will consider it for my final decision.
I thank the reviewers for their rebuttal and will raise my score to 7.
Thanks for the thoughtful response.

There are several reasons to believe that deep learning (DL) methods have the opportunity to outperform cryoSPARC. Among them, cryoAI with amortized inference has been shown to be faster than cryoSPARC for large datasets; e.g. Fig. 3 (left) in the cryoAI paper shows that cryoAI requires less time than cryoSPARC to obtain a given resolution for large numbers of particles. In turn, our inference in the amortized stage is faster than cryoAI due to encoder/decoder modifications (see rebuttal PDF, Table 1); e.g. based on the resolution-time plots in Fig. 3 in our paper, our method reaches 10A resolution on the Spike and HSP datasets ~4x faster than cryoAI (in 11.0 and 5.3 minutes vs 41.7 and 31.4 minutes, respectively). This advantage over cryoSPARC is significant as cryo-EM datasets continue to increase in size, now regularly with more than 1M particles.

A further advantage of DL is the potential for sample heterogeneity. With our semi-amortized inference, DL methods can now meet or exceed the performance of finely-honed methods like cryoSPARC for homogeneous reconstruction. This opens the door for DL and the inference of heterogeneity latents, going well beyond the capabilities of cryoSPARC. For context, cryoSPARC handles heterogeneity via (1) discrete 3D classification (to which homogeneous methods are easily extended), (2) 3D variability analysis, which is linear and lacks expressiveness, and (3) 3DFlex, which uses DL to learn non-linear variations, but is limited to conformational variability of large substructures. In contrast, our semi-amortized DL method can be extended to both conformational and compositional heterogeneity (a la cryoDRGN and cryoFIRE).

Indeed, to realize the full potential of cryo-EM in practical structural biology research, it is essential to be able to model structural dynamics in heterogeneous datasets. There are open challenges in heterogeneous reconstruction, such as how to handle multi-modality in the posterior over the latent variables and the conflation of pose and conformational states. In this respect, extension of our multi-head encoder and semi-amortization to heterogeneity inference should be helpful, and we plan to investigate these lines in future work.
Thanks for the authors rebuttal! My understanding is that this work deals with the task of homogeneous ab-initio reconstruction, exactly the same with CryoAI. Although the results are improved compared to cryoAI, the method proposed in the paper does not have a clear advantage over cryoSPARC, let alone the ability of cryoSPARC to handle heterogeneous reconstruction.

Can the authors provide some insights into the significance of their proposed method given cryoSPARC? What is the next step for the proposed method or more generally for deep learning methods? How long will it take for them to be applied in practical structural biology research?
Thank you! I would like to improve my rating regarding to these discussions.
We’re glad the rebuttal addressed your concerns, and we appreciate your quick response and thoughtful engagement.\
Regarding your question, the semi-amortized method inherits some benefits from amortized methods like cryoAI in terms of runtime for large datasets (e.g. Fig. 3 (left) in the cryoAI paper shows that cryoAI requires less time than cryoSPARC to obtain a given resolution for large numbers of particles). And our inference in the amortized stage (see rebuttal PDF, Table 1) is faster than cryoAI due to encoder/decoder modifications. More specifically, based on the resolution-time plots in Fig 3 in our paper, our method reaches 10A resolution on the Spike and HSP datasets ~4x faster than cryoAI (in 11.0 and 5.3 minutes vs 41.7 and 31.4 minutes, respectively). This advantage over cryoSPARC is significant as cryo-EM datasets regularly contain more than 1M particles.\
More generally, the cryo-EM community has been excited about deep learning mainly due to its potential to handle heterogeneous experimental data where inference of latents is critical along with pose. The fact that deep learning methods (like our semi-amortized inference) can now meet or exceed the performance of finely-honed methods like cryoSPARC, will help enable improved inference methods on heterogeneous data, going well beyond the current capabilities of cryoSPARC (e.g. with its linear methods for heterogeneity).
Thanks for authors' detailed rebuttal! The authors have thoroughly addressed my concerns based on my review comments. However, as shown in Figure 1 of the uploaded PDF, the performance of the proposed method appears to be similar to that of cryoSPARC. Could the authors please further elaborate on the advantages of the proposed method compared to cryoSPARC, such as faster speed, automation, or other benefits? If the performance is indeed comparable at this stage, what potential improvements could be made?