# COmic: Convolutional Kernel Networks for Interpretable End-to-End Learning
on (Multi-)Omics Data

Jonas C. Ditz
Methods in Medical Informatics
Department of Computer Science
University of Tübingen
Tübingen, Germany
<EMAIL_ADDRESS>
& Bernhard Reuter
Methods in Medical Informatics
Department of Computer Science
University of Tübingen
Tübingen, Germany
<EMAIL_ADDRESS>
& Nico Pfeifer
Methods in Medical Informatics
Department of Computer Science
University of Tübingen
Tübingen, Germany
<EMAIL_ADDRESS>

###### Abstract

Motivation: The size of available omics datasets is steadily increasing with
technological advancement in recent years. While this increase in sample size
can be used to improve the performance of relevant prediction tasks in
healthcare, models that are optimized for large datasets usually operate as
black boxes. In high stakes scenarios, like healthcare, using a black-box
model poses safety and security issues. Without an explanation about molecular
factors and phenotypes that affected the prediction, healthcare providers are
left with no choice but to blindly trust the models. We propose a new type of
artificial neural network, named Convolutional Omics Kernel Network (COmic).
By combining convolutional kernel networks with pathway-induced kernels, our
method enables robust and interpretable end-to-end learning on omics datasets
ranging in size from a few hundred to several hundreds of thousands of
samples. Furthermore, COmic can be easily adapted to utilize multi-omics data.
Results: We evaluated the performance capabilities of COmic on six different
breast cancer cohorts. Additionally, we trained COmic models on multi-omics
data using the METABRIC cohort. Our models performed either better or similar
to competitors on both tasks. We show how the use of pathway-induced Laplacian
kernels opens the black-box nature of neural networks and results in
intrinsically interpretable models that eliminate the need for post-hoc
explanation models.
Availability: Datasets, labels, and pathway-induced graph Laplacians used for
the single-omics tasks can be downloaded here. While datasets and graph
Laplacians for the METABRIC cohort can be downloaded from the above mentioned
repository, the labels have to be downloaded from cBioPortal. COmic source
code as well as all scripts necessary to reproduce the experiments and
analysis are publicly available at https://github.com/jditz/comics.

##  1 Introduction

In recent years, artificial neural networks (ANNs) show promising performance
when employed to learn correlations between data points and outcome variables.
They combine feature extraction and prediction training in a single end-to-end
learning scheme lowering the necessary amount of labour put into feature
engineering and can be used on very large datasets with relative ease. With
the advent of big data and high-throughput data generation techniques in
computational biology and healthcare, resulting in an increased number of data
points available for the training of prediction models, the use of ANNs in
these fields has vastly increased. In computational biology, ANNs showed
promising performance capabilities when applied to prediction tasks in
regulatory genomics (Alipanahi et al.,, 2015; Zhou and Troyanskaya,, 2015;
Kelley et al.,, 2016) and in biological image analysis (Ronneberger et al.,,
2015; Pärnamaa and Parts,, 2017; Ferrari et al.,, 2017). Furthermore, several
authors showed the potential of ANNs in healthcare scenarios such as diagnosis
(Arieno et al.,, 2019; De Fauw et al.,, 2018), drug discovery (Álvarez-
Machancoses and Fernández-Martínez,, 2019; Fleming,, 2018), epidemiology (Hay
et al.,, 2013), personalized medicine (Miotto et al.,, 2016), and operational
efficiency (Nelson et al.,, 2019). However, utilizing ANNs for prediction
tasks usually comes with two shortcomings: First, a large amount of data is
needed to robustly train a deep neural network and, second, neural network
models operate as black-boxes. While the first problem can be tackled by
shrinking the complexity of the neural network, which increases the stability
of the model but often leads to a decrease in performance, the second
shortcoming is most often addressed using post-hoc interpretation models. This
technique involves solving a secondary task that utilizes a pre-trained
prediction model such that the computed solution provides a humanly
understandable interpretation for the results computed by the prediction
model. Commonly used methods include Shapley additive explanation (SHAP,
Lundberg and Lee, (2017)), counterfactual explanation using generative models
(Stepin et al.,, 2021), and saliency methods like Layer-wise Relevance
Propagation (LRP, Montavon et al., (2019)), Deep Taylor Decomposition (DTD,
Montavon et al., (2017)), GuidedBP (Springenberg et al.,, 2014), or DeepLIFT
(Shrikumar et al.,, 2017). Using post-hoc interpretation methods can provide
additional information to improve understanding and advance scientific
knowledge in low-risk scenarios but they have several properties that render
their use in high-risk scenarios potentially problematic. Most post-hoc
interpretation methods are unfaithful to the computations of the original
model (Rudin,, 2019). Furthermore, many saliency methods ignore information
provided by deeper layers of ANNs (Sixt et al.,, 2020). Recent work showed
that post-hoc methods are limited in adversarial contexts (Bordt et al.,,
2022) and can be exploited to provide seemingly plausible but misleading
explanations (Lipton,, 2018). In healthcare, decisions that are made based on
wrong or misleading explanations have the potential to cause harm to patients.

Kernel methods can provide both robustness on small datasets and
interpretation capabilities within the domain of the data. These methods
utilize the kernel trick to solve a prediction task by implicitly projecting
data into the reproducing kernel Hilbert space (RKHS) of a kernel function and
solve the classification or regression problem within the RKHS. While the use
of a kernel functions does not always guarantee interpretation capabilities,
there are several kernel functions for biological data that result in
interpretable models, e.g., the oligo kernel for sequences (Meinicke et al.,,
2004) or the pathway-induced kernel for omics data (Manica et al.,, 2019).
Combining kernel functions with ANNs is a promising direction to increase the
robustness of ANN models on small datasets and several efforts in that
direction have been published in recent years (Cho and Saul,, 2009; Bo et
al.,, 2011; Mairal et al.,, 2014; Mairal,, 2016). Chen and colleagues showed
the feasibility of kernel networks for biological sequences by using a
relaxation of the mismatch kernel (Eskin et al.,, 2003) to build convolutional
and recurrent neural network architectures (Chen et al., 2019a, ; Chen et al.,
2019b, ). Furthermore, they showed how to use convolutional kernel neural
networks on graph-structured data like protein structures (Chen et al.,,
2020). While these models showed promising results and increased robustness,
the choice of the kernel function resulted in models that are not
intrinsically interpretable. However, we recently showed that a carefully
chosen kernel function results in intrinsically interpretable kernel networks
for biological sequence data (Ditz et al.,, 2021). With this work we introduce
Convolutional Omics Kernel Networks (COmic), a neural network architecure that
allows for intrinsically interpretable end-to-end learning on (multi-)omics
data. This is achieved by using a kernel function based on graph Laplacians of
biological networks to project input samples into a subspace of the
corresponding reproducing kernel Hilbert space (RKHS) with a variant of the
Nyström method. Using max-pooling combined with strictly linear layers for
classification results in COmic models that provide global interpretation,
while attention layers can be used to create COmic models that provide local
interpretation. In this manuscript, we use the definition most commonly found
in the interpretable ML literature for global and local interpretation
(Molnar,, 2020). In simple words, global interpretation can be used to answer
the question "How does the trained model make predictions?", while local
interpretation can be used to answer the question "Why did the model make a
certain prediction for a specific input?".

We show the performance and interpretation capabilities of COmic models on six
different breast cancer microarray cohorts. These cohorts contain microarray
gene expression data from patients with breast cancer and were stratified on
the occurrence of a relapse within 5 years. We compare our proposed method to
15 previously published approaches including several methods based on support
vector machines (SVMs) like network-based SVMs (Zhu et al.,, 2009), recursive
feature elimination SVMs (Guyon et al.,, 2002), and graph diffusion kernels
for SVMs (Rapaport et al.,, 2007; Gao et al.,, 2009) as well as classification
by average pathway expression (Guo et al.,, 2005), classification by
significant hub genes (Taylor et al.,, 2009), classification by pathway
activity (Lee et al.,, 2008), and pathway-induced multiple kernel learning
(PIMKL, Manica et al., (2019)). We show how the projection into a subspace of
the RKHS of pathway-induced kernels in combination with linear and attention
layers leads to global and local interpretations, respectively. Furthermore,
we use the METABRIC cohort (Curtis et al.,, 2012) to show how COmic models can
be used on multi-omics data. On the METABRIC breast cancer cohort, we
predicted disease-free survival using gene expression (mRNA) and copy number
alteration (CNA) data.

This work is structured as follows. We first introduce COmic by describing the
pathway-induced kernel and define the necessary network architecture to build
a COmic model. Afterwards, we show how to achieve a globally interpretable
COmic model using strictly linear layers and a locally interpretable COmic
model using attention layers. We evaluate COmic models on six breast cancer
cohorts and show how COmic models can be utilized for multi-omics data. With
this manuscript, we introduce a new kernel network architecture that can be
both robustly trained on small-scale (multi-)omics datasets and easily
utilized for prediction tasks on (multi-)omics datasets with several hundreds
of thousands of data points. Furthermore, our method results in intrinsically
interpretable models offering global and local interpretations of prediction
results.

Figure 1: Schematic of the proposed interpretable COmic models. Top: Pooling-
based COmic model. The kernel embedding of each involved pathway are reduced
to a single dimension by using a one-dimensional max pooling operation. The
output of each pooling layer is concatenated and prediction is performed using
a strictly linear fully-connected layer. The pooling-based COmic models are
globally interpretable similar to PIMKL models by utilizing the weights of the
fully-connected layer. Bottom: Attention-based COmic model. The kernel
embeddings of each involved pathway are transformed into a bag of instances of
a multiple instance learning problem. Attention weights for each instance are
calculated using each pathway’s kernel embedding and a matrix multiplication
between the bag of instances and the attention weights is performed. The
output is used for prediction with a strictly linear fully-connected layer.
The attention-based COmic models can be locally interpreted by utilizing the
attention weights.

##  2 Convolutional Omics Kernel Networks

In the following section, we describe the theoretical background of
convolutional kernel networks for prediction tasks on omics-based datasets.

###  2.1 Pathway-Induced Kernel Functions

The foundation of pathway-induced kernel functions are so-called graph
Laplacian matrices. To define these matrices we first assume $G=(V,E)$ to be
an undirected graph with vertices $V=\\{v_{1},\ldots,v_{n}\\}$ and edges $E$.
Furthermore, $G$ is assumed to be a weighted graph with weight matrix
$W\in\mathbb{R}^{n\times n}$, where $w_{ij}=w_{ji}\geq 0$ describes the weight
of the edge between vertices $v_{i}$ and $v_{j}$. The degree of each vertex
$v_{i}\in V$ is defined as $d_{i}=\sum_{j=1}^{n}w_{ij}$. The diagonal matrix
with the degrees $d_{1},\cdots,d_{n}$ on the diagonal is called the degree
matrix $D$. The unnormalized graph Laplacian $L\in\mathbb{R}^{n\times n}$ is
defined as (Von Luxburg,, 2007):

$L\coloneqq D-W$ (1)

Since an unnormalized graph Laplacian has undesirable mathematical properties
in case of very broadly distributed degrees within $G$ (Von Luxburg,, 2007),
we use a normalized graph Laplacian instead, which is defined as:

$L_{\text{sym}}\coloneqq D^{-\frac{1}{2}}LD^{-\frac{1}{2}}$ (2)

Similar to previous manuscripts, see e.g., (Chen et al.,, 2011; Manica et
al.,, 2019), we use molecular interaction networks (MIN) as graphs underlying
the normalized graph Laplacians. Using known interaction networks allows to
define a kernel function that computes the similarity of molecular measures
(gene expression, DNA methylation, etc.) under the assumed interactions
defined by the network. Given two molecular measures $x_{i}\in\mathbb{R}^{n}$
and $x_{j}\in\mathbb{R}^{n}$, we define the kernel function as

$K_{\text{MIN}}(x_{i},x_{j})=x_{i}^{T}L_{\text{MIN}}x_{j},$ (3)

where $L_{\text{MIN}}$ is the normalized graph Laplacian (as defined in Eq. 2)
of a molecular interaction network.

Manica and colleagues proposed to use pathway-specific sub-networks instead of
whole interaction networks for computing normalized graph Laplacians (Manica
et al.,, 2019). This method allows for a more tailored induction of prior
knowledge into a prediction task. The authors call this approach pathway-
induced (PI) kernel functions. Here, the similarity between two molecular
measures is not computed using a single graph Laplacian but with a set of $p$
graph Laplacians $\mathbf{L}=\\{L_{\text{PI}_{1}},\ldots,L_{\text{PI}_{p}}\\}$
each defined over a pathway-specific sub-network of the molecular interaction
network. Therefore the pathway-induced kernel for two molecular measures
$x_{i}\in\mathbb{R}^{n}$ and $x_{j}\in\mathbb{R}^{n}$ is not a single function
but a set of functions defined as

$K_{\text{PI}}(x_{i},x_{j})=\\{K_{\text{PI}_{1}}(x_{i,\text{PI}_{1}},x_{j,\text{PI}_{1}}),\ldots,K_{\text{PI}_{p}}(x_{i,\text{PI}_{p}},x_{j,\text{PI}_{p}})\\}$
(4)

with

$K_{\text{PI}_{r}}(x_{i,\text{PI}_{r}},x_{j,\text{PI}_{r}})=x_{i,\text{PI}_{r}}^{T}L_{\text{PI}_{r}}x_{j,\text{PI}_{r}},$
(5)

where $L_{\text{PI}_{r}}\in\mathbb{R}^{d\times d}$ is the symmetric graph
Laplacian of the $r$th pathway-specific sub-network with $d$ nodes and
$x_{i,\text{PI}_{r}}\in\mathbb{R}^{d}$ and
$x_{j,\text{PI}_{r}}\in\mathbb{R}^{d}$ are vectors containing only the signal
values of the molecular measures that correspond to nodes within the pathway-
specific sub-network described by $L_{\text{PI}_{r}}$. Manica and colleagues
used multiple kernel learning (MKL) to combine the set of pathway-induced
kernel functions into a single learning framework. In contrast to prior work,
we are using a variant of the Nyström method to formulate an explicit
parametrization of an orthogonal projection onto a finite-dimensional subspace
of a pathway-induced kernel’s RKHS. This enables us to define pathway-induced
kernel layers that can be incorporated into artificial neural networks. By
using a neural network architecture as the basis for COmic, the resulting
models can be tailored to specific datasets (single- or multi-omics) as well
as the desired form of interpretation. We will show that in the following
sections.

###  2.2 Convolutional Kernel Layer projects onto a finite-dimensional RKHS-
Subspace

Convolutional kernel networks make use of a variant of the Nyström method to
project input samples into a finite-dimensional subspace of the RKHS
$\mathcal{H}$ of a kernel function. To achieve this, a set of $q$ anchor
points $z_{1},\ldots,z_{q}$ is used to define a $q$-dimensional subspace
$\mathcal{E}$ of $\mathcal{H}$. The anchor points lie in the input space of
the kernel function and the RKHS subspace is defined as

$\mathcal{E}=\text{Span}(\phi_{z_{1}},\ldots,\phi_{z_{q}}),$ (6)

where $\phi(z_{i})$ denotes the image of the $i$th anchor point under the
kernel function. The orthogonal projection of input points onto $\mathcal{E}$
admits an explicit parametrization that utilizes the kernel trick to avoid
explicitly calculating the images $\phi(z_{i})$ (Mairal,, 2016; Williams and
Seeger,, 2001; Zhang et al.,, 2008). For an input $x$, i.e., a molecular
measure in case of omics data, the explicit parametrization
$\psi(x)\in\mathbb{R}^{q}$ is defined as

$\psi(x)=K_{ZZ}^{-\frac{1}{2}}K_{Z}(x),$ (7)

where $K_{ZZ}=(K(z_{i},z_{j}))_{i=1,\ldots,q;j=1,\ldots,q}$ is the gram matrix
formed by the anchor points, $K_{ZZ}^{-\frac{1}{2}}$ denotes the
(pseudo-)inverse square root of the Gram matrix, and
$K_{Z}(x)=\left[K(x,z_{1}),\ldots,K(x,z_{p})\right]^{T}$. As shown in Figure
1, each pathway-induced kernel function has to be modelled with a separate
orthogonal projection. This means that a COmic model utilizing $p$ pathway-
induced kernel functions maps each input onto $p$ representations
$\psi_{\text{PI}_{1}},\ldots,\psi_{\text{PI}_{p}}\in\mathbb{R}^{q}$. These
representations are then used to solve the prediction task for the input. In
the next section, we show two different approaches to combine the
representations leading to globally or locally interpretable models,
respectively.

The anchor points can be initialized using $k$-means on all input samples with
the number of clusters set to the number of anchor points. Afterwards, the
anchor points are optimized with the end-to-end learning scheme used to train
the whole network. For all experiments described in this manuscript, anchor
points were initialized using $k$-means++ (Vassilvitskii and Arthur,, 2006).

###  2.3 Globally and Locally Interpretable COmic Models

Globally interpretable COmic models are based on multi-kernel learning (MKL).
A simple approach to MKL is finding an optimal linear combination of all
utilized kernels. This approach learns a weight for each kernel and,
therefore, can be used to determine the influence each kernel has on the
prediction outcome. Since kernels are directly associated with pathways in
PIMKL, Manica and colleagues show that MKL weights can be used to determine
the importance of different pathways for a prediction (Manica et al.,, 2019).
We can embed a similar weighted sum of pathway-induced kernels into the
architecture of COmic models. Each kernel embedding produced by the PI-kernel
layer described in section 2.2 is passed into a one-dimensional max pooling
layer. This results in a single activation
$A_{r}=\max\left(\psi_{PI_{r}}\right)$ for each pathway-induced kernel, where
$\max\left(\psi_{PI_{r}}\right)$ denotes the maximum value in vector
$\psi_{PI_{r}}$. This activation is high, if the input is similar to one of
the learned anchor points, and low otherwise. By concatenating all activations
and passing them into a strictly linear fully-connected layer, the model
learns a single weight for each pathway-induced kernel and the prediction is
calculated as a weighted sum of all kernels, i.e.,

$\hat{y}=\sum_{r=1}^{p}w_{r}A_{r},$ (8)

where $\hat{y}$ is the overall prediction, $w_{r}\in\mathbb{R}$ is the weight
and $A_{r}\in\mathbb{R}$ is the activation of the $r$th pathway-induced
kernel. We call this architecture pooling-based COmic model. In contrast to
the MKL approach, the weights can become negative. This enhances the
interpretation capabilities of pooling-based COmic models, since we cannot
only infer if a pathway is important for the prediction task but also with
which class each pathway is associated by looking at the sign of the weight.
The top part of Figure 1 shows a schematic of a pooling-based COmic model.

Locally interpretable COmic models are based on multiple instance learning
(MIL). In MIL, each sample is represented as a bag of instances with a single
label per bag (Dietterich et al.,, 1997; Maron and Lozano-Pérez,, 1997; Oquab
et al.,, 2014). There are two general approaches to solve an MIL problem: the
instance-level approach and the embedding-level approach. In the instance-
level approach, an instance-level classifier predicts a score for each of the
instances in the bag. Afterwards, scores are aggregated by MIL pooling to
compute the prediction for the bag. In the embedding-level approach, a low-
dimensional embedding of each instance is computed and MIL pooling is used on
the embedded instances to create a bag representation. This representation is
used by a bag-level classifier to provide the prediction. While it was shown
that the embedding-level approach leads to better performances (Wang et al.,,
2018), the instance-level approach leads to interpretable models (Liu et al.,,
2012). Ilse and colleagues proposed an MIL-model based on neural networks that
combines the strength of both approaches, called attention-based multiple
instance learning (Ilse et al.,, 2018). Their approach can be utilized for
COmic models in the following way. The output of our proposed PI-kernel layer
can be viewed as a bag of low-dimensional instances
$H=\\{\psi_{PI_{1}},\ldots,\psi_{PI_{p}}\\}$, where each
$\psi_{PI_{r}}\in\mathbb{R}^{q}$ is the projection onto a $q$-dimensional
subspace of the RKHS of one pathway-induced kernel. Attention-based MIL
pooling is then used to compute the bag representation, i.e.,

$\tilde{\psi}=\sum_{r=1}^{p}a_{r}\psi_{PI_{r}},$ (9)

where

$a_{r}=\frac{\exp\\{w^{T}\tanh(V\psi_{PI_{r}}^{T})\\}}{\sum_{j=1}^{p}\exp\\{w^{T}\tanh(V\psi_{PI_{j}}^{T})\\}}.$
(10)

$w\in\mathbb{R}^{l\times 1}$ and $V\in\mathbb{R}^{l\times m}$ are parameters
of the attention layer. As noticed by Ilse and colleagues, the $\tanh(\cdot)$
non-linearity introduces a potential limitation due to the fact that it is
roughly linear only for $x\in[-1,1]$. This limitation can be reduced by using
a gating mechanism (Dauphin et al.,, 2017). In this case, the attention
weights are calculated as

$a_{r}=\frac{\exp\\{w^{T}(\tanh(V\psi_{PI_{r}}^{T})\odot\text{sigm}(U\psi_{PI_{r}}^{T}))\\}}{\sum_{j=1}^{p}\exp\\{w^{T}(\tanh(V\psi_{PI_{j}}^{T})\odot\text{sigm}(U\psi_{PI_{j}}^{T})\\}}.$
(11)

Again, $w\in\mathbb{R}^{l\times 1}$, $V\in\mathbb{R}^{l\times m}$, and
$U\in\mathbb{R}^{l\times m}$ are parameters of the attention layer. In both
cases, the training of all attention layer parameters is part of the end-to-
end training routine for the whole network and, hence, does not introduce the
need for additional measures. We call this architecture attention-based COmic
model. Since the attention weights $a_{p}$ are input specific, they enhance a
model with local interpretation capabilities. The bottom part of Figure 1
shows a schematic of an attention-based COmic model.

Figure 2: A: Cross-validation performance of COmic models compared to previously published methods. The boxplots show the ten mean auROC validation scores of a 10-times repeated 10-fold cross-validation over each of the six breast cancer cohorts. Performance of COmic models are shown in orange. The center line of each box indicates the median. The height of the boxes represents the inter quartile range (IQR) with the upper and lower whiskers set to 1.5 times the IQR. Outliers are depicted by black diamonds. Notches represent the confidence interval (CI) around the median and were calculated using bootstrapping with 10000 iterations. B: Visualizing the global interpretation capabilities of a pooling-based COmic model. Each box represents one of the 50 pathways and was created using the pathway weights of the models trained on the six publicly available breast cancer cohorts: GSE11121, GSE1456, GSE2034, GSE2990, GSE4922, and GSE7390. The boxplots are defined as in (A) but without notches (CIs not shown). C: Visualizing the local interpretation capabilities of an attention-based COmic model. Each heat-map shows the attention weights for each of the 50 pathways for three different patients. The model was trained on the GSE11121 cohort. Patient 1 was correctly classified to have a metastasis free survival (DMFS) above five years. Patient 2 was correctly classified to have a DMFS below five years. Patient 3 was wrongly classified to have a DMFS above five years while the DMFS of patient 3 was actually below five years. More examples can be found in the supplement. D: Mean training time of pooling-based and attention-based COmic models for differently sized datasets. The number of samples is 100, 1000, 10000, and 100000, respectively. Training was repeated five times per dataset and the stars represent the mean training time. The blue and green lines show the results for a fixed batch size of 32 samples per batch. The red and yellow lines show the results for an adaptive batch size of 1% of the dataset size (i.e., the batch size was 1 for the dataset with 100 samples and 1000 for the dataset with 100000 samples). Each model was trained for 200 epochs. Table 1: Total number of patients, number of patients in each class, and sources of the datasets used in single- and multi-omics prediction experiments. The first six rows contain information about the single-omics datasets used to train COmic models and compare the results to previously published methods. The last line contains information about the METABRIC multi-omics dataset used in multi-omics prediction experiments. The single-omics classes are DMFS/RFS below 5 years and above 5 years while the multi-omics classes are RFS NO and RFS YES. Dataset | Patients | DMFS/RFS | Source
---|---|---|---
|  |  $<$ 5y / NO |  $\geq$ 5y / YES |
GSE11121 | 181 | 28 | 153 | (Schmidt et al.,, 2008)
GSE1456 | 153 | 34 | 119 | (Pawitan et al.,, 2005)
GSE2034 | 275 | 93 | 182 | (Wang et al.,, 2005)
GSE2990 | 158 | 42 | 116 | (Sotiriou et al.,, 2006)
GSE4922 | 228 | 69 | 159 | (Ivshina et al.,, 2006)
GSE7390 | 191 | 56 | 135 | (Desmedt et al.,, 2007)
METABRIC | 1980 | 803 | 1177 | (Curtis et al.,, 2012)

##  3 Experiments on Cancer Benchmark Data

To assess the performance and interpretation capabilities of our COmic models
we use publicly available cancer benchmarks. The evaluation involves tasks on
single-omics data as well as multi-omics data.

###  3.1 Single-Omics Prediction on Breast Cancer Benchmark Cohorts

We trained COmic models on six different public breast cancer Affymetrix
HGU133A microarray datasets (GSE11121, GSE1456, GSE2034, GSE2990, GSE4922, and
GSE7390) that were previously used to benchmark knowledge-based classification
methods that use interaction network priors. The task was to predict for each
patient if metastasis free survival (DMFS) or relapse free survival (RFS)
exceeded five years. On GSE11121 and GSE4922, the end point was DMFS while RFS
was considered for all other cohorts. Details about the datasets can be found
in Table 1. Both, pooling-based and attention-based COmic models, used 50
different pathways to build kernel layers with 30 anchor points each. We used
the Laplacians derived from a merge between KEGG pathways and Pathway Commons
that were publicly released by Manica and colleagues (Manica et al., (2019),
see original manuscript and corresponding supplementary material for details).
Furthermore, we used gated attention together with an attention dimension of
128 for the attention-based COmic models. Networks were trained for 200 epochs
with the Adam optimizer (Kingma and Ba,, 2014) using the class-balanced loss
function (Cui et al.,, 2019). The batch size was set to 32. All models
presented in this work were trained on a single NVIDIA GeForce GTX 1080 Ti. We
used the area under the receiver operating characteristic (auROC) as our
performance measure to be comparable to previously published results on the
benchmarks. Competitors’ performances shown in Figure 2A are taken from
(Manica et al.,, 2019), for the PIMKL model, and (Cun and Fröhlich,, 2012),
for all other competitors.

As shown in Figure 2A, COmic models either outperformed competitors or
performed similar to previously published methods. Notably, the globally
interpretable pooling-based COmic models were able to achieve a small
improvement in terms of auROC compared to all other models. On the other hand,
the locally interpretable attention-based models achieved a similar
performance as the previously best-performing model, PIMKL. We derived
exemplary visualizations to evaluate the interpretation capabilities of COmic
models. Since the pooling-based variant learns a molecular signature by
weighting each pathway, we assessed the stability of this signature across the
six breast cancer benchmarks. Each box in Figure 2B represents one of the 50
pathways used for the prediction task and are created from the six
corresponding weights learned by the models trained on the different datasets.
The pathway signature remains quite stable over the six different datasets and
high (absolute) weights are associated with known cancer pathways like
androgen response (Pietri et al.,, 2016), hedgehog signaling (Jamieson et
al.,, 2020), notch signaling (Farnie and Clarke,, 2007), and MYC target (Xu et
al.,, 2010). With the introduction of attention-based COmic models, we
introduce models with the capability of providing local interpretations, i.e.,
visualizations that provide insights into the decision process for a specific
sample. We show an exemplary visualization of attention weights for three
different, randomly chosen patients in the GSE11121 dataset in Figure 2C. For
patient 1, the DMFS was correctly predicted to exceed 5 years. Patient 2 was
correctly classified to have a DMFS below 5 years and patient 3 was wrongly
classified to have a DMFS above 5 years while the actual DMFS of patient 3 was
shorter than 5 years. The highest attention weights are associated with known
cancer pathways. For patient 2, the highest amount of attention is given to
hedgehog signaling. Androgen response gets the highest attention for patient 1
and 3. More examples can be found in the supplement.

One key advantage of artificial neural networks over kernel methods is their
applicability on datasets with a vast number of samples. In the following, we
will investigate, if our kernel networks provide the same applicability to
large-scale datasets. Thus, we created simulated omics datasets of four
different sizes: 100 samples, 1000 samples, 10000 samples, and 100000 samples.
We then repeatedly trained pooling-based and attention-based COmic models on
each simulated dataset five times and calculated the mean training time.
Figure 2D shows the results. Since the batch size is usually chosen based on
the number of samples in the training set, we calculated the mean training
time for two different batch sizes. The blue and green lines show the training
times of models trained with a fixed batch size of 32 samples per batch. The
red and yellow lines show the training times of models with an adaptive batch
size of one percent of the total sample count, i.e., each batch included a
single sample, in case of the smallest simulated dataset, and 1000 samples ,
in case of the largest simulated dataset. The results show that COmic models
can be easily trained on datasets with several hundreds of thousands of
samples with the training time being linear depended on the number of samples.
Furthermore, choosing an appropriate batch size can improve the training time
by more than 50% on large-scale datasets.

###  3.2 Multi-Omics Prediction on the METABRIC Benchmark Cohort

Figure 3: Left: Cross-validation performance of COmic and PIMKL models on the
METABRIC cohort. The boxplots show the ten mean auROC validation scores of a
10-times repeated 10-fold cross-validation. Performances of COmic models are
shown in orange. The center line of each box indicates the median. The height
of the boxes represents the inter-quartile range (IQR) with the upper and
lower whiskers set to 1.5 times the IQR. Notches represent the confidence
interval (CI) around the median and were calculated using bootstrapping with
10000 iterations. Right: Mean training time of multi-omics COmic models.
Artificial datasets with 2, 3, and 4 different omics modalities and 1000 data
points per modality were investigated. Batch size and number of epochs were
fixed as described for the METABRIC cohort. Training of models was repeated
five times and mean times are indicated by stars.

Since our proposed kernel layer can be incorporated into any ANN, COmic models
can be flexibly expanded to multi-omics datasets. One possibility is to
directly add the pathway kernels for the additional omics datatypes to the
kernel layer, thereby increasing the number of graph Laplacians in the kernel
layer. Another simple approach is to create sub-networks for each omics type,
i.e., combine the output of pooling- or attention-based single-omics COmic
models with a simple fully connected network. There are numerous other ways to
expand COmic models to multi-omics data and, since our proposed approach is
knowledge-driven, the individual solution has to be selected with the context
of the data in mind. Similar to the authors of PIMKL, we chose the METABRIC
cohort to investigate the practicality of applying COmic models to multi-omics
datasets. The METABRIC breast cancer cohort contains gene expression (mRNA)
and copy number alteration (CNA) data. We performed the same prediction task
as in (Manica et al.,, 2019), i.e., using molecular measures to predict
whether a patient had recurrent cancer.

While we used pooling-based models with the same hyperparameters as described
above for the single-omics prediction experiments, the creation of a multi-
omics COmic model for the METABRIC cohort had to be done with caution. The
issue with CNA data is that this datatype is tremendously sparse. To
compensate for this sparseness, we build a network that used pooling-based
kernel layers to compute an embedding for each datatype that is robust enough
for sparse data and afterwards used a gated attention layer as described in
section 2.3 to make the prediction. The pooling-based kernel layers used the
same hyperparameters as the pooling-based COmic models in the single-omics
experiments. For the gated attention layer we used an attention dimensionality
of 4. Networks were trained for 200 epochs with the Adam optimizer (Kingma and
Ba,, 2014) using the class-balanced loss function (Cui et al.,, 2019). The
batch size was set to 32. We used the auROC as our performance measure to
enable the comparison to previously published PIMKL results on the METABRIC
cohort. Additionally, we investigated the computational efficiency of COmic
with regard to an increasing number of omics modalities.

The results of our experiments on the METABRIC cohort can be found on the left
side of Figure 3. The shown PIMKL performance is taken from (Manica et al.,,
2019). As expected, neither PIMKL nor COmic achieved good performance on the
sparse CNA data. COmic models slightly outperformed PIMKL on the single-omics
prediction task using gene expression data. While PIMKL shows a slightly
decreased performance on the multi-omics prediction task, COmic seems to have
the same performance on the multi-omics data as on the gene expression data
alone with an increase in variance. The runtime analysis (right side of Figure
3) shows a linear dependency on the number of omics modalities.

##  4 Discussion

Kernel methods allow to induce prior knowledge into a prediction task
resulting in increased robustness and the introduction of interpretation
capabilities. In this work we propose COmic, a method to incorporate pathway-
induced kernel functions into convolutional kernel networks. We are able to
create learning models that can be robustly trained on small-scale datasets
and scale very well with the number of samples. Thus, they can be efficiently
applied to large datasets with hundreds of thousands of samples. Furthermore,
our models provide global and local interpretations of predictions made on
molecular measures due to the pathway-induced kernel function.

We used six different breast cancer cohorts to compare the performance of
COmic models to previously proposed methods that use prior knowledge for
prediction tasks with molecular measures as input data. The results presented
in Figure 2A show that our method reaches state-of-the-art performance on
classifying patients based on their DMFS/RFS from gene expression data:
Compared to the considered competitors, COmic performs similar or even better.
However, COmic models have the advantage that the time needed to train a model
scales linearly with the number of samples (see Figure 2D). This enables the
use of COmic models on datasets with hundreds of thousands of samples. We
provide evidence that our method can be readily applied on large datasets by
training models on simulated single-omics data with sizes ranging from 100 to
100,000 samples. Although datasets and patient cohorts used in computational
biology and medicine traditionally have smaller sample sizes, high-throughput
methods and the nowadays more frequently used big data paradigm will result in
increasing sample counts in biological and medical datasets. At this day, TCGA
already contains data from more than 85,000 patients. While methods that can
deal with large datasets are usually deployed as black-box models, our method
provides increased insight into the decision making process.

Using single-omics datasets strongly limits the decision process for diagnosis
of a majority of diseases. Nowadays, it is well known that multi-omics
information has to be incorporated to get a complete image of the
pathomechanism causing a certain disease. Therefore, methods that are limited
to a single datatype face serious constraints if employed as a decision-
support system or to deepen knowledge about a pathomechanism. Our proposed
method does not face the limitation of only using single-omics data as we show
in our experiment with the METABRIC multi-omics cohort. The results show again
that our method improves single-omics prediction as demonstrated by the
performance on the gene expression data. The lower performance that both
methods, PIMKL and COmic, show on the copy number alteration data can be
explained by the sparseness of CNA data. Sparse data poses serious problems
for prediction models (Li et al.,, 2016) and both methods are not specifically
designed for sparse data. However, we can show in Figure 3 that COmic models
are able to achieve slightly better performance than PIMKL models on the
multi-omics prediction task. This indicates that our approach could be
advantageously used on multi-omics data, while the flexibility of the
architecture (as described in section 3.2) enables researchers to tailor COmic
models for specific datasets using domain expertise.

Computing an interpretation of a machine learning method, either with post-hoc
methods or through intrinsically interpretable models, is beneficial if and
only if the interpretation serves a purpose. This purpose cannot be defined in
general as it is highly dependent on the task, the data, and the user that is
presented with the obtained interpretation. For the presented experiments, we
investigated if the inherent interpretation capabilities of our COmic models
are able to learn biological meaningful concepts directly from data. First, we
considered the global interpretation capabilities of COmic. Here, COmic models
assign a weight to each of the used pathways and the weights reflect the role
that each pathway plays in classifying an input sample, i.e., a patient. We
trained COmic models on six different single-omics breast cancer cohorts.
Since we expect the biological processes in the cohorts to share high
similarities, the weight signatures of all models should be similar if the
COmic method is able to learn meaningful pathways from data. As shown in
Figure 2B, this assumption is indeed well fulfilled with all six models having
similar weight signatures. Furthermore, pathways with a high weight assigned
to them are mainly known cancer-related pathways. Therefore, COmic models are
able to learn biological meaningful pathway weights. Although the previously
published PIMKL method also has a global interpretation capability, our method
is able to learn pathways that are important for both, the negative and the
positive class, due to the fact that the learned weights can be positive or
negative. PIMKL only learns positive weights.

While global interpretation is useful to gain insights into a dataset, local
interpretation can be used to get insights into the decision a model makes for
a specific input. Attention-based COmic models can provide this insight
utilizing the attention weights that are computed for each input sample
separately. These weights directly determine the influence that each pathway
has on the decision made by the model. We can visualize these influence using
a heatmap (as shown in Figure 2C) to quickly see which pathways played an
important role in the decision made. We randomly selected three samples from
the GSE11121 dataset to evaluate if the attention weights are biological
meaningful. Similar to the weight signatures of the globally interpretable
COmic models, the attention weights of the locally interpretable COmic models
highlighted known cancer-related pathways. Interestingly, the selected patient
with a DMFS below five years has attention weights that are strongly focused
on a single pathway. This is true for all correctly classified patients with a
DFMS below five years (see supplement). On the other hand, patients with a
DMFS below five years that were wrongly classified to have a DMFS above five
years show attention weight patterns similar to those of patients with a DMFS
above five years (see patient 3 in Figure 2 and additional examples in the
supplement). This could indicate that the wrongly classified patients exhibit
a different mechanism causing a DMFS below five years, compared to the
correctly classified ones, which was not learned by the model. The local
interpretation capabilities of COmic models can help to directly show possible
directions to further investigate the data. Furthermore, the results of our
experiments strongly suggest that both COmic model types are able to generate
biologically meaningful interpretations. We chose heatmaps to visualize
attention weights, since it appeared convenient for the considered prediction
task on the studied dataset. However, different forms of explanations can be
computed with attention weights, e.g., counterfactual explanation (Tran et
al.,, 2021) and adversarial explanation (Kitada and Iyatomi,, 2021). The most
suitable form of explanation is highly dependent on the application, target
user group, and the goal aimed at by the explanation. Therefore, the chosen
visualization should be understood as an example and not a general application
recommendation.

COmic models have a few hyperparameters that can be optimized using
appropriate methods like, e.g., grid search or random search. These
hyperparameters include the number of anchor points, the attention type, the
dimensionality of the attention layer’s parameters $V$ and $U$, and the choice
of pathways used for kernel layers. Furthermore, different initialization
procedures for the anchor points can be explored, e.g., a parameter-free
clustering that combines initializing anchor points with optimizing the number
of anchor points for each pathway-induced kernel layer. We recommend to
explore hyperparameter optimization when applying COmic models. However,
minimizing energy consumption is a pressing concern that should be considered
in every line of research nowadays. Therefore, we limited the computations
performed for this work to the minimum required to support our claims. The
hyperparameters for all models presented in this work were chosen by combining
prior experience about kernel networks with domain expertise. Interestingly,
this computation-free approach to hyperparameter selection already leads to
competitive performance of our method on the considered prediction tasks.

##  5 Conclusion

The introduced convolutional omics kernel networks utilize prior knowledge by
pathway-induced kernel functions to provide robust end-to-end learning on
small- to large-scale molecular measure datasets. Furthermore, utilizing
pathway-induced kernel functions makes our method intrinsically interpretable
with the ability to provide global and local interpretations.

We show the competitive performance of our method on six different single-
omics breast cancer cohorts while providing new interpretation capabilities
that exceed the possibilities of previously proposed methods. Furthermore, we
show that COmic models can be readily adapted to multi-omics datasets.

On a larger scale, we show that incorporating a carefully crafted kernel
function into an artificial neural network allows to robustly train ANNs on
small-scale datasets as they frequently occur in computational biology and
medicine. On the other hand, our method enables scientist to utilize kernel
functions for large datasets as they arise more frequently with the increasing
use of high-throughput methods and big data.

## Acknowledgments and Disclosure of Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC number 2064/1 – Project
number 390727645. This research was supported by the German Federal Ministry
of Education and Research (BMBF) project ’Training Center Machine Learning,
Tübingen’ with grant number 01|S17054. This work was supported by the German
Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ:
01IS18039A.

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##  Appendix A Societal and Environmental Impact

Medical data are notoriously biased against minorities and there are numerous
examples of machine learning models that learn this biases and have a severe
deterioration in performance with regard to minorities (see e.g., Morley et
al., (2020)). We did not include a statement on the societal impact of our
work into the main manuscript due to the fact that we did not have meta
information about ethnicity of patients included in the used benchmarks.
Therefore, it was not feasible to investigate if the prediction performance
and interpretation capabilities of COmic models change for minorities.
However, we encourage researchers that want to apply COmic models on real-
world data to investigate potential bias in their results.

All experiments were conducted using a single NVIDIA GeForce GTX 1080 Ti GPU.
All experiments together required a approximated total of 146 hours of
computing time. This resulted in total emissions of 15.77 kg CO2e, which is
equivalent to burning 7.9 kg of coal. To compensate this emissions, 0.26 tree
seedlings have to sequester carbon for 10 years. These estimations were
calculated using the Machine Learning Impact
calculator111https://mlco2.github.io/impact/ by Lacoste and colleagues
(Lacoste et al.,, 2019).

##  Appendix B Additional Examples of the Local Interpretation Abilities of
Attention-Based COmic Models

We present additional examples of the local interpretation capabilities of our
attention-based COmic models. Figure 4 A shows a set of 20 randomly selected
patients from the GSE11121 cohort that were correctly predicted to have a
metastasis free survival of more than five years. Figure 4 B shows all
patients from the GSE11121 cohort that were correctly predicted to have a
metastasis free survival of less then five years. Figure 4 C shows all
patients from the GSE11121 cohort that were wrongly predicted to have a
metastasis free survival of more than five years when their real metastasis
free survival was less than five years. The order of the pathway attention
weights are the same as in the corresponding figure in the main manuscript.

We can see that the group of correctly classified patients with a metastasis
free survival of more than five years show a similar pattern with a relative
high attention weight on many different pathways. On the other hand, patients
that were correctly classified to have a metastasis free survival of less than
five years show attention weights that are focused on a single pathways or two
pathways at most. Furthermore, the focused pathways are always the same,
hedgehog signalling (pathway 29) and androgen response (pathway 16). The
hedgehog signalling pathway is known to be important in tumor metastasis (Li
et al.,, 2006). Furthermore, the androgen receptor plays an important role in
the breast development cycle and also is known to affect breast cancer
development and metastasis (Pietri et al.,, 2016; Arce-Salinas et al.,, 2014).
The focus of attention weights on these two pathways in patients with shorter
metastasis free survival reflects the known facts about the role that both
pathways play in this process. However, there is an interesting pattern that
can be observed for patients with a metastasis free survival of less than five
years that were wrongly classified to have a metastasis free survival above
five years. While some have the clear focus on either hedgehog signalling or
androgen response and, therefore, strongly suggest that they are simple
wrongly classified patients, most of them shows similar attention weight
patterns to patients with an actual metastasis free survival above five years.
This could hint at the fact that the group of wrongly classified patients with
strong attention weight similarities to patients with longer metastasis free
survival have a different mechanism that causes metastasis than correctly
classified patients with short metastasis free survival. And this different
mechanism is not learned by the model. In any case, the interpretation
indicates that there are different groups of wrongly classified patients and
these groups should be investigated further. These findings support our claim
of the benefits of using intrinsically interpretable models due to the fact
that these different groups in wrongly classified patients can be easily
noticed directly from the model without further polluting the prediction task
with necessary assumptions and additional computations for post-hoc
interpretation models.

Figure 4: Additional examples of the local interpretation capability of
attention-based COmic models. A: Randomly selected samples from the GSE11121
cohort that belong to patients that had a metastasis free survival of more
than five years. All samples represent patients that were correctly predicted
to have a metastasis free survavival of more than five years. B: All samples
from the GSE11121 cohort that belong to patients that had a metastasis free
survival of less than five years and were correctly predicted to have a
metastasis free survival of less than five years. C: All samples from the
GSE11121 cohort that belong to patients that had a metastasis free survival of
less than five years and were wrongly predicted to have a metastasis free
survival of more than five years.