# VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on
Unregistered Data

Emmanuel Hartman1<EMAIL_ADDRESS>Emery Pierson2 ( emery.pierson@univ-
lille.fr)
Department of Mathematics E. Hartman was supported by NSF grant DMS-1953244.
E. Pierson has been partially supported by by the Austrian Science Fund (grant
no P 35813-N).  Florida State University  Tallahassee  USA1
Institute of Mathematics  University of Vienna  Vienna  Austria 2

###### Abstract

We present a novel geometric deep learning layer that leverages the varifold
gradient (VariGrad) to compute feature vector representations of 3D geometric
data. These feature vectors can be used in a variety of downstream learning
tasks such as classification, registration, and shape reconstruction. Our
model’s use of parameterization independent varifold representations of
geometric data allows our model to be both trained and tested on data
independent of the given sampling or parameterization. We demonstrate the
efficiency, generalizability, and robustness to resampling demonstrated by the
proposed VariGrad layer.

##  1 Introduction

Geometric deep learning is an exciting field that has been rapidly evolving in
recent years. It involves applying deep learning techniques to problems that
involve geometric data such as point clouds, meshes, graphs, and curves. The
ability to effectively learn from geometric data has numerous applications in
fields such as computer graphics, computer vision, and robotics. In this
project, we aim to contribute to the development of geometric deep learning by
proposing a novel layer for deep neural networks. Our layer leverages the
gradient of the varifold norm to define a vector field on a template shape
which is then seen as a feature vector defined by the input. This method has
the potential to improve the accuracy of deep learning models on geometric
data, which could have a significant impact on a wide range of applications.
In the following article, we will describe our method in detail and present
experimental results that demonstrate its effectiveness for 3D curves and
shape graphs.

###  1.1 Background & Related Work

Figure 1:  Illustration of the proposed methodology. The input geometric data
and template shape are transformed into varifold representations and the
gradient of the varifold norm is taken with respect to the vertices of the
template. This gradient is passed through graph convolutional and edge pooling
layers with respect to the template edges. Resulting feature vectors can be
fed into traditional deep learning frameworks for reconstruction and
classification tasks.

The ability to compute similarity metrics between geometric objects is crucial
in many applications. However, traditional methods that rely on aligning
parameterizations can be computationally expensive and may not be suitable for
large or complex datasets. Varifold representations have emerged as a powerful
tool for computing parameterization invariant similarity terms for geometric
objects. Unlike traditional approaches, varifold representations are
independent of the parameterization of the data, making them well-suited for
cases where objects do not share a point-to-point correspondence. The space of
varifolds has been equipped with a reproducing kernel Hilbert space (RKHS)
norm. The benefits of using this norm are that it is robust to noisy data and
can be efficiently computed. The RKHS norm is also differentiable, making it
suitable as a loss function for optimization-based shape matching [SBC19,
HSK∗23], or deep learning models for shape reconstruction [BPAD23].

Most of the pioneering works use well-defined shape descriptors to obtain a
suitable representation of curves or shape graphs. Indeed, it is theoretically
possible to retrieve a curve from its geometric moments, or its affine
integral signature [FKK10], that have been applied in the context of curve
classification. However, retrieving curves from such features implies solving
costly optimization problems [KS21]. The Square Root Velocity Field framework
has been used in elastic matching [SKJJ10] on 3D curves. It has been used on
face-extracted curves for face identification [VvJD∗11] and on extremal human
curves [SWD14] for human pose retrieval. However, this elastic matching
approach requires solving the problem of parameterization for each pair of
curves. Moreover, their extension to shape graphs remains an active area of
research [GSS21, GBNS22, SBC22].

In the recent years, a few works have shown the superiority of deep learning-
based methods over the previous approaches. However, some approaches require
strong hypotheses on the data, such as a common parameterization [HSC∗21], or
an ordering of the curve or shape graph[HE18, XJB21]. Meanwhile, a few
approaches using convolutional neural networks on binary images have been
proposed for sketch representation [QYC∗22, MKDM22]. Not only those approaches
don’t take profit from the geometric information of curves, but they also do
not extend to 3D data. Moreover, most of those approaches require training on
large datasets [SBHH16, HE18] to be efficient.

Geometric deep learning [BBL∗17] tries to circumvent those problems compared
to the previous approaches. In particular, a few methods address the problem
of dealing with data that lacks point-to-point correspondences or fixed
shapes. Those methods interpret point clouds with an arbitrary number of
inputs as a feature vector of a single-shaped object. PointNet [CSKG17] is a
notable example of such methods, designed specifically for processing and
classifying point clouds. PointNet employs a shared multi-layer perceptron
(MLP) network to transform individual points into a high-dimensional feature
space. The feature vectors are then aggregated using a symmetric function,
such as max pooling, to generate a fixed-length representation of the entire
point cloud. This representation can be fed into a fully connected neural
network for classification or other downstream tasks. To overcome the
limitation of solely relying on point cloud information without considering
connectivity and local relationships between points, the Dynamic Graph
Convolutional Neural Network (DGCNN) [WSL∗19] method was introduced. DGCNN
utilizes k-nearest neighbors (kNN) to establish edges between points, enabling
the construction of a graph representation. By employing graph convolutional
and edge pooling layers, DGCNN generates high-dimensional feature vectors for
each point and aggregates them using a symmetric function. This results in a
fixed-length representation of the entire point cloud that depends on the
local relationships between points, which can be fed into a fully connected
neural network for classification or other tasks. While some other
architectures [MKK21, SACO22, WNEH22] have shown more expressivity on 3D point
clouds and surfaces, they rely on surface-specific operators such as the
Laplace-Beltrami operator [SACO22], which cannot be used for 3D curves or
shape graphs.

###  1.2 Contributions

In contrast, our proposed method directly operates on curves and shape graphs
rather than their vertices as point clouds. We introduce a novel approach that
leverages varifold representations to construct consistent-sized feature
vectors, enabling the representation of geometric data irrespective of its
sampling or registration. By utilizing varifold representations, our method
captures the intrinsic properties of curves and shape graphs, leading to
enhanced representations for geometric data. We propose a novel solution to
this problem that leverages the differentiability of the varifold similarity
metric. We compute the gradient of the distance between a template shape and
the input shapes with respect to the points of the template shape. The
resulting gradient is a vector field on the template shape that describes the
optimal direction of deformation of the vertices of the template to minimize
the distance between the shapes. This vector field can be interpreted as a
feature vector for each input shape whose dimension is fixed regardless of the
discretization or parameterization of the input shape. Traditional deep
learning layers can process this feature vector, which is represented as a
vector field lying on our template shape. Additionally, the vector field can
be convolved with respect to the geometry of our template shape. For open
curves, this is a 1D convolution, and for closed curves, this convolution is
performed with cyclic padding. For surfaces and shape graphs, a graph
convolution can be performed with respect to the underlying connectivity of
the template shape. This approach provides a robust and flexible method for
converting unregistered geometric objects into a fixed-dimensional feature
vector that can be utilized in traditional and geometric deep-learning models.

##  2 Feature Vectors for 3D Curves and Shape Graphs

We focus on discrete 3D curves and shape graphs which are composed of vertices
$V\subseteq\mathbb{R}^{3}$ and linear edges $E$ connecting these vertices.
However, a major challenge in working with such geometric data is the reliance
on specific parameterizations, which can make downstream learning tasks and
analysis difficult. To address this challenge, we adopt varifold
representations as a powerful tool for characterizing and studying geometric
objects without being dependent on their parameterizations.

###  2.1 Varifold Representations of 3D Curves and Shape Graphs

Figure 2: Network Architectures for Downstream Tasks: A diagram of the
classifier and auto-encoder networks used for numerical experiments.
Unparameterized 3D curves or shape graphs are given as inputs and feature
vectors are extracted via a feature vector architecture. These feature vector
architectures are then combined with the downstream architectures described in
this figure and the combined networks are trained for their respective tasks.

In recent years, varifold representations of geometric data have seen many
applications in the field of shape analysis [SBC19, KCC17, CT13, CCG∗20]. The
space of varifolds is given by Borel measures on the Cartesian product of
$\mathbb{R}^{3}$ and the unit sphere $S^{2}$ denoted by
$\mathcal{M}(\mathbb{R}^{3}\times S^{2})$. Each discrete shape graph,
$\mathcal{C}=(V,E)$, is mapped to a finitely supported measure
$\mu\in\mathcal{M}(\mathbb{R}^{3}\times S^{2})$ where each support corresponds
to each edge and is given by the centroid and normalized tangent vector of the
edge. The mass of the support associated with an edge in the varifold
representation is determined by its length. This mapping ensures that
regardless of the specific parameterization or ordering of points and edges in
a shape graph, it will be identified with the same varifold representation.
The space of varifolds is equipped with a Reproducing Kernel Hilbert Space
(RKHS) norm, we define a suitable kernel function that measures the similarity
between varifolds. The kernel function captures the inner product structure in
the space of varifolds, allowing us to compute distances and perform various
operations. This kernel function can be defined coordinate-wise by taking the
product of a Gaussian kernel on $\mathbb{R}^{3}$ and a Binet kernel on
$S^{2}$. The inner product defined by this kernel is given by

$\langle\mu,\nu\rangle_{Var}=\int_{\mathbb{R}^{3}\times
S^{2}}\int_{\mathbb{R}^{3}\times S^{2}}e^{-a\|x_{1}-x_{2}\|^{2}}\langle
u_{1},u_{2}\rangle^{2}d\mu\,d\nu$

where $a$ is a balancing parameter. From this inner product, a distance on the
space of varifolds is given by

$\|\mu-\nu\|^{2}=\langle\mu,\mu\rangle_{Var}+\langle\nu,\nu\rangle_{Var}-2\langle\mu,\nu\rangle_{Var}.$

We define the dissimilarity between two discrete shape graphs
$\mathcal{C}_{1}=(V_{1},E_{1})$ and $\mathcal{C}_{2}=(V_{2},E_{2})$ as the
varifold distance between the associated finitely supported varifolds
$\mu_{1},\mu_{2}$ respectively. In particular, we write the dissimilarity
between two shape graphs as
$d_{Var}(\mathcal{C}_{1},\mathcal{C}_{2})=\|\mu_{1}-\mu_{2}\|$. As these
varifolds are finitely supported the inner product can be approximated by a
sum over $E_{1}$ and $E_{2}$ given by

$\langle\mu_{1},\mu_{2}\rangle_{Var}=\sum_{e_{i}\in E_{1}}\sum_{e_{j}\in
E_{2}}e^{-a\|c_{i}-c_{j}\|^{2}}\langle u_{i},u_{j}\rangle^{2}l_{i}l_{j}.$

Here $c_{i},c_{j}$ denote the centroids of $e_{i},e_{j}$, $u_{i},u_{j}$ denote
the normalized tangent vectors of $e_{i},e_{j}$, and $l_{i},l_{j}$ denote the
lengths of $e_{i},e_{j}$.

By utilizing varifold representations equipped with a RKHS norm, we can
effectively and efficiently capture the essential geometric information of 3D
curves and shape graphs in a parameterization-independent manner. This allows
for meaningful and efficient comparisons, analysis, and processing of
geometric data, as well as facilitating tasks such as shape matching,
recognition, and reconstruction. Moreover, the invariance of varifold
representations to parameterization changes provides robustness and
consistency in geometric deep learning tasks.

###  2.2 VariGrad Feature Vector Architecture

Our architecture is designed for data-driven applications where the geometric
data used for training or applying the model can be matched to a template
shape $\mathcal{T}$. Leveraging this template shape, we compute the gradient
of the varifold distance between each input shape and the template shape,
specifically with respect to the vertices of the template shape. This
computation results in a vector field on the template that can be viewed as a
representation of the input. This data is fed into a sequence of convolutional
layers where the convolution is performed with respect to the template shape
[KW17]. The resulting vector field is concatenated into a fixed-dimension
representation of the input data regardless of the sampling of the input data.
A visual representation of the VariGrad architecture is given in Figure 1.
Moreover, we have made a Pytorch implementation of this framework as well as
the trained VariGrad models available on Github111https://github.com/emmanuel-
hartman/Pytorch_VariGrad.

Our framework sees improved results compared to methods that combine pointwise
Multi-Layer Perceptron (MLP) and feature point aggregation. This is mainly due
to the utilization of the geometric prior given by the template shape on the
input data. Moreover, rather than treating the points individually our method
is able to efficiently process whole curves and shape graphs even in the
unregistered setting, thanks to their varifold representations. Moreover, as
they use MLP to embed each point in the point cloud in a high dimensional
space, PointNet and DGCNN require more trainable parameters than the graph
edge convolutions [KW17] used by VariGrad.

##  3 Experimental Results

The following experiments aim to compare the performance of the proposed
VariGrad layer to the commonly used PointNet and DGCNN architectures in
geometric deep learning applications where point-to-point correspondences are
unknown. Each experiment constructs models that utilize the PointNet, DGCNN,
and VariGrad architectures respectively to learn a consistent size feature
vector. The feature vectors from these architecturesare used in further
learning tasks such as classification or shape reconstruction. Other than the
differences in resources required by the different feature vector
architectures these models use the same trainable parameters, loss functions,
optimization parameters, and number of training epochs.

###  3.1 Datasets

To evaluate the performance of the methods, similar networks will be trained
using these layers on two different datasets. We consider one dataset of 3D
curves and one dataset of 3D shape graphs that are extracted from 3D surface
data. We first extract landmarks from the shapes available in those datasets
and compute the surface geodesics on the given shapes, giving curves that
represent the pose of the 3D shapes. An illustration of the process is
visualized in Figure 3. We use this geometric data as input of the different
models. In both datasets, the extraction process produces unregistered curves
and shape graphs, meaning that there is no prior point correspondence present
in the data. For both of the datasets we consider, we perform a random split
between training and testing set, ensuring that the classes (face or body
identities) are well-balanced between the two splits. Moreover, we take the
template shape to be a random shape from the corresponding dataset. The first
dataset comprises boundary curves of human faces extracted from the COMA
dataset of human face motions [RBSB18]. The extraction process is performed on
all 20465 elements of the COMA dataset, resulting in 20465 closed curves in
three dimensions. These curves are unparameterized and have a variety of
samplings due to the extraction process with numbers of vertices ranging from
64 to 96. This dataset is then separated into a training set of 18506 curves
and a validation set of 1959 curves. Along with each curve, a label indicating
which of the eight identities present in the COMA dataset from which the curve
was extracted. The second dataset consists of 41220 shape graphs representing
human pose, extracted from the Dynamic FAUST dataset of 4D human motion
scans[BRPMB17]. Again these shape graphs are unparameterized where the numbers
of vertices in the shape graphs range from 111 to 356. Similar to the dataset
of curves, the data is split into 35000 training samples and 6220 testing
samples and an identity label is included for each shape graph in the dataset,
corresponding to one of the twelve identities. By training and comparing the
performance of the networks on these datasets, insights can be gained
regarding the suitability and effectiveness of the VariGrad layer when
compared to the state-of-the-art PointNet and DGCNN methods.

Figure 3: Examples of curve and shape graph extraction used for the creation
of the example datasets. Surface meshes are given as inputs, landmarks are
identified, geodesics along the surface are found between the landmark points,
and finally these geodesic curves are concatenated to create the curves and
shape graphs.

###  3.2 Classification

We evaluate the classification models based on their classification accuracy
on the testing data. The results show that the model utilizing the VariGrad
architecture outperforms the PointNet and DGCNN architectures. This difference
in performance is likely due to the sparser set of points provided by the
vertices of the three-dimensional curves considered in this experiment. The
lower number of points may not be sufficient for the PointNet and DGCNN
architectures to learn representative feature vectors. The downstream network
is an MLP, for which the architecture is outlined in Figure 2. Here $N$
denotes the dimension of the feature vector recovered by the respective
architectures and $C$ denotes the number of classes in the data set. The input
and hidden fully connected layers are followed by a ReLU activation function
and a batch normalization layer.

| Face Curve Data | Shape Graph Data
---|---|---
| Classification | Train Time | Classification | Train Time
| Accuracy | (s / batch) | Accuracy | (s / batch)
PointNet | 29.36 | 0.157 | 78.08 | 0.148
DGCNN | 62.71 | 0.099 | 83.29 | 0.113
VariGrad | 84.37 | 0.045 | 83.31 | 0.046
Table 1: Classifications Results: We present the results of the classification
network utilizing three different feature vector architectures. We present the
percentage of correct identity classifications by our model on an unseen
testing set. Further, we present the mean training time per batch (10 samples)
of each network to underscore the efficiency of the VariGrad approach.

We train models using each feature vector architecture on both the COMA curve
training set and DFAUST shape graph training set. We report the results of
these experiments in Table 1 along with the average training time of each
model. In both the cases of curves and shape graphs, the models using the
VariGrad feature vector architecture produce higher classification accuracies
than the models utilizing the PointNet or DGCNN architectures. Moreover, the
reliance on less trainable parameters means that the average training time of
the VariGrad architecture is significantly lower than that of PointNet and
DGCNN.

###  3.3 Shape Reconstruction

Figure 4: Qualitative comparison of shape reconstruction results for three
examples from the testing set of the human pose shape graph dataset. We
display the target shape graph (orange), with the reconstructions of PointNet
(purple), DGCNN (green), and VariGrad (blue).

For the task of shape reconstruction, three fully connected autoencoder models
are trained to learn a latent code representation of the shapes based on the
feature vectors produced by PointNet, DGCNN, and VariGrad, respectively. The
auto-encoder architecture is outlined in Figure 2 where $N$ denotes the
dimension of the feature vector recovered by the respective architectures and
$|V|$ denotes the number of desired vertices for the shape reconstruction.

| Face Curve Data | Shape Graph Data
---|---|---
| Mean | Train Time | Mean | Train Time
| Error | (s / batch) | Error | (s / batch)
PointNet | 0.0053 | 0.149 | 0.0039 | 0.174
DGCNN | 0.0010 | 0.113 | 0.0015 | 0.133
VariGrad | 0.0004 | 0.073 | 0.0005 | 0.072
Table 2: Shape Reconstruction Results: We present the results of the shape
auto-encoder network utilizing three different feature vector architectures.
We present the mean varifold error for an unseen testing set as well a the
mean training time per batch (10 samples).

Again, we train models using each feature vector architecture on both the COMA
curve training set and DFAUST shape graph training set. To evaluate each
network we compute the mean of the squared varifold norm between unseen
testing sets of shapes and the reconstructions produced by each network. We
report the results of these experiments in Table 2 with the average training
times of each model. Again, the VariGrad architecture requires less training
time than PointNet or DGCNN. Meanwhile, the VariGrad architecture produces a
lower mean error than PointNet or DGCNN for both curves and shape graphs. For
the testing set of shape graphs, we show a qualitative comparison of the
reconstructions produced by each of the feature vector architectures in Figure
4. This comparison confirms that VariGrad not only produces numerically
superior results but also visually more accurate reconstructions of 3D
geometric data.

###  3.4 Generalizability and Reparameterization Invariance

The final set of experiments we present demonstrates the generalizability of
our model and the invariance of our model to reparameterizations of the input
data. To do this, we consider two additional testing sets of shape graphs.
First, we consider resampled versions of the DFAUST shape graph testing set.
For this dataset, we apply random reparameterizations to each curve in the
shape graph. This includes identity labels so that we can evaluate both the
reconstruction and classification networks on the data. Second, we extract 90
shape graphs from the FAUST dataset of 3D scans of human poses. This dataset
includes less extreme poses than the DFAUST shape graph dataset but does
include some variability in the rigid alignment of the shape graphs not seen
in the training data.

| Reparameterized DFAUST | FAUST
---|---|---
| Classification | Reconstruction | Reconstruction
| Accuracy | Mean Error | Mean Error
PointNet | 49.98 | 0.0040 | 0.0061
DGCNN | 50.63 | 0.0049 | 0.0064
VariGrad | 80.91 | 0.0005 | 0.0040
Table 3: Generalizability Results: Here we test our pre-trained shape graph
reconstruction and classification models on two additional datasets which
include variability unseen in the training.  Figure 5: Qualitative comparison
of each model’s invariance to reparameterization. We display 100 resampled
versions of a single shape graph (orange) and the 100 reconstructions produced
by PointNet (purple), DGCNN (grean), and VariGrad (blue).

In Table 3, we report the evaluation metrics for the trained classification
and reconstruction models applied to the reparameterized DFAUST shape graphs
and for the trained reconstruction model applied to the FAUST shape graphs. In
these results, we show that the VariGrad architecture produces superior
results for both classification and reconstruction tasks. In particular, for
the reparameterized DFAUST data, we report similar results to the original
testing set in both classification and reconstruction tasks. Meanwhile, the
DGCNN architecture sees a significant decrease in classification and
reconstruction accuracy for the reparameterized data.

Finally, we propose an experiment to evaluate qualitatively the
parameterization invariance of the different models. We created 100
reparameterizations of a single randomly selected shape graph from the DFAUST
shape graph data set and applied them as input for the trained autoencoders.
We collected the 100 outputs for each method and display them in Figure 5. We
observe that our model keeps a very low variation of output shapes, as opposed
to both PointNet and DGCNN. This visually confirms the invariance of our model
to resampling and reparameterization.

##  4 Conclusion

In this paper, we presented VariGrad, a new feature vector architecture for
learning tasks on geometric data such as curves and shape graphs. As we
demonstrated in the experiments, by leveraging the varifold representation,
this architecture produces superior results in downstream learning tasks.
Moreover, varifold representations are not dependent on the given
parameterization of the data and thus the VariGrad layer is robust to
reparameterization and resampling of geometric data. We plan in the short
future to extend this work to new applications, such as hand sketches.
However, as the topology of hand sketches often vary in a same class of
drawnings, the choice of Varigrad’s template shape would have to be done more
carefully. Moreover, as the varifold representation of curves has a natural
extension to surfaces and point clouds, we plan to focus on extending VariGrad
to such objects in the future.

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