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# Constructing compact signatures for individual fingerprinting of brain
connectomes

Vikram Ravindra
Department of Computer Science
Purdue University
W Lafayette, IN 47906
<EMAIL_ADDRESS>
Petros Drineas
Department of Computer Science
Purdue University
W Lafayette, IN 47906
<EMAIL_ADDRESS>
Ananth Grama
Department of Computer Science
Purdue University
W Lafayette, IN 47906
<EMAIL_ADDRESS>

###### Abstract

Recent neuroimaging studies have shown that functional connectomes are unique
to individuals, i.e., two distinct fMRIs taken over different sessions of the
same subject are more similar in terms of their connectomes than those from
two different subjects. In this study, we present significant new results that
identify, for the first time, specific parts of resting state and task-
specific connectomes that code the unique signatures. We show that a very
small part of the connectome codes the signatures. A network of these features
is shown to achieve excellent training and test accuracy in matching imaging
datasets. We show that these features are statistically significant, robust to
perturbations, invariant across populations, and are localized to a small
number of structural regions of the brain. Furthermore, we show that for task-
specific connectomes, the regions identified by our method are consistent with
their known functional characterization. We present a new matrix sampling
technique to derive computationally efficient and accurate methods for
identifying the discriminating sub-connectome and support all of our claims
using state-of-the-art statistical tests and computational techniques.

_K_ eywords Fingerprinting Brain Connectome $\cdot$ Individual Signatures in
Connectomes $\cdot$ Matrix Sampling $\cdot$ Sparse Decompositions

A number of functional MRI studies focus on differences between sets of
cohorts (e.g., healthy vs. diseased). These studies characterize an invariant
signal within each cohort and identify significant changes in these signals
across cohorts. Invariant signals within cohorts are designed to suppress
individual-level heterogeneity of the functional connectomes, and can be
computed using methods that range from simple averaging to sophisticated
matching techniques, to identify statistically significantly conserved
components within the cohort.

Clinically observed functional and/or structural variability between
individual brains typically manifests in structural and functional
connectomes. It has been shown that functional connectomes of an individual
express higher degree of similarity than connectomes of different individuals.
It has also been shown that restricting such analyses to regions constituting
well-known networks such as fronto-parietal networks, amplifies the
individual-specific signature. This suggests that the signal corresponding to
identifiability is spatially selective in its expression.

In this paper, we present a novel technique for identifying interpretable
individual-specific signatures in resting and functional brain networks. This
is in contrast to prior efforts, which computed individual signatures as low-
rank representations of the matrix corresponding to the connectome. While
prior results demonstrated the existence of the signatures, the computed
signatures are not physiologically relevant or even interpretable, since they
do not map back to regions of the brain. In contrast, we identify a highly
accurate and compact set of features from the connectome that map directly to
regions of the brain. By not restricting our search for features to specific
regions of the brain, we are able to identify a comprehensive, yet highly
specific set of features. We show that these sets yield stronger
discriminatory signals, resulting in higher accuracy of identification. We
also demonstrate the use of our method to isolate signatures that encode
information specific to the task being performed by an individual. We
demonstrate the statistical significance and robustness of our signatures, and
show that the signatures correspond to regions of the brain that are
consistent across subjects.

## Results

The goal of this effort is to characterize the uniqueness of the functional
connectome of an individual in terms of its accuracy, underlying features, and
associated task-specificity. We define the innate uniqueness property as
identifiability. We define a _signature_ as a pattern that encodes
identifiability. In this effort, we isolate small regions on the human cortex
that strongly express these signatures. In this context, we use the terms
_signatures_ and _fingerprints_ interchangeably. We present detailed
experiments to characterize the accuracy of these signatures in terms of their
ability to identify individuals, as well as the task they are performing.

We used the functional MRI images of the unrelated subset of subjects,
acquired as part of the WU-Minn Project by the Human Connectome Project (HCP)
Consortium. The images were acquired over two sessions on separate days. The
first day included a resting session (REST1), followed by working memory (WM),
motor task (MOTOR) and language (LANG). The second day included another
resting session (REST2), social cognition (SOCIAL), relational processing
(RELATION), and emotion processing (EMOTION). Each of the rest and task
sessions included a left-right (LR) encoding and a right-left (RL) encoding.
We follow the the minimum pre-processing pipeline designed by HCP. We used the
Atlas designed by Glasser et al. (2016) to parcellate the cortex into 360
anatomically and neuro-physiologically relevant regions and constructed
region-wise similarity matrices. We then vectorize these correlation matrices,
after discarding symmetric components (see Online Methods for details).

In our experiments, we use two groups ($G_{1}$ and $G_{2}$) of connectomes.
Each group has exactly one data point per subject. Our task is to successfully
match the connectivity matrices belonging to a subject across the two groups.
In the first set of experiments, where we use resting state connectivity
matrices, the two groups have correlation matrices from REST1 and REST2,
respectively. In the experiment that investigates identifiability of
individuals while performing tasks, the first group consists of LR encoding
and the second group consists of RL encoding. In the final experiment that
focuses on task identifiability, the first group consists of REST1 and the LR
encodings of seven tasks, whereas the second group consists of REST2 and the
RL encodings of the same seven tasks. Each group matrix is constructed by
stacking the vectorized correlation matrices. Thus each data point in this
matrix corresponds to a subject and each feature is a measure of coherence in
activation between two regions. In all the experiments that follow, we use the
principal features subspace method, which uses _leverage scores_ to select
features (see Online Methods for more details).

### A small set of features encode individual signatures

Our first set of experiments is designed to demonstrate our key result, that a
small number of features encode resting-state signatures that are largely
unique to individuals. This is in contrast to prior techniques that do not
identify either structural or functional features associated with individual
signatures, just that such signatures exist in the aggregate datasets.

To demonstrate this result, we use the group matrix $G_{1}$ to identify
features corresponding to the top 100 leverage scores (from among over 64K
scores). We show later that accuracy of identification saturates at less than
100 features. Then, pairwise correlation scores are calculated between data
points of the two groups in this restricted feature-space. The purpose of this
experiment is to demonstrate that two Functional Connectomes of the same
subject are more similar than two Functional Connectomes belonging to
different subjects using only  this small feature set.

We define accuracy of identification as the percentage of instances where
individuals are correctly matched across the two groups, $G_{1}$ and $G_{2}$.
Note that a match is identified as the highest correlation of an individual in
$G_{2}$ over all individuals in $G_{1}$. To ensure robustness, we repeat this
experiment for 1000 different (random) subsets of subjects. Table 1 presents
accuracy for subsets of different sizes. The accuracy obtained in the reduced
space is comparable to results obtained by using the full correlation
matrices, or by using a low rank matrix approximation.

To characterize the statistical significance of the selected features, we
repeat the experiment with the same number of features chosen uniformly at
random $10^{6}$ times. The average accuracy was much lower (approximately 55%,
see Table 2). None of these instances yielded accuracy values higher that that
for our leverage-score based feature selection. This suggests high statistical
significance of our identified features (empirical $p$-value$<10^{-6}$).

Training/test split | Training Set Accuracy(%, mean $\pm$ std) | Test Set Accuracy(%, mean $\pm$ std)
---|---|---
80/20 | $96.23\pm 2.24$ | $93.11\pm 3.61$
50/50 | $96.30\pm 2.59$ | $92.94\pm 3.82$
30/70 | $96.81\pm 3.07$ | $90.23\pm 4.30$
20/80 | $97.01\pm 3.22$ | $87.60\pm 5.27$
10/90 | $97.72\pm 2.65$ | $81.86\pm 7.15$

Table 1: Accuracy of principal features subspace method for different splits
of training and test samples. The results show consistently high training
accuracy across a range of training set sizes. As expected, training accuracy
increases with smaller training sets; however, test accuracy goes down as
training set size is reduced, due to overfitting.

### The feature set is robust across individuals

Our first set of experiments identified a set of features from individuals in
Group 1, and used these features to draw correspondence (identification)
between the two groups. Note that in these experiments, the identification of
signatures and identification of subjects is on the same set of individuals.
In our second set of experiments, we show that this feature-set is invariant
across individuals. Specifically, we identify features using one set of
subjects (features with highest leverage scores in the training set), and use
these features to draw correspondences between a distinct set of subjects (the
test set) across the two groups. The result of this experiment is presented in
Table 2. The high test accuracy indicates that features relevant for the
training set are also discriminative for the test set (p-value: $<10^{-6}$).
This suggests an anatomical and/or physiological basis for the features
selected.

Feature Selection Method | Training Set Accuracy(%, mean $\pm$ std) | Test Set Accuracy(%, mean $\pm$ std)
---|---|---
Leverage Score | $96.23\pm 2.24$ | $93.11\pm 3.61$
Random | $54.62\pm 7.76$ | $54.52\pm 7.52$

Table 2: Performance of leverage-score based feature selection technique,
compared to selecting features uniformly at random . The significantly higher
test and training set accuracy demonstrates the performance of our principal
subspace feature selection method.

To further highlight the implication of this result, it is possible to apply a
computational procedure to extract features from training and test sets. The
computational procedure may be a complex function defined over all features.
This is in fact the case with prior methods. In contrast, what our experiment
shows is that no such computational procedure is required – rather, a robust
selection of sample-invariant features is highly discriminative and accurate.

### The size of the discriminating feature set is compact

Our first set of experiments demonstrated that a small set of 100 features is
robust and statistically significant in its discriminative power. We now seek
to determine the smallest such set that can code individual signatures with
high accuracy. We design an experiment in which the number of features is
successively increased (in order of decreasing leverage scores) and the test
accuracy is measured at each step. The result of this experiment is presented
in Figure 1. We note that the test accuracy essentially plateaus using 61
features at $93\%$. The maximum accuracy ($95.5\%$) is achieved using 95
features.

Figure 1: Variation of test accuracy as a function of feature set size. The
rapid convergence in test accuracy demonstrates that a small set of features
codes the discriminating signature.

It is important to note that this experiment relies on a leverage score
ordering of the features. It is indeed possible that an alternate ordering may
yield a smaller feature set. However, our results on the suitability of
leverage scores for feature selection, combined with the rapid plateauing of
accuracy, strongly suggest that a feature set comprised of 61 to 95 features
(from among over 64K features) is enough to code individual signatures with
high accuracy.

### A small number of structural regions encode the signature

Our previous experiments used entries in the correlation matrix (edges in
correlation network among regions of the brain) as features. We now
investigate whether a small number of structural features (regions in the
brain) are capable of coding individual signatures with high discriminating
power.

In this experiment, we compute the top 100 features with the highest leverage
scores for a randomly selected subset of 80 subjects. We repeat this procedure
1000 times. We then extract the high confidence features using a hyper-
geometric p-value cutoff of $10^{-20}$. The results are shown in Fig 2. We
observe that the prefrontal cortex and the parietal regions strongly encode
the signature. This is in agreement with Finn et al. (2017), who showed that
the fronto-parietal network has the most discriminating power. However, in
contrast to previous methods, we are able to make this observation solely by
analyzing the resting state connectivity matrices of one session in the
training set.

In order to find the regions based on our parcellation scheme, we select the
regions that are over-represented in the top features across all 1000 tests
(hypergeometric p-value $<10^{-20}$). This process resulted in 12 high-
confidence regions, which, as we show, encode the signature. To do this, we
restricted ourselves to work with the $24\times 24$ connectivity matrix,
corresponding to both hemispheres of the previously identified regions. We
then constructed the vectorized representation of this smaller matrix for the
test subjects and found the accuracy to be $94.05$ ($\pm 1.22$). In
comparison, identifiability accuracy for 12 randomly chosen regions was
$41.47$ ($\pm 12.25$, for p-value $<10^{-6}$). The regions in this set are
listed in Table 3. Overall, the signature expressing regions occupy about 4.5%
of the cortex.

Structurally, five of the 12 regions (namely Area Anterior 9-46v, Area 9-46v,
Inferior 6-8 transitional area, Area Anterior 10p, and Area 9 anterior) belong
to the Dorsolateral Prefrontal Cortex. Furthermore, Intraparietal dorsal,
Intraparietal 0, and Lateral Intra Parietal Ventral belong to the Parietal
Cortex (see Supplementary Material in Glasser et al. (2016)). This shows that
regions coding the signature are physiologically localized.

Figure 2: High confidence edges that encode resting-state signature in the human connectome. The connectivity map shows that the signature is strongly expressed in the prefrontal cortex and the parietal cortex. For illustrative purposes, we show edges when at least one terminal node has a degree of 30. Area Number | Area Notation | Region
---|---|---
141 | Temporo-Parieto-Occipital Junction 3 | Parietal
11 | Premotor Eye Field | Premotor
95 | Intraparietal dorsal | Parietal
85 | Area Anterior 9-46v | Prefrontal
86 | Area 9-46v | Prefrontal
97 | Inferior 6-8 transitional area | Prefrontal
146 | Intra Parietal 0 | Parietal
48 | Lateral Intra Parietal Ventral | Parietal
89 | Area Anterior 10p | Prefrontal
87 | Area 9 anterior | Prefrontal
137 | PHT | Lateral Temporal
79 | IFJa | Inferior Frontal

Table 3: Top brain regions associated with discriminating features for
resting-state connectomes. We observe that most of the signature expressing
regions are in the prefrontal and the parietal cortex.

### Our method significantly outperforms state of the art methods in terms of
test accuracy

In this set of experiments, we compare the accuracy of our method with other
state-of-the-art techniques. Finn et al. (2017) compute all pairwise Pearson
correlation coefficients, where each data point is represented by all elements
of the upper diagonal of the time-series correlation matrix. We randomly
choose 50 out of 100 subjects and compute pairwise similarity; the accuracy of
Finn et al.’s method is $88.65$ ($\pm 1.76$) averaged over 1000 trials. Amico
and Goñi (2018) first denoise the data by retaining a subset of the principal
components. As before, we randomly choose 50 subjects and compute the training
set accuracy (see Table 4). We observe that removing the top few principal
components helps to improve accuracy, since these components correspond to
signals that are common to all subjects, as noted in Amico and Goñi (2018).
The test set accuracy is computed by removing the top principal components of
the training dataset from the test dataset. This reveals that the common
signal of the training dataset is an artifact of those particular subjects,
and not a pattern that provides significant insights about the functional
behaviour of the brain. In all cases, our method yields comparable training
set accuracy, significantly higher test set accuracy, while relying on a small
set of structurally and functionally interpretable features.

Principal Components | Training Set Accuracy(%, mean $\pm$ std) | Test Set Accuracy(%, mean $\pm$ std)
---|---|---
All (Same as Finn et al. (2017)) | $88.65\pm 1.76$ | $88.98\pm 1.72$
2:end | $94.30\pm 1.35$ | $71.76\pm 8.76$
11:end | $96.74\pm 1.00$ | $69.61\pm 8.94$
21:end | $95.03\pm 1.90$ | $69.44\pm 8.99$
31:end | $71.97\pm 6.08$ | $68.95\pm 9.07$
41:end | $2.77\pm 1.74$ | $65.70\pm 9.59$
Principal Features Subspace | $96.23\pm 2.24$ | $93.11\pm 3.61$

Table 4: Comparison of training and test accuracy for various methods. Numbers
in the first row use the entire matrix; the next five rows use a subset of the
principal components; the last row corresponds to our method. Our method
achieves almost optimal training set accuracy and significantly better test
set accuracy compared to competing methods.

### Individuals performing tasks can be distinguished

In this experiment, we demonstrate that our method for finding discriminating
edges extends to the case of task-based functional networks. For each task, we
construct the two group matrices and partition subjects into training and test
sets as described before. We compute the leverage scores separately for each
task using the respective training subsets, restrict the feature space to top
leverage scores, and report predictions on the test set in the same manner as
in the resting state experiments. The results (averaged over 1000 trials) are
shown in Figure 3. The performance of our approach is comparable to using the
full correlation matrix in all but two tasks, namely MOTOR and WM. This
indicates that our method recognizes features that important for most tasks.
As before, we localize the features to physical regions on the brain. In order
to do this, we only use features whose leverage scores are consistently
sufficiently high to remain in the reduced feature space across all 1000
trials (hyper-geometric p-value $<10^{-10}$).

Figure 3: Prediction accuracy of individuals performing tasks.

We discuss the results for each task. In the LANGUAGE task, the statistically
significant regions include the Inferior Frontal Cortex and surrounding
regions (Orbital and Frontal Cortex and Dorsolateral Prefrontal Cortex). They
also include regions in the Anterior Cingulate and Medial Prefrontal Cortex,
as well as areas that surround the Superior Temporal Cortex (the Medial and
Lateral Temporal Cortex). These results are consistent with the regions of
interest for LANGUAGE, identified by Barch et al. (2013). Furthermore, the
task involves auditory inputs, and requires communication via pressing a
button. This explains the Premotor Cortex and the Auditory Association Cortex
being recognized as regions of interest by our approach. The high confidence
edges for the task are shown in Figure 4.

In EMOTION, the regions implicated in the signature are found in the
Prefrontal Cortex and neighboring regions (Inferior Frontal Cortex, Orbital
and Polar Frontal Cortex), as well as areas surrounding the Insula (Lateral
and Temporal Parietal Cortex). These results are consistent with the results
of Barch et al. (2013), except for the fact that we do not identify the Medial
Temporal Cortex as a region of interest. The connectivity graph for EMOTION is
shown in Figure 5. In GAMBLING, we find the expected regions of interest in
the Prefrontal cortex, as well as the Orbito-frontal cortex. The other area of
interest is the Sub-cortical Striatum, which is not included in our
parcellation scheme. In this experiment, the subjects are shown visual stimuli
and are required to communicate their decision by pressing a button (Figure
6). This explains the fact that the regions in Visual and Motor Cortex are
being recognized as potential regions of interest. The results for SOCIAL
tasks are shown in Figure 9: we identify all relevant regions of interest
(Temporal Parietal Junction and Medial Prefrontal Cortex, along with the
visual cortex and the motor cortex). In the RELATIONAL task, we identify
regions in the prefrontal cortex, but also in the parietal cortex, as shown in
Figure 8. In the MOTOR task (Figure 7), we do not identify the regions of
interest (Somatosensory and Motor cortex). This explains the poor prediction
accuracy for this task. Similarly, we do not identify the Entorhinal Cortex or
the Hippocampus in WM, which again explains the poorer prediction accuracy for
this task.

While the regions obtained by our method broadly agree with the regions that
are expected to be activated during respective tasks, we note that a general
pipeline of tasks for fMRI would factor in additional information, such as
presence/absence of various stimuli, duration of each block, and performance
metrics associated with the tasks performed. The connectivity profiles of each
of the tasks are sufficiently distinct, suggesting that we can identify tasks,
which is indeed our main conclusion.

Figure 4: High confidence edges that encode signature while performing the
language task of HCP. The regions with high edge density are in good agreement
with the regions of interest (ROIs) for language Barch et al. (2013). For
illustrative purposes, we show edges when at least one terminal node has a
degree of 30. Figure 5: High confidence edges that encode signature while
performing the emotion processing task of HCP. The regions with high edge
density are in good agreement with the ROIs for emotion processing Barch et
al. (2013). For illustrative purposes, we show edges when at least one
terminal node has a degree of 30. Figure 6: High confidence edges that encode
signature while performing the gambling task of HCP. The regions with high
edge density are in good agreement with the ROIs for gambling Barch et al.
(2013). For illustrative purposes, we show edges when at least one terminal
node has a degree of 30. Figure 7: High confidence edges that encode signature
while performing the language task of HCP. Our method does not find the
regions implicated in this case. For illustrative purposes, we show edges when
at least one terminal node has a degree of 30. Figure 8: High confidence edges
that encode signature while performing the relational processing task of HCP.
The regions with high edge density are in good agreement with the ROIs for
relational processing. For illustrative purposes, we show edges when at least
one terminal node has a degree of 30. Figure 9: High confidence edges that
encode signature while performing the social processing task of HCP. The
regions with high edge density are in good agreement with the ROIs for social
processing. For illustrative purposes, we show edges when at least one
terminal node has a degree of 30. Figure 10: High confidence edges that encode
signature while performing the working memory task of HCP. Our method does not
find the regions implicated in this case. For illustrative purposes, we show
edges when at least one terminal node has a degree of 30.

### Task-identifying signatures are strongly localized to the Occipital Lobe

In this final experiment, we aim to identify the tasks performed by an
individual. For each subject, we create two group matrices. The first group
matrix contains the LR encoding of all tasks and the second group matrix
contains the corresponding RL encoding. Here, a match is successful if the
same tasks across the two groups are more similar to each other than with any
other task. As before, we split the subjects randomly into train and test sets
of size 80 and 20 respectively. For each subject in the train set, we find the
features corresponding to the top leverage scores. This gives us 80 sets of
features. We find the set of statistically significant features for the entire
training set using a hyper-geometric p-value test (threshold $10^{-50}$) to
identify recurring features across all subjects. These statistically
significant features are used to restrict the feature space in the test set.
The accuracy, in terms of correctly identified tasks in the test set, averaged
across 1000 trials was $91.93$ ($\pm 1.34\%$). In contrast, the same
experiment using the entire connectome yielded accuracy of $84.95$ ($\pm
2.2\%$).

We also observe through this experiment that the regions expressing these
task-identifying signatures are strongly localized to the cortex. In Figure
11, we show the high confidence edges. For clarity of visualization, we only
show edges where at least one of the end nodes has a sufficiently high degree
(> 30). This figure illustrates our hypothesis clearly. A significant fraction
of the high confidence edges are connected to the Occipital region, with a
strong connectivity to the Parietal region. The Occipital region contains the
different layers of the visual cortex. These layers processes various visual
stimuli that a subject is exposed to, during the course of the different
tasks. Similarly, the Parietal lobe is responsible for integrating sensory
information for the visual cortex. Our results suggest that the signature
captures the difference in processing different stimuli, which is integral to
the description of any task. Finally, we note that the regions are dependent
on the task set: for instance, the Occipital lobe is not significantly
implicated in a set of tasks with no visual stimulus.

Figure 11: Task identifiability for a single subject. These results show that
the edges that distinguish tasks in the Human Connectome Project are strongly
localized to the occipital lobe.

## Discussion

In this paper, we use a novel matrix sampling method to answer critical
questions related to individual and task-specific signatures in the human
brain connectome. We show that a very small number of features in a functional
connectome codes for the identity of an individual. Furthermore, we show that
these features (edges in the connectome) are robust, statistically
significant, and invariant across individuals. The regions corresponding to
these features are consistent with existing literature – both for resting-
state and task, strongly supporting the physiological basis for our method. We
also show that the features that code for specific tasks are localized
strongly to the Occipital Lobe.

The core contribution of our method lies in the fact that it can predict
relevant features, without requiring multiple scans. In doing so, it
generalizes the application of brain signatures beyond the HCP dataset.
Specifically, it can be used in any study, where the goal is to find
individual differences in a cohort. Furthermore, since the method is built on
sound theoretical foundation, its computational and statistical underpinnings
are well-characterized.

The success of leverage scores in picking relevant features for brain
signatures suggests that it can also be used in conventional fMRI studies,
where the aim is to find differences between cases and controls. The key
distinction between the two setups is that the former requires unsupervised
feature selection, whereas the latter requires supervised feature selection.
We note that this necessitates new sampling techniques, along with associated
theoretical guarantees.

## Methods and Materials

### Dataset

The images used in this study are collected as part of the Healthy Young Adult
study in the WU-Minn Project Essen et al. (2013) by the Human Connectome
Project Consortium. The data includes 3T structural and functional MRI for
1113 adults, 7T resting and task magnetoencephalography (MEG) from 184
subjects, and 3T and 7T diffusion data. Full details of the data can be found
in vanEssen et al. Essen et al. (2012).

The resting state functional MRI is acquired in two sessions, on two separate
days. The resting state data in each session lasted approximately 30 minutes
(15 minutes for L-R and R-L phase encoding). The spatial resolution was
$2\times 2\times 2\ mm^{3}$ and temporal resolution (TR) of 720 ms. In this
study, we use the L-R encoding from the 100 “unrelated” subjects, which is
part of the S900 release. For a detailed description of the acquisition
protocol see Smith et al. Smith et al. (2013).

In each session, the resting state session was followed by tasks. On the first
day, the tasks included working memory, gambling and motor; the second day
included language, social cognition, relational processing and emotional
processing. The task functional MRI data also have the L-R encoding and the
R-L encoding. However, the duration of each session varied considerably,
ranging from 176 frames in emotion processing to 401 frames in working memory,
with TR of 5520 ms. For a detailed description of the protocol see Barch et
al. Barch et al. (2013).

#### Preprocessing

The preprocessing steps follow the minimum pre-processing pipeline prescribed
by Glasser et al. Glasser et al. (2013) and Smith et al. Smith et al. (2013).
The HCP pre-processing steps include spatial pre-processing and temporal pre-
processing. In resting state functional MRIs, we also perform global signal
regression.

The goal of spatial pre-processing is to remove artifacts, while leaving
useful information intact. It includes correction for spatial distortions due
to gradient non-linearity (of magnetic field), followed by correction for head
motion of subject during data acquisition, and registration to the structural
MRI image. Since brains of different subjects are of different sizes, the
image is transformed to 2mm MNI co-ordinate system. This is followed by
normalizing the 4D image to global mean and masking out non-brain voxels.
Finally, the cortical surface vertices and the sub-cortical volumetric
vertices are combined into the standard grayordinate space in a CIFTI dense
time-series format. The surface cortical voxels are aligned using non-linear
surface registration and non-linear volume registration for volume structures,
thereby achieving spatial correspondence between images of different subjects.
The rationale behind the grayordinate space is explained in Glasser et al.
Glasser et al. (2013). In this study, we only use time series from cortical
vertices.

Temporal pre-processing is particularly important for resting state analyses,
as they rely on correlation between voxel time series. These can get corrupted
by temporal artifacts that span across several voxels (in contrast, task
analyses have pre-determined timed events, which make them robust to temporal
artifactual influences). The HCP temporal pre-processing pipeline includes a
minimal high pass filter (cutoff of 2000s) and a slow roll off below that
point, which achieves a linear de-trending of data. For task functional MRI,
the cutoff was 200s.

In the case of resting state functional MRI, we follow this by global signal
regression, a procedure that regresses out the mean time series out of every
voxel. Since resting state analyses have shown that fluctuations at low
frequencies are due to haemodynamic responses to neural activation, we applied
a bandpass filter (0.008Hz to 0.1Hz).

#### Brain Atlas

The description of a functional connectome as a network requires the
identification of a set of nodes and a set of edges between these nodes. A
brain atlas defines parcellation schemes that help achieve the first
requirement. Ideally, parcels are non-overlapping, locally functionally
distinct areas. The variation within a parcel should be dominated by the
variation between parcels. While parcellation is an abstraction, it allows for
analyses of a structurally and functionally relevant region $\times$ region
network, rather than a raw voxel $\times$ voxel network.

In our work, the cortical structures are parcellated in accordance with the
atlas of Glasser et al. Glasser et al. (2016). This atlas consists of 360
regions (180 in each hemisphere), bound by sharp edges on the basis of
anatomy, function, and/or topology. Each of the 180 regions are also
classified into 22 larger regions, which allows for analysis at a coarser
level, if needed. We find that other parcellation schemes either have too few
regions, or are annotated on the basis of fewer samples. In contrast, the
boundaries in Glasser et al. Glasser et al. (2016) are drawn using resting
state and task fMRI images from two groups of 210 subjects, all of which is
part of the previously described Human Connectome Project.

### Methodology

#### Experimental Setup

The time series data (i.e., a 2D matrix of gray-ordinate voxel $\times$ time-
points) is z-score normalized. Following this, voxels belonging to the same
parcel are averaged to yield a region $\times$ time-points matrix. The Pearson
Correlation of all pairs of time series is then computed, resulting in a
region $\times$ region correlation matrix. This process is repeated for both
sessions of all subjects. Then, the upper triangular matrix for the first
session for all subjects is vectorized and stacked next to each other into a
subjects $\times$ feature matrix, where feature is an entry in the correlation
matrix. A similar matrix is constructed for the second session as well.

We first discuss the concept of row and column sampling as a general means of
feature selection and, in particular, leverage score sampling. We then discuss
the use of leverage scores to select features for brain fingerprinting.

#### Row/column sampling

Given matrix $A$, an individual entry $a_{i,j}$ corresponds to the weight of
the $i$th edge of subject $j$. The problem of identifying the most
discriminating subset of features then translates to selecting non-zeros from
the correlation matrix (row-column pairs) that are most discriminative in
terms of individual signatures. In contrast to conventional dimensionality
reduction techniques such as Singular Value Decomposition (SVD), the ability
to choose rows or columns from a data matrix directly translates to feature
selection, especially when features have physical meaning. Retaining such
features in the matrix sketch can make the underlying physical phenomenon
explainable, while also de-noising the data by eliminating non-discriminating
features and noise. Furthermore, in contrast to methods that use within-
subject and between-subject metrics to reduce dimensions as in Byrge et al.
Byrge and Kennedy (2019), we do not require data from both sessions to
identify important edges.

Given a matrix $A\in\mathbb{R}^{m\times n}$, one can use the following
randomized meta algorithm to create a sketch matrix $\tilde{A}$ that retains
$s$ ($<m$) rows.

function row_sample(A,s)

Let $\tilde{A}$ be an empty matrix

for $t=1$ to $s$ do

Randomly sample a row according to the distribution $P$

Let $A_{i_{t},\star}$ be the sampled row, with corresponding probability
$p_{i}$

Set $\tilde{A}_{t,\star}=\frac{1}{\sqrt{sp_{i}}}A_{i_{t},\star}$

end for

return $\tilde{A}$

end function

The algorithm samples $s$ rows of $A$ in independent, identically distributed
trials according to $P$s. The re-scaling of $\tilde{A}$ ensures that
$\tilde{A}^{T}\tilde{A}$ is unbiased, i.e.,
$\mathbb{E}[(\tilde{A}^{T}A)_{i,j}]=(A^{T}A)_{i,j},\forall i\in\\{1\dots
m\\},j\in\\{1\dots n\\}$ Drineas et al. (2006).

The only unspecified detail is the choice of distribution $P$. A simple choice
is to sample rows uniformly, however, this yields poor results. An intuitively
better choice for the distribution relies on the matrix $A$ itself – assigning
higher weights to more important elements. A simple non-uniform distribution
is based on $l_{2}$ sampling, which can be defined as:

$p_{i}=\frac{||A_{i,\star}||_{2}^{2}}{\sum_{i}||A_{i,\star}||_{2}^{2}}=\frac{||A_{i,\star}||_{2}^{2}}{||A_{i,\star}||_{F}}.$
(1)

Using norm-squared sampling, Drineas et al. Drineas et al. (2006) prove that:

$\mathbb{E}[||A^{T}A-\tilde{A}^{T}\tilde{A}||_{F}]\leq\frac{1}{\sqrt{s}}||A||_{F}^{2}.$
(2)

The bounds in Equation 2 imply that $\tilde{A}$ can be used as a proxy for
$A$. However, this approximation introduces an additive error, which depends
on $||A||_{F}$. To achieve better bounds, one can make use of the knowledge of
column space of $A$. The associated sampling technique is called leverage
score sampling Drineas and Mahoney (2016).

##### Leverage Score Sampling.

Let $A\in\mathbb{R}^{m\times n}$, with $m\gg n$. Let $U\in\mathbb{R}^{m\times
n}$ be any orthonormal matrix spanning the column space of $A$. Then,
$U^{T}U=I$ and $UU^{T}=P_{A}$, which is an $m$-dimensional projection matrix
onto the span of $A$. Then, the probabilities $p_{i}$ are defined as:

$p_{i}=\frac{||U_{i,\star}||_{2}^{2}}{\sum_{i}||U_{i,\star}||_{2}^{2}}=\frac{1}{n}(P_{A})_{i,i}\qquad\forall
i\in\\{1\dots m\\}.$ (3)

The values of $p_{i}$s in Equation 3 are known as statistical leverage scores.

In our matrix of vectorized edge values, each row contains values expressing
functional connectivity markers between the same pair of physical regions on
the brain cortex. Hence, leverage scores are indicative of relative importance
of edges in discriminating samples. While the randomized approach discussed
here helps in understanding the process, we find that a deterministic approach
performs well in practice. We call this sketching process Principal Features
Subspace Method.

#### Principal Features Subspace Method

As before, let $A$ be the data matrix of connectomes, and $U$ be the
orthonormal matrix that spans the column space of $A$. Additionally, let $t$
be the number of features that need to be retained. An example of such a
matrix $U$ is constructed using left singular vectors from a Singular Value
Decomposition (SVD) of $A$. We can then compute the leverage($l$) scores of
$A$ as:

$l_{i}=||U_{i,\star}||_{2}^{2},\qquad\forall i\in\\{1\dots m\\}.$ (4)

We sort the leverage scores and retain the features corresponding to the top
$t$ leverage scores. We call this subspace the principal features subspace. In
contrast to prior randomized approaches, we select features in a deterministic
manner; Cohen et al. Cohen et al. (2015) provide theoretical bounds for this
selection process.

Starting from the matrix of vectorized edge values $A$, we compute the left
singular vectors using SVD. The ordering of edges according to their leverage
scores, if robust across different groups, is indicative of a set of features
that can accurately fingerprint an individual’s functional connectome. In this
case, for a given parcellation scheme, and for a given measure of region-
region coherence, we need to apply SVD just once to determine the relevant
edges. We present results from our scheme applied to 100 MRI samples to
demonstrate powerful new results on the compactness and robustness of the sub-
connectome coding individual fingerprints.

### Related Methods

There is significant recent work in characterizing population-level
differences from neuroimaging datasets. Given the focus of our work on
individual-level differences, we restrict ourselves to significant results in
this space, and use them to motivate our new results.

Finn et al. Finn et al. (2017) divide the identification task into two steps:
whole brain and network specific. In the first step, they perform
identification on the basis of the region-wise correlation matrix of the whole
brain, and report a success rate of approximately 93%. Following this, they
divide the brain into eight functional networks: medial frontal, fronto-
parietal, default mode, subcortical-cerebellum, motor, visual I, visual II,
and visual association, on the basis of an atlas constructed from the Yale
Dataset (268 nodes covering the whole brain). Restricting analysis to each of
these regions, they perform the identification task. From this analysis, they
find that the networks medial frontal and fronto parietal carry most
discriminating features, yielding accuracies as high as 98%. Similar results
have also been reported by Miranda-Dominguez et al.Miranda-Dominguez et al.
(2014) and Mueller et al. Mueller et al. (2013). However, these results
require significant prior knowledge on composite of parcels known to have
functional/structural coherence.

Our effort takes a similar two-level approach, viewing identification at the
whole brain and network levels. Instead of using prior knowledge, we show that
the Principal Features Subspace yields a comparable identification training
accuracy of approximately 96%. Using the features obtained in the training set
on the test set, we find that our test accuracy is around 93%, which is
significantly higher than that from Finn et al. Finn et al. (2017). In
contrast to Finn et al. we restrict the analysis in the second step to the
regions that are computationally identified from our first step. The regions
obtained from our first step are in fact consistent with those used by Finn et
al. in the second step, in that we find regions in the parietal and frontal
cortex being over-represented in our top nodes (9 out of 12 belong to fronto-
parietal regions). However, the network identified by our method is
significantly smaller than the region-specific network of Finn et al. It is
shown to be robust, statistically significant, and invariant on test subjects.
Restricting ourselves to 24 parcels (which may belong to different networks),
we report an accuracy of approximately 96%.

There have been recent attempts aimed at identifying discriminating edges,
using the so-called differential power introduced by Finn et al. and improved
greatly by Byrge et al. Byrge and Kennedy (2019). In comparison to Byrge et
al., our approach has two distinct advantages. First, our principal features
subspace method uses the concept of leverage scores which has strong
theoretical guarantees Drineas et al. (2006); Drineas and Mahoney (2016);
Cohen et al. (2015); Papailiopoulos et al. (2014); these guarantees are not
provided by differential power. Second, our approach does not require any
labels in order to find discriminating edges. This significantly increases the
applicability of our method, since most studies do not have multiple scans of
a subject. As acquisition protocols may vary from study to study, it isn’t
possible to use the features from HCP in other studies, where it may be
necessary to find subject-level unique patterns. Our setup simply requires two
sets of images, with the guarantee that each subject has one scan in each set.
We use only one set to find discriminative edges and the second set to confirm
the efficacy of our approach. In this context, the definition of differential
power is not applicable.

Column/row sampling using randomized techniques have significant benefits, in
addition to de-noising and avoiding overfitting. Other commonly used de-
noising techniques transform the data matrix into a combination of linearly
independent (orthogonal) components, using methods such as Principal Component
Analysis (PCA) or Singular Value Decomposition (SVD). De-noising can be
achieved by discarding the components corresponding to noise, and working with
a low rank approximation of the matrix, as explored in context of brain
fingerprinting by Amico et al. Amico and Goñi (2018).

Using the “unrelated” subset of HCP subjects, Amico et al. Amico and Goñi
(2018) report that using the original data matrix (simply computing
correlations with known samples and selecting the most correlated subject for
identification) as a fingerprint yielded an accuracy of around 87%. The
process of de-noising using low rank approximations yields better results –
Amico et al.Amico and Goñi (2018) report an accuracy of 91% for brain
fingerprinting resting state fMRIs.

As discussed earlier, the advantage of the principal features subspace method
is that it provides a small set of regions, thereby de-noising data while also
making the connectivity matrix much smaller. It also identifies the signature
as a set of biologically relevant features.

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