L EARNING H YPERBOLIC R EPRESENTATIONS FOR U NSUPERVISED 3D S EGMENTATION
Anonymous authors
Paper under double-blind review
A BSTRACT
There exists a need for unsupervised 3D segmentation on complex volumetric data, particularly when annotation ability is limited or discovery of new categories is desired.
Using the observation that much of 3D volumetric data is innately hierarchical, we propose learning effective representations of 3D patches for unsupervised segmentation through a variational autoencoder (VAE) with a hyperbolic latent space and a proposed gyroplane convolutional layer, which better models the underlying hierarchical structure within a 3D image.
We also introduce a hierarchical triplet loss and multi-scale patch sampling scheme to embed relationships across varying levels of granularity.
We demonstrate the effectiveness of our hyperbolic representations for unsupervised 3D segmentation on a hierarchical toy dataset, BraTS whole tumor dataset, and cryogenic electron microscopy data.
I NTRODUCTION
Recent advances in technology have greatly increased both the availability of 3D data, as well as the need to process and learn from 3D data.
In particular, technologies such as magnetic resonance imaging and cryogenic electron microscopy (cryo-EM) have led to greater availability of 3D voxel data.
Deep learning is a promising technique to do so, but producing annotations for 3D data can be extremely expensive, especially for richer tasks such as segmentation in dense voxel grids.
In some cases, labels may also be impossible to produce due to the limitations of current knowledge, or may introduce bias if we want to conduct scientiﬁc discovery.
Unsupervised learning, which does not require annotations, is a promising approach for overcoming these limitations.
In this work, we tackle the challenging problem of unsupervised segmentation on complex 3D voxel data by addressing the essential challenge of representation learning.
We expand from prior literature in the hyperbolic domain that conducts classiﬁcation in simple data to the task of segmentation in 3D images, which requires signiﬁcantly more representation discriminability.
In order to learn effective representations, we need to capture the structure of our input data.
We observe that 3D images often have inherent hierarchical structure: as a biomedical example, a cryo-EM tomogram of a cell has a hierarchy that at the highest level comprises the entire cell; at a ﬁner level comprises organelles such as the mitochondria and nucleus; and at an even ﬁner level comprises sub-structures such as the nucleolus of a nucleus or proteins within organelles.
For downstream analysis, we are typically interested in the unsupervised discovery and segmentation of structures spanning multiple levels of hierarchy.
However, prior work on representation learning for unsupervised 3D segmentation does not explicitly model hierarchical structure between different regions of a 3D image.
We argue that this hampers the ability to leverage hierarchical relationships to improve segmentation in complex 3D images.
Our key insight is that we can utilize a hyperbolic embedding space to learn effective hierarchical representations of voxel regions in 3D images.
Hyperbolic representations have been proposed as a continuous way to represent hierarchical data, as trees can be embedded in hyperbolic space with arbitrarily low error (Sarkar, 2011).
These methods have shown promise for modeling data types such as natural language word taxonomies (Nickel & Kiela, 2017; 2018), graphs (Nickel & Kiela, 2017; Mathieu et al., 2019; Ovinnikov, 2019; Chami et al., 2019), as well as simple MNIST (LeCun et al., 2010) image data for classiﬁcation (Mathieu et al., 2019).
To the best of our knowledge, our work is the ﬁrst to introduce learning hyperbolic representations to capture hierarchical structure among subregions of complex 3D images, and to utilize the learned hyperbolic representations to perform a complex computer vision task such as segmentation.
Our approach for learning hyperbolic representations of 3D voxel grid data is based on several key innovations.
First, to handle larger and more complex 3D data such as biomedical images, we propose a hyperbolic 3D convolutional VAE along with a new gyroplane convolutional layer that respects hyperbolic geometry.
Second, we enhance our VAE training objective with a novel self-supervised hierarchical triplet loss that helps our model learn hierarchical structure within the
VAE’s hyperbolic latent space.
Finally, since our goal in segmentation is to learn hierarchy within voxel regions of 3D input, we present a multi-scale sampling scheme such that our 3D VAE can simultaneously embed hierarchical relationships across varying levels of granularity.
In summary, our key contributions are as follows:
• We introduce a hyperbolic 3D convolutional VAE with a novel gyroplane convolutional layer that scales the learning of hyperbolic representations to complex 3D data.
• We propose a multi-scale sampling scheme and hierarchical triplet loss in order to encode hierarchical structure in the latent space and perform 3D unsupervised segmentation.
• We demonstrate the effectiveness of our approach through experiments on a synthetic 3D toy dataset, the Brain Tumor Segmentation (BraTS) dataset (Menze et al., 2014; Bakas et al., 2017; 2018), and cryo-EM data.
R ELATED W ORK
Segmentation on 3D voxel data Since 3D voxel grids are dense, computer vision tasks such as supervised segmentation are commonly performed using deep learning architectures with 3D convolutional layers (Chen et al., 2016; Dou et al., 2017; Hesamian et al., 2019; Zheng et al., 2019).
However, due to the challenges of obtaining voxel-level segmentations in 3D, there has been signiﬁcant effort in ﬁnding semi-supervised approaches, including using labels only from several fully annotated 2D slices of an input volume (C¸ic¸ek et al., 2016), using a smaller set of segmentations with joint segmentation and registration (Xu & Niethammer, 2019), and using one segmented input in conjunction with other unlabelled data (Zhao et al., 2019).
Unsupervised approaches for 3D segmentation are useful not only for further reducing the manual annotation effort required, but also for scientiﬁc discovery tasks where we lack the sufﬁcient knowledge to provide representative training examples for structures of interest.
Moriya et al.
(2018) extends to 3D data an iterative approach of feature learning followed by clustering (Yang et al., 2016).
Nalepa et al.
(2020) uses a 3D convolutional autoencoder architecture and performs clustering of the latent representations.
Another approach, (Dalca et al., 2018), uses a network pre-trained on manual segmentations from a separate dataset to perform unsupervised segmentation of 3D biomedical images.
However, this limits applicability to areas where we already have a dataset with manual annotations and makes it unsuitable for unbiased unsupervised discovery.
Gur et al.
and Kitrungrotsakul et al.
(2019) developed unsupervised methods for 3D segmentation of vessel structures, but these are specialized and do not generalize to the segmentation of other structures.
Beyond unsupervised 3D segmentation, there has been work such as Ji et al.
(2019) that performs unsupervised 2D segmentation based on a mutual information objective, and Caron et al.
(2018), which proposes using the clustered output of an encoder as pseudo-labels.
While these methods can be applied to 2D slices of a 3D volume to perform 3D segmentation, they generally suffer limitations due to insufﬁcient modeling of the 3D spatial information.
None of the aforementioned approaches explicitly model hierarchical structure, which is the main focus of our work.
Hyperbolic representations A recent line of work has employed hyperbolic space to model hierarchical structure, with the intuition that tree structures can be naturally embedded into continuous hyperbolic space (Nickel & Kiela, 2017).
Several works have proposed hyperbolic variational autoencoders (VAEs) as an unsupervised method to learn hyperbolic representations.
Ovinnikov (2019) proposes a Wasserstein autoencoder on the Poincar´e ball model of hyperbolic geometry.
Nagano et al.
(2019) proposes a VAE on the hyperboloid model of hyperbolic geometry where the last layer of the encoder is an exponential map, and derives a reparametrisable sampling scheme for the wrapped normal distribution, which they use for the prior and posterior.
Mathieu et al.
(2019) proposes a VAE on the Poincar´e ball model of hyperbolic geometry.
In addition to having the last layer of the encoder be an exponential map, Mathieu et al.
(2019) also proposes to have the ﬁrst layer of the decoder be the gyroplane layer proposed by Ganea et al.
(2018) in order to better handle the geometry of the hyperbolic latent space, and applies their model to MNIST image classiﬁcation.
Our work differs by introducing an approach for learning hyperbolic representations that models the hierarchy between sub-volumes of complex 3D images, and uses a novel hierarchical triplet loss and sampling scheme to capture relationships among multiple levels of granularity in a given input.
In addition, a related ﬁeld of study has sought to generalize traditional Euclidean neural networks or their components to non-Euclidean spaces.
Ganea et al.
(2018) proposes hyperbolic feed-forward and recurrent architectures based on the theory of gyrovector spaces.
Building on this work, Chami et al.
(2019) propose a hyperbolic graph convolutional network.
Other works such as Bachmann et al.
; Becigneul & Ganea (2019); Gu et al.
(2019) have also proposed learning with a product space of manifolds.
Our work generalizes a layer of Ganea et al.
(2018) in order to create and use a new hyperbolic convolutional layer, which we call the gyroplane convolutional layer.
P RELIMINARIES
Hyperbolic Space Hyperbolic space is a non-Euclidean space with constant negative curvature.
Curvature is a measure of the deviation of the geometry from a ﬂat plane (Chami et al., 2019).
There are ﬁve equivalent models of hyperbolic geometry. Following previous work (Mathieu et al., 2019; Ganea et al., 2018; Lou et al., 2020), we use the Poincar ´ e ball model.
Hyperbolic space can be considered the continuous version of trees (Nickel & Kiela, 2017), making it a natural choice for embedding hierarchical data.
Trees can be embedded in the Poincar ´ e ball with arbitrarily low error (Sarkar, 2011), and like the leaves of a tree, the area of a disc in the Poincar
´ e ball increases exponentially with the radius.
Unlike trees, hyperbolic space is smooth, permitting deep learning.
Poincar´e
ball model of hyperbolic geometry The Poincar ´ e ball (of curvature c = − 1 ) is the open ball of radius 1 centered at the origin equipped with the metric tensor g p =
( λ x ) 2 g e , where the conformal factor λ x = 2 1 −|| x ||
and g e is Euclidean metric tensor (i.e., the usual dot product).
Formally, this makes the Poincar ´ e ball a Riemannian manifold .
The distance d p between points on the Poincar ´ e ball is given by:
[Equation]
The exponential and logarithm maps are a useful way to map from Euclidean space to the Poincar ´ e ball and vice versa (in general, to map from a tangent space to a Riemannian manifold and vice versa).
On the Poincar ´ e ball, the exponential and logarithm maps have the closed forms
[Equation]
where ⊕ denotes Mobius addition , which was ﬁrst introduced by Ungar (2001) as a way to deﬁne vector operations on hyperbolic space (see Appendix).
M ETHODS
In this section, we describe our approach for learning hyperbolic representations of subvolumes (3D patches) from 3D voxel grid data.
We propose a model that comprises a 3D convolutional variational autoencoder (VAE) with hyperbolic representation space and a new gyroplane convolutional layer, along with a novel hierarchical triplet loss and a multi-scale sampling scheme that facilitates learning hierarchical structure within the hyperbolic latent space.
To produce segmentations, we cluster the learned hyperbolic representations.
In Section 4.1, we describe our VAE framework as well as our proposed gyroplane convolutional layer and hierarchical triplet loss.
In Section 4.2, we introduce our approach of hyperbolic clustering for segmentation.
4. U NSUPERVISED HYPERBOLIC REPRESENTATION LEARNING 3D Hyperbolic VAE
framework The VAE framework (Kingma & Welling, 2013; Rezende et al., 2014) is widely used for unsupervised representation learning, but requires new innovations to lean effective hierarchical representations in 3D image data.
Our proposed hyperbolic VAE consists of a 3D convolutional encoder which maps sampled 3D patches of the input volume into hyperbolic space and produces the parameters of the variational posterior, as well as a 3D convolutional decoder which reconstructs the patch from sampled latent hyperbolic representations.
The last layer of the encoder is an exponential map that ensures that the output is in hyperbolic space, and the ﬁrst layer of the decoder is our proposed gyroplane convolutional layer which maps hyperbolic space to
Euclidean space.
This ensures that both the encoder and decoder respect the hyperbolic geometry of the latent space.
We use the wrapped normal distribution as our prior and posterior distribution (see Appendix).
Figure 1 illustrates an overview of this VAE framework.
Our variational autoencoder takes as input a patch of ﬁxed size m × m × m . This allows our model to learn representations of subvolumes that can subsequently be used to perform voxel-level segmentation of the whole 3D volume.
To learn hierarchical structure in the 3D scene of each input, we generate training examples using a multi-scale sampling scheme that samples patches of size r × r × r , where r is randomly sampled.
We use two sampling schemes, one for input of smaller sizes and one for input of larger sizes.
In both schemes, for a given 3D volume, we sample i patch centers v i uniformly.
In the sampling scheme for smaller inputs, we sample r ∼ U ( (cid:96) min , (cid:96) max ) , where (cid:96) min , (cid:96) max are hyperparameters.
The patch is then upsampled or downsampled to size m × m × m .
For larger inputs, we observe that semantic changes tend to occur on a logarithmic scale, so
we instead ﬁrst sample e ∼ U ( (cid:96) min , (cid:96) max ) and then set r = 2 e .
This sampling scheme is motivated by the intuition that for larger patches, a small change in r is less likely to correspond to signiﬁcant semantic difference.
Gyroplane convolutional layer Since R n = R × . . .
× R , high-dimensional Euclidean spaces can be decomposed into a product of low-dimensional Euclidean spaces.
An equivalent decomposition does not hold for arbitrary Riemannian manifolds, making it difﬁcult to generalize the usual (Euclidean) convolutional layer to arbitrary Riemannian manifolds.
For manifolds that are products of manifolds, we can generalize the usual convolution by replacing the Euclidean afﬁne transformation with an afﬁne transformation on the manifold.
For the Poincar´e ball, one analogue of the Euclidean afﬁne transformation is the gyroplane operator f a,p (see Appendix).
The details are as follows: for simplicity, suppose x is a 4D tensor containing elements of the Poincar ´ e ball and our kernel size is k × k × k , with an odd k value.
Our gyroplane convolutional layer is deﬁned as:
[Equation]
Our gyroplane convolutional layer can be extended in the same way as Euclidean convolutional layers to incorporate even kernel size k , input and output channels, padding, stride, and dilation.
Our model’s encoder mean output ( µ in Figure 1) can be interpreted as a product of Poincar´e balls, justifying our deﬁnition and use of the gyroplane convolutional layer.
Hierarchical triplet loss As our model is trained on patches of the whole 3D volume, the hierarchical structure of the volume is not readily apparent from the individual inputs.
To help the model infer hierarchical structure, we provide self-supervision in the form of a hierarchical triplet loss where positive examples are sub-patches of an anchor patch and negative examples are patches that do not overlap with the anchor patch.
To sample 3D patches for the triplet loss, we ﬁrst generate an anchor patch centered at voxel v with size r × r × r according to one of the above sampling schemes.
A positive child patch is generated as a smaller sub-patch of the anchor patch as follows: the positive child patch is centered at v with size r child × r child × r child , where r child ∼ U ( (cid:96) min , r − r gap ) , and r gap is a hyperparameter representing the gap in size between the anchor size and the child size.
A negative child is a patch of size r child × r child × r child centered at v neg , where v neg is sampled uniformly from the set of voxels w such that a patch of size r child × r child × r child centered at w does not overlap with the anchor patch. 3D ConvEncoder 3D ConvDecoder μ 𝞂
Z
Exponentialmap Softplus Gyroplane convolutionallayer Wrapped normal distribution EuclideanHyperbolic
X
Input volume
Sampled 3D patch
X pos neg μ p μ pos μ neg
Hierarchical triplet loss
Sampled anchor, positive child, and negative child from input volume
X’ Our choice of positive and negative patches is motivated by the compositional hierarchy of 3D volumes. Our hierarchical triplet loss encourages the anchor patch and a sub-patch (positive child) to have similar representations, while encouraging the anchor patch and a distant patch (negative child) to have dissimilar representations. In hyperbolic space, this has the interpretation of belonging to the same hierarchy and belonging to different hierarchies, respectively.
We learn hierarchy within a 3D image through this triplet loss.
The hierarchical triplet loss can be formulated with any dissimilarity measure d between the encoder outputs µ (see Figure 1) of the anchor µ p , positive child µ pos , and negative child µ neg .
For our model, we take d to be the Poincar ´ e ball distance d p and deﬁne our triplet loss with margin α as:
[Equation]
This formulation can be extended to any metric space by taking the dissimilarity measure d to be the space’s metric.
In particular, for our ablations using an Euclidean latent space we take the dissimilarity measure d to be the Euclidean distance.
Optimization We optimize a loss function that can be decomposed as an evidence lower bound (ELBO) loss and our new hierarchical triplet loss that encourages the learning of hierarchical structure in the latent representations.
The total loss can be formulated as L total =
L ELBO + βL triplet , where β is a hyperparameter that controls the strength of the triplet loss.
4. S EGMENTATION BY CLUSTERING REPRESENTATIONS
Hyperbolic clustering In 3D segmentation, we seek to assign each voxel v a segmentation label s v ∈ { 1 , . . . , n } , where n is the number of segmentation classes.
We perform segmentation by clustering the representations of patches centered at each voxel.
We ﬁrst generate latent representations µ v for each voxel v by running our trained VAE on a patch of ﬁxed size p × p × p centered at v , upsampled or downsampled to encoder input size m × m × m if necessary.
We then cluster the µ v into n clusters, and produce a segmentation by assigning each v the cluster label of µ v .
Clustering is done using a k -means algorithm that respects hyperbolic geometry, which we derive by replacing the Euclidean centroid and distance computations of classical k -means with their appropriate counterparts in Riemannian geometry, the Fr´echet mean and manifold distance.
We calculate the Fr´echet mean using the algorithm of Lou et al.
E XPERIMENTS
Though our method is general to any 3D voxelized grid data, we evaluate on several biomedical datasets due to the availability of annotated 3D voxel data in the ﬁeld.
We evaluate our method quantitatively on both a synthetic 3D toy dataset simulating biological image data, as well as the BraTS tumor segmentation dataset.
Our biologically-inspired toy dataset allows quantitative evaluation of segmentation at multiple levels of hierarchy, while the BraTS dataset is a well-known benchmark for 3D MRI segmentation.
We also demonstrate the use of unsupervised segmentation for discovering new biological features in real-world cryo-EM data.
For all models, the encoder of our variational autoencoder is comprised of four 3D convolutional layers with kernel size 5 of increasing ﬁlter depth { 16 , 32 , 64 , 128 } .
The decoder is of the same structure, except with decreasing ﬁlter depth and a gyroplane convolutional layer as the initial layer.
We use β = 1 e 3 as the weighting factor between L ELBO and L triplet and α = 0 .
2 as the triplet margin, and train the model using the Adam optimizer (Kingma & Ba, 2014).
We ﬁx the representation dimension to be d = 2 .
For training on the toy dataset, we sample 3D volume sizes uniformly, and for BraTS and the cryo-EM dataset we sample using an exponential scale (see Section 4.1).
For inference, we obtain the latent representations of 5 × 5 × 5 patches densely across the full volume, and then perform hyperbolic k -means clustering, where the number of clusters k is a hyperparameter that controls the granularity of the segmentation.
For quantitative evaluation, we then use the
Hungarian algorithm (Kuhn, 1955) to match each predicted segmentation class with a corresponding ground truth label.
5. B IOLOGICALLY - INSPIRED TOY DATASET
Since most 3D image datasets are not annotated at multiple levels of hierarchy, we ﬁrst generate a hierarchical toy dataset to enable more thorough evaluation of the effectiveness of our model for unsupervised 3D segmentation.
We note that datasets such as ShapeNet (Chang et al., 2015) are unsuitable since they contain 3D shape models instead of 3D voxel grid data, which is the focus of our work.
Our toy dataset is inspired by cryo-EM tomograms of cells.
Each volume in our toy dataset contains multiple levels of hierarchy with objects at each level determined by texture and size.
Figure 2 shows an example input volume with sampled slices shown.
Each 3D image of our toy dataset consists of a background and a large sphere which represents a cell, which we will refer to as Level 1 of the image’s hierarchy.
The large sphere contains a mediumsize sphere and cube meant to represent cellular substructures such as vesicles, which we will refer to as Level 2 .
In turn, each of these shapes contains two smaller objects of the same shape in Level 3 .
The color, size, and location of each shape vary randomly.
We also apply biologically realistic noise in the form of pink noise.
More details can be found in the Appendix.
To measure the ability of our model to capture the hierarchy of the toy dataset, we separately evaluate on the three levels of hierarchy deﬁned above and use the average class DICE score to compare segmentation performance.
Since our model is unsupervised, segmentation classes are assigned to ground truth labels using the Hungarian algorithm.
See results in Table 1 and Table 6.
Comparison with prior approaches Table 1 shows quantitative comparison of our method with prior state-of-the-art 3D unsupervised, 2D unsupervised (which we extend to 3D), and semisupervised models.
As unsupervised 3D segmentation is a relatively unexplored ﬁeld, we provide these baselines with different levels of supervision for additional reference.
C¸ic¸ek et al.
(2016) was trained with 2% of the ground truth slices in each of the xy , yz , and xz planes, and Zhao et al.
(2019) was trained with one fully annotated atlas.
Ji et al.
(2019) was implemented using the authors’ original code and extrapolated to 3D. For Nalepa et al.
(2020) and Moriya et al.
(2018), we re-implemented their methods as the original code was unavailable.
Our model performs signiﬁ- [Table] [Table] cantly better at all levels of hierarchy compared to its unsupervised counterparts, and comparably to the semi-supervised approach of Zhao et al.
Ablation Table 6 presents ablation studies on the hierarchical toy dataset comparing our contributions: Euclidean vs. hyperbolic representations, the addition of our gyroplane convolutional layer, and the addition of our hierarchical triplet loss.
The Base Euclidean conﬁguration is the 3D convolutional VAE with Euclidean latent space, no gyroplane convolutional layer, and trained with just the ELBO loss.
The Triplet Euclidean conﬁguration adds the hierarchical triplet loss to the base
Euclidean conﬁguration.
The Base Hyperbolic conﬁguration is the same as the Base Euclidean conﬁguration except with hyperbolic latent space.
The Triplet conﬁguration is the hyperbolic analogue of the Euclidean Triplet conﬁguration, and GyroConv conﬁgurations have the addition of the gyroplane convolutional layer.
Hyperbolic representations outperform their Euclidean counterparts in all experiments.
We attribute this to the more efﬁcient and better organization of hyperbolic representations.
When we introduce the hierarchical triplet loss, performance improves signiﬁcantly for our hyperbolic models, but performance for our Euclidean model does not improve as much, likely due to information loss in representing hierarchical input.
Introducing the gyroplane convolutional layer shows clear improvement over our Base Hyperbolic model, which shows the beneﬁt of having a layer that respects the geometry of the latent space.
The combination of the triplet loss and gyroplane convolutional layer exhibits the most gain over the Base Hyperbolic model, but only small gains over the model with just the added triplet loss.
This shows the importance of the our triplet loss for learning effective hierarchical representations.
5. B RAIN T UMOR S EGMENTATION CHALLENGE DATASET
The BraTS 2019 dataset is a public, well-established benchmark dataset containing 3D MRI scans of brain tumors along with per-voxel ground truth annotations of tumor segmentation masks.
The scans are of dimension 200 × 200 × 155 and have four modalities; we use the FLAIR modality, which is the most commonly used one-modality input.
We use the same evaluation metric as in the BraTS challenge, and compare DICE score on whole tumor (WT) segmentation, which is detectable solely from FLAIR.
There are 259 high grade glioma (HGG) labelled training examples, which we split into 180 train examples, 39 validation examples, and 40 test examples.
We do not use the ofﬁcial validation or test sets because the ground truth annotations for these sets are not publicly available.
Table 3 shows the comparison of our results against prior work; we train all baselines on the speciﬁed data split for fair comparison.
The only exception is the current state-of-the-art fully- supervised result (Jiang et al., 2019) in Table 3, which also uses all 4 modalities.
We show this for reference as an upper bound; the reported number is trained on the full train set and evaluated on the BraTS test set.
Our best model performs signiﬁcantly better than the unsupervised baselines, and in addition outperforms one 3D semi-supervised model.
This illustrates the ability of our hyperbolic latent representations to effectively capture the hierarchical structure in individual brain scans.
We use a granular segmentation with three clusters for quantitative evaluation in order to capture the tumor, brain, and background, then use the Hungarian algorithm for assignment.
In addition, we also show qualitative results for our model (see Figure 3), which include byproduct segmentations from the same model with different numbers of clusters speciﬁed, showcasing additionally discovered features in the scan that could also be clinically useful. [Table] et al
[Table]
5. C RYOGENIC ELECTRON MICROSCOPY TOMOGRAMS
Finally, we show an example of unsupervised 3D segmentation in a real-world scenario where unsupervised discovery is important.
Cryogenic electron microscopy is a technique that images cells at cryogenic temperatures with a beam of electrons.
The value of each voxel is the electron density at that location, and is created through reconstruction from tilt slices of ± 70 degrees from electron microscopy.
Cryo-EM tomograms are a rich source of biological data, capturing many subcellular features that are unknown or unexplored.
We train our model on three 512 × 512 × 250 cryo-E0M tomograms of cells collected from a research laboratory, and run inference on a fourth tomogram.
Figure 3 shows segmentations produced by our model on a mitochondria from the evaluation tomogram, using the proposed hyperbolic embedding space vs. Euclidean embedding space, and at a coarse and ﬁner level of granularity.
Unlike the Euclidean approach, the hyperbolic approach discovers a ﬁne-grained class corresponding to small features on the mitochondria, which may be macromolecular aggregates.
As an example of performing unsupervised discovery with our model, the discovered features can now be investigated for their chemical identities and functions.
[Figure]
C ONCLUSION
We propose a method for learning hyperbolic representations of subvolumes in 3D voxel grid data, that is based on a hyperbolic 3D convolutional VAE with a new gyroplane convolutional layer that respects hyperbolic geometry.
We enhance the VAE training objective with a self-supervised hierarchical triplet loss that facilitates learning hierarchical structure within the VAE’s hyperbolic latent space and a multi-scale sampling scheme.
We demonstrate that hyperbolic clustering of learned voxel-level representations can be used to achieve state-of-the-art unsupervised 3D segmentation, on a hierarchical toy dataset and the BraTS dataset.
We also illustrate the promise of using our model for unsupervised scientiﬁc discovery on an example of cryogenic electron microscopy data.
A A PPENDIX A. R IEMANNIAN M ANIFOLDS
In this section we give a more complete introduction to Riemannian manifolds, of which hyperbolic space is an example.
Riemannian manifolds are spaces that locally resemble Euclidean space.
To deﬁne this mathematically, we ﬁrst introduce a manifold as a set of points M that locally resembles the Euclidean space R n .
Associated with each point x ∈ M is a vector space called the tangent space at x , denoted T x M , which is the space of all directions a curve on the manifold M can tangentially pass through point x .
A metric tensor g deﬁnes an inner product g x on every tangent space, and a Riemannian manifold is a manifold M together with a metric tensor g .
Distance on a Riemannian manifold as can deﬁned as the following.
Let γ : [ a, b ] → M be a curve on the manifold M .
The length of γ is deﬁned to be (cid:82) b a | γ (cid:48) ( t ) | γ ( t ) dt and denoted L ( γ ) .
The distance between any two points x , y on the manifold is deﬁned as d M ( x , y ) =
inf L ( γ ) , where the inf is taken over all curves γ that begin at x and end at y .
This distance makes M a metric space.
The exponential map exp x
( v ) :
T x M → M is a useful way to map vectors from the (Euclidean) tangent space to the manifold.
The exponential map is deﬁned as exp x
( v ) =
γ (1) , where γ is the unique geodesic, the shortest possible curve between two points, starting at x with starting direction v .
Intuitively, one can think of the exponential map as telling us how to travel one step starting from a point x on the manifold in the v direction.
The logarithmic map log v ( x ) : M → T x M is the inverse of the exponential map, and maps vectors back to Euclidean space.
A. G YROVECTOR O PERATIONS IN THE P OINCAR ´ E B ALL
Gyrovector operations were ﬁrst introduced by Ungar (2008) to generalize the Euclidean theory of vector spaces to hyperbolic space.
Mobius addition is the Poincar ´ e ball analogue of vector addition in Euclidean spaces.
The closed-form expression for Mobius addition on the Poincar ´ e ball with negative curvature c is Mathieu et al.
[Equation]
As one might hope anticipate, when c =
we recover Euclidean vector addition.
Additionally, the analogue of Euclidean vector subtraction is Mobius subtraction, which is deﬁned as x (cid:9) c
y = x ⊕ c ( − y ) , and the analogue of Euclidean scalar multiplication is Mobius scalar multiplication, which can be deﬁned for a scalar r as (Ganea et al., 2018):
[Equation]
(6) where we also recover Euclidean scalar multiplication when c = 0 .
In this paper, we only consider the Poincar ´ e ball with ﬁxed constant negative curvature c = 1 , which allows us to drop the dependence on c .
A. W RAPPED N ORMAL D ISTRIBUTION
The importance of the normal distribution in Euclidean space has led to many attempts to generalize the normal distribution to Riemannian manifolds.
The wrapped normal distribution is one popular way to do this (Mathieu et al., 2019; Nagano et al., 2019).
The wrapped normal distribution can be deﬁned on an arbitrary Riemannian manifold as the push-forward measure obtained by mapping the normal distribution in Euclidean space along the manifold’s exponential map.
The probability density function of the wrapped normal with mean µ and covariance Σ is:
[Equation]
where the subscripts P, E indicate whether the distribution is over the Poincar´e ball or Euclidean space, respectively.
To use the wrapped normal in a VAE, we require both a way to sample from the wrapped normal as well as a way to train its parameters.
Mathieu et al.
(2019) provides a reparametrization and sampling scheme for the wrapped normal on the Poincar´e ball.
A. G YROPLANE O PERATOR
The gyroplane layer can be thought of as a hyperbolic afﬁne transformation, and is motivated by the fact we can express a Euclidean afﬁne transformation as (cid:104) a, z − p (cid:105)
= sign( (cid:104) a, z − p (cid:105) )
|| a || d E
( z, H a,p ) (Ganea et al., 2018), where d E is Euclidean distance and H a,p = { z ∈ R p |(cid:104) a, z − p (cid:105) = 0 } .
H a,p is called the decision hyperplane.
Ganea et al.
(2018) deﬁned the gyroplane operator
f a,p from this formulation by replacing each component with its hyperbolic equivalent:
[Equation]
where H a,p is the hyperbolic decision boundary H a,p = { z ∈ B|
[Equation]
a, log p ( z ) (cid:105) = 0 } , and the distance to the hyperbolic decision boundary d p ( z, H a,p ) is
[Equation]
A. T OY D ATASET
Our biologically-inspired toy dataset has 120 total volumes, which we split into 80 train examples, 20 validation examples, and 20 test examples.
Each toy volume in our dataset is 50 × 50 × 50 and contains multiple levels of hierarchy.
The ﬁrst level of hierarchy ( Level 1 ) is a an outer sphere centered in the volume of radius r ∼ N (25 , 1) .
Using a cell analogy, this represents the outer cell.
The second level ( Level 2 ) consists of spheres (“vesicles”) and cuboids (“mitochondria”), both of which are textured, hollow, and completely contained within the outer cell wall.
The positions are randomly sampled with radius of r ∼ N (8 , 0 . 5) and with side length of s ∼ 2 · N (8 , 0 . 5) .
In the third level ( Level 3 ) we introduce small spheres and cuboids (“proteins”) in the vesicle spheres and mitochondria cubloids respectively.
The Level 3 proteins are randomly positioned with radius of r ∼ N (2 , 0 . 2) and with side length of s ∼ 2 · N (3 , 0 . 15) .
Each instance of a shape with a particular size is also given its own unique texture to mimic the different organelles of the cell.
The color of each object is chosen randomly, according to a standard normal distribution.
We also apply pink noise with magnitude m = 0 .
25 to the volume as it is commonly seen in biological data.
For the toy dataset, our implementations of the proposed models take between ﬁve to eight hours to train on a single Titan RTX GPU for both Euclidean and Hyperbolic variants.
We note that for our current implementation, hyperbolic k-means clustering takes on the order of a few hours versus minutes for Euclidean k-means.
However, this is because we are using our own unoptimized implementation based on recent research in fast Frechet mean algorithms, and standard packages such as scikit-learn do not include hyperbolic k-means algorithms.
The Euclidean k-means algorithms in these packages are heavily optimized with parallelization.
We anticipate that such optimization would bring the hyperbolic k-means’s runtime to the order of the Euclidean k-means, as the computational complexity of the algorithms are similar in practice.
A. B RA TS D ATASET
We also conduct an ablation study on the BraTS dataset with each of our added components with error bars over 4 independent runs.
Results are shown in Table 4.
We can see that our best Hyperbolic model outperforms our best Euclidean model signiﬁcantly.
The addition of the triplet loss improved both Euclidean and Hyperbolic models, while the Hyperbolic models see more improvement due to ability to encode hierarchy in hyperbolic latent space.
Our gyroplane convolutional layer also improves performance, while both of our additions jointly improve upon our Hyperbolic baseline, showing the beneﬁt of these added components to learning effective representations.
We include the average and 95 percentile Hausdorff distance as complementary evaluation metrics on the BraTS dataset.
See Table 5.
We show performance on our method compared to other unsupervised baselines; our model outperforms all prior methods on both metrics.
[Table] [Table] A. E VALUATION We use DICE score to quantitatively evaluate segmentation performance.
The DICE score is deﬁned as the following:
[Equation]
where TP is the number of true positives, FN is the number of false negatives, and FP is the number of false positives.
For our toy dataset, we ﬁrst assign predicted classes to ground truth labels using the Hungarian algorithm Kuhn (1955) [Table] then evaluate using the average class DICE score.
For the BraTS dataset Menze et al.
(2014); Bakas et al.
(2017; 2018), we evaluate DICE of the whole tumor segmentation following ofﬁcial evaluation guidelines.
We also use Hausdorff distance to evaluate the worst-case performance of our model. For two sets of points A, B , the directed Hausdorff distance from A to B is deﬁned as [Equation] where d is any distance function.
We will take d to be the Euclidean distance.
The Hausdorff distance is then deﬁned to be
[Equation]
The ofﬁcial BraTS evaluation uses 95 percentile Hausdorff distance as measure of model robustness (Bakas et al., 2018).
A. H YPERPARAMETERS
We use a single set of hyperparameters on all of our evaluation datasets, and these hyperparameters are not tuned on any of the evaluation datasets.
In order to obtain a reasonable set of hyperparameters, we created a completely separate synthetic dataset on which we trained models and tuned hyperparameters.
This synthetic dataset was created in a similar manner to our toy dataset; however, we designed it to have different and fewer objects, simpler nesting structure, no noise, and fewer textures.
The application of this single set of hyperparameters to our evaluation datasets — our toy dataset, the BraTS dataset, and the cryogenic electron microscopy dataset, demonstrates the robustness of our approach.
With the synthesis dataset, we tuned over a range of hyperparameter values using its validation set.
This includes weight of triplet loss β
= { 10 − 2 , 10 − 1 , 1 , 10 1 , 10 2 , 10 3 , 10 4 , 10 5 } with the ﬁnal [Figure] weight used as β = 10 3 .
The patch size for inference was tuned with range p = { 5 , 10 , 15 , 20 , 40 } with chosen size as 5 × 5 × 5 .
The number of epochs was tuned with range e = { 3 , 5 , 8 , 10 , 12 , 15 } with ﬁnal epoch number 8 .
The BraTS 2019 dataset Menze et al.
(2014); Bakas et al.
(2017; 2018) can be downloaded following directions from https://www.med.upenn.edu/cbica/brats2019/registration.
html .
We will release our toy dataset with the ﬁnal code release.
A. M ULTI - PATCH S AMPLING
Our method is designed to model the compositional hierarchy of 3D data, where we often ﬁnd visual substructures contained within other structures.
Based on this idea, we sample triplets of 3D volume patches that capture this notion of hierarchical structure.
Triplets are sampled through the following process: First, we sample a 3D patch of data to be the anchor element, and consider this to be the parent in the triplet.
Second, we sample a smaller patch of data that is completely contained within the parent patch, and consider this to be the positive child patch.
Then, we sample a smaller patch of data that does not overlap with the anchor patch, and consider this to be the negative child patch.
See Section 4.1 for further details on sampling procedure.
We input the (parent, positive child, negative child) tuples into our hierarchical triplet loss, where the loss encourages the anchor parent and positive child to have closer representations relative to the anchor and the negative child.
See
Figure 4 for an overview.
A. L ATENT D IMENSION A BLATION
In Section 5.1, Section 5.2, and Section 5.3, our experiments were all run with latent dimension of two.
To show the effect of higher latent space dimensions, we report an ablation study for both hyperbolic and Euclidean representations.
As expected, the performance increases with dimension for our model with Euclidean latent space, but our model with hyperbolic latent space still outperforms the Euclidean model at all tested dimensions.
[Table]