# LEARNING SHARED NEURAL MANIFOLDS FROM
## MULTI-SUBJECT FMRI DATA

**Anonymous authors**
**Paper under double-blind review**

ABSTRACT

Functional magnetic resonance imaging (fMRI) is a notoriously noisy measurement
of brain activity because of the large variations between individuals, signals marred
by environmental differences during collection, and spatiotemporal averaging required by the measurement resolution. In addition, the data is extremely high
dimensional, with the space of the activity typically having much lower intrinsic
dimension. In order to understand the connection between stimuli of interest and
brain activity, and analyze differences and commonalities between subjects, it
becomes important to learn a meaningful embedding of the data that denoises,
and reveals its intrinsic structure. Specifically, we assume that while noise varies
significantly between individuals, true responses to stimuli will share common,
low-dimensional features between subjects which are jointly discoverable. Similar
approaches have been exploited previously but they have mainly used linear methods such as PCA and shared response modeling (SRM). In contrast, we propose a
neural network called MRMD-AE (manifold-regularized multiple decoder, autoencoder), that learns a common embedding from multiple subjects in an experiment
while retaining the ability to decode to individual raw fMRI signals. We show that
our learned common space represents an extensible manifold (where new points
not seen during training can be mapped), improves the classification accuracy of
stimulus features of unseen timepoints, as well as improves cross-subject translation of fMRI signals. We believe this framework can be used for many downstream
applications such as guided brain-computer interface (BCI) training in the future.

1 INTRODUCTION

Brain activation dynamics, as measured by fMRI, exist in an extremely high-dimensional collection
space and contain high levels of noise. Noise sources include measurement imprecisions and physiological factors such as a subject’s blood pressure variability, head motion, and respiration (Brooks
et al., 2013; Mumford & Poldrack, 2007). Moreover, even after accounting for the idiosyncrasies
in noise, neural responses evoked by the same stimulus can vary greatly between different subjects.
This makes using fMRI to extract meaningful signals or general trends that are task-relevant quite
difficult. Experiments involving fMRI are expensive and usually involve pilot studies conducted in a
smaller scale with simpler stimuli and on a smaller group of subjects. Thus, it is desirable to have a
method that can learn common trends from pilot data and can generalize to a larger group of subjects
or more complex tasks.

Like for other high dimensional biomedical data, it is broadly acknowledged that neural activity
actually lies in a lower-dimensional latent space. Moreover we hypothesize that several features
of these latent spaces are shared across subjects given that measurements are usually task-based
and are taken when subjects experience the same stimuli. Dimensionality reduction methods have
been used to discover such low-dimensional spaces and facilitate understanding the underlying brain
activities. Methods such as principal component analysis (PCA) and factor analysis have been the
common choice (Smith et al., 2014). However, these linear methods are sensitive to noise and restrict
interactions in the latent space. Nonlinear dimensionality reduction algorithms are shown to better
capture the geometry of high-dimensional biological data (Moon et al., 2019; Van der Maaten &
Hinton, 2008) and account for dynamics in neural activities (Gao et al., 2020). PHATE (Moon et al.,


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2019), a diffusion based manifold learning method, was shown to capture both global and local
geometry of complex biological data, which appears to be a good candidate to model fMRI data.

Using a nonlinear manifold such as PHATE to reduce the dimensionality of data is appealing as it can
extract meaningful signals, denoise the data, and accelerate downstream analysis. However, unlike a
method like PCA which learns a projection operator that can be applied to new data, manifold learning
methods are fixed to the input data and do not naturally extend to new data from new tasks or from new
subjects. To extend an existing manifold to out-of-sample data, landmark interpolation or Nyström
extensions are commonly used. However it is usually unsatisfactory as shown by decremented
performance(Appendix Fig. 1). To tackle these shortcomings, we turn to neural networks such as
autoencoders that provide nonlinear dimensionality reduction via a learned parametric function that
is readily applicable to new data. While our key goal is learning an informative low-dimensional data
manifold shared across multiple subjects, this is not something automatically done by autoencoders.
First, there is nothing enforcing autoencoder latent embeddings to respect data manifold geometry,
and indeed they often just spread out points for ease of decoding. Second, individual variation often
dominates embeddings which prevents a common space to be automatically learned.

To address these issues, we propose a manifold-regularized multi-decoder autoencoder (MRMD-AE)
that can process fMRI data from a group of subjects and extract common latent space representations
that respect individual data geometry. Key features of the MRMD-AE include: (a) A common encoder
that projects fMRI data from any subject on to a shared latent space, and subject-specific decoders
that learn to reconstruct data for each subject faithfully. (b) A manifold-geometry regularization of
the latent space based on a precomputed PHATE embedding. (c) Penalties for distances between
individual patient encodings of common stimuli to ensure that the latent space is not split into
individual embeddings.

We show results on three types of tasks in two different datasets. First, we show that we are able
to extend the manifold embedding to new timepoints of data not used in training. This shows the
_extensibility of our manifold in contrast to non-neural network based manifold learning methods._
Second, we show that our latent space improves the ability to classify or infer stimulus features based
on subject fMRI measurements. Surprisingly, the multiple decoder improves classification accuracy
over a common decoder, showing that this framework allows the network to separate common from
individual variations. Finally, we show that even untrained, translation between subjects has increased
accuracy on withheld timepoints. Further, we show highly improved cross subject translation after
training for this task.

2 RELATED WORK

**Manifold Learning** Manifold learning assumes that high-dimensional ambient data X ∈ R[d] lies
on a low dimensional manifold. Given a set of datapoints measured in the ambient space, assumed to
be sampled from a manifold M, manifold learning optimizes a low dimensional Euclidean space
that encodes the intrinsic geometry of M and the original data. Popular manifold learning methods
include diffusion maps (Coifman & Lafon, 2006), Laplacian Eigenmaps (Belkin & Niyogi, 2003)
and PHATE (Moon et al., 2019). We note that multiscale manifold learning methods (Wolf et al.,
2012; Brugnone et al., 2019; Kuchroo et al., 2020) can be used to consider multiple manifolds that
capture relations and structure in data at different resolutions. Other methods, such as tSNE (Van der
Maaten & Hinton, 2008) and UMAP (McInnes et al., 2018), relax the manifold assumption to focus
on neighborhood preservation in low dimensions. While these are often considered as variations of
manifold learning, they are mostly suitable for visualization of clustered data that does not necessarily
have global structure, and typically do not preserve overarching trends or relations between data
regions (see, e.g., Moon et al., 2019; Gigante et al., 2019).

**SAUCIE** In (Amodio et al., 2019), SAUCIE (Sparse Autoencoder for Clustering, Imputation, and
Embedding) is an autoencoder-based generative model that manipulated internal representations to
force the network to produce specific desired transformations in the data. By imposing these structure
on the latent representations of data, it can quickly extract information about datasets more massive
than alternative approaches can easily handle. Further, this framework establishes that multiple
regularizations, in conjunction, can be used to encourage interpretable latent representations that
accentuate biologically reliable relations in data coming from multi-patient cohorts, as demonstrated,
e.g., in immune response profiling (Zhao et al., 2020). Inspired by the success of this approach in


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immunology, we apply a similar multi-objective neural network approach here to process multisubject fMRI data and alleviate some of the main challenges associated with this data type.

**SRM** Shared response modeling (SRM) is a linear method to account for differences in functional
topographies in fMRI data in a low-dimensional space. SRM uses a singular value decomposition
approach to learn a set of low-dimensional signals (shared response) common to multiple subject’s
fMRI data during the same experiment. SRM learns subject-specific transformation to align subject
data with the shared model using an orthonormally-constrained weight matrix, similar to that of
PCA (Chen et al., 2015).

3 PRELIMINARIES
**Manifold learning and dimensionality reduction with PHATE** To capture the complex interaction in the latent space and balanced global and local structures of highly noisy fMRI data, we use
Potential of Heat-diffusion for Affinity-based Transition Embedding (PHATE) (Moon et al., 2019), a
manifold learning method based on diffusion geometry (Coifman & Lafon, 2006). PHATE computes
the diffusion operator which is the probability transition matrix P between data point pairs via an
_α−decay kernel. Then by raising P to the power of t, the PHATE optimal diffusion time scale, it_
simulates a t−step random walk over the data graph. Diffusion potential distances D is extracted
from P _[t], which accounts for both global and local data geometry. Finally, metric multi-dimensional_
scaling (metric MDS) (Abdi, 2007), is applied to D to get the PHATE embedding of input data.
PHATE has been shown to effectively capture geometry of high-dimensional biological data and was
shown to preserve manifold structure in fMRI data in (Rieck et al., 2020). We utilize the extensible
framework of PHATE and use PHATE embedding of the training data as regularization applied to
the manifold embedding layer as shown in Fig. 1. Note that we use PHATE over other popular
dimensionality reduction methods such as tSNE or UMAP because of its established ability to retain
manifold and trajectory structure in biomedical data (Moon et al., 2019; Kuchroo et al., 2020; Chung
et al., 2020; Pappalardo et al., 2020).

**Geometry-regularized Autoencoders (GRAE)** The latent space of the vanilla autoencoders are
usually difficult to interpret and not necessarily reflect data geometry (see, e.g., Mishne et al., 2019;
Moor et al., 2020; Duque et al., 2020). In order to encourage meaningful data geometry to emerge,
or be retained, in latent representations, the training process can be enhanced with appropriate
regularizations, often derived from manifold learning principles. For example,Jia et al. (2015) and Yu
et al. (2013) propose regularization terms that penalize inaccurate preservation of neighborhood
relationships in the ambient space. Recent work by Duque et al. (2020) has shown that the geometry
extracted by PHATE can be leveraged to formulate Geometry-Regularized Autoencoders (GRAE)
that faithfully capture the instrinsic data geometry, while demonstrating advantages in extendibility
and invertibility compared to previous approaches. In this work we adapt this approach to fMRI
data in conjunction with adjusted architecture and additional regularizations that address further
challenges encountered in these settings.

**Cross-subject batch effects in fMRI** Relative to other neuroimaging methods, fMRI is noninvasive, minimal risk, and has broad coverage with high spatial resolution. fMRI experiments commonly
use naturalistic stimuli such as movies to probe an array of cognitive mechanisms in conditions that
closely mimic the human experience of the world. However, the data generated by such experiments
are notoriously high-dimensional and complex, as fMRI is rife with noise from sources like the
scanner, subject movement, and the measured blood-oxygen-level-dependent (BOLD) signal, which
is a secondary effect of neural activity. This makes it difficult to aggregate data across sessions, either
within or between subjects in a study. Beyond the noise sources of fMRI data, another problem with
aggregating data across subjects to uncover something common about brain functioning pertains to
the idiosyncratic nature of functional topographies in the brain. In many brain regions, shared stimuli
will reliably evoke similar patterns of neural activity across subjects (Nastase et al., 2019). However,
that signal of interest could be located in slightly different anatomical locations across subjects or
otherwise obscured by noise, causing a mismatch between functional and anatomical locations for a
signal. A large body of neuroimaging research centers around rectifying this problem with a variety
of linear functional alignment methods by learning either high-dimensional (Haxby et al., 2011;
Busch et al., 2021) or reduced-dimensional (Chen et al., 2015) common spaces. A solution to this
issue must consider both the complexity of the data at the individual subject level, where there is
biological and environmental noise, and the broader cross-subject representational idiosyncrasies.


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4 MULTI-DECODER MANIFOLD-REGULARIZED AUTOENCODER

4.1 PROBLEM SETUP

RLetthese stimuli. We seek to learn a low dimensional latent spaceassociated map[q] _X: k = = 1 {X, . . ., mk : k f : = 1 R}, j, . . ., m[q]_ = 1→ R, . . ., n[d]} be a set of common stimuli forand inverse maps[1], be observed fMRI neuronal activations for gj : R[d] _→_ R m[q], timepoints, and let M ⊂ j = 1, . . ., nR[d] (where, such that: (a) The n d Y subjects over ≪j = {qY) withj,k ∈
mappings corresponding to each stimulus Xk in subjects Yi and Yj are close to one another, i.e.,
_f_ (Yi,k) _f_ (Yj,k) 2 _ϵ. (b) Stimulus features (for each Xk) can be computed by a function_
_∥f_ (Yj,k) _−_ _f_ (Xk) of the low dimensional latent space. (c) Stimulus features can be inferred in∥ _≤_
_7→_ _[′]_
an unseen/untrained subject Yn+1 from the same latent space via f (Yn+1,k) 7→ _f_ _[′](Xk), at any_
timepoint k. (d) Subject-specific responses can be predicted on unseen stimulus Xm+1 (e.g., at
held-out timepoint m + 1) based on the shared latent space, i.e., gj(f (Yi,m+1)) = Yj,m+1 for
subjects i, j. Note here that each Yi,k is a high dimensional measurement consisting of raw activation
values at hundreds to thousands of voxels depending on the respective region of interest (ROI). By
contrast, we expect f (Yi) to be vastly lower dimensional. We analyzed the spectral entropy of
PHATE diffusion potential of the fMRI activation data used in the following experiments and found
the intrinsic dimensions of the data ranging 10 to 40 (Appendix Table 1).

4.2 MRDM-AE ARCHITECTURE

Figure 1: Multi-decoder Manifold Regularized Autoencoder

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embedding layer

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input common encoder 𝑓 individual decoders 𝑔! reconstruction

**Single Encoder – Multiple Decoders** To achieve the properties outlined in the previous section,
we propose a manifold-regularized common-encoder, multiple-decoder model (Figure 1). The
common encoder f (Yj,k) = Enc(Yj,k) takes any set of neuronal activities from any subject and maps
them in a many-to-one fashion to the same latent space L, with the same stimulus prediction. Our
model features several decoders gj(f (Y·,k)) = Dec[j](f (Y·,k)) ≈ _Yj,k, to decode back to subject_
specific responses. The rationale for using a common encoder is for the model to learn a common
fMRI language from which subject specific responses can be derived. While the fMRI data exhibits
strong batch effect between subjects, the common encoder builds invariance in the latent space
to individual noise in the encoded space. The individual decoders enable incorporating subjectdependent interpretation of the latent information from the common encoder. Thus this framework
pushes individual noise into the decoding space. In particular, the decoders can learn to reconstruct
the subject-specific signals and recreate batch effect to reconstruct data. The flexibility provided
by individual decoders allow the single common encoder to only retain input-dependent variability
between subject responses that is sufficient for the decoders to reconstruct the measurements. This
base model is trained with a reconstruction error penalty for each subject seen at training time.


_L[reconstruction]f,g1,...,gn_ [(][Y][1][, . . ., Y][n][) =]


_Yj,k_ _gj(f_ (Yj,k))
_j,k_ _∥_ _−_ _∥[2]_

X


1During preprocessing, functional images were aligned to the common template based on anatomical
landmarks and one time point of a fMRI region of interest is a vector of length equal to the number of voxels.


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**Embedding Alignment Penalty** Despite the common encoder (and individual decoders), the
network may learn to separate subject-specific responses into their own latent sub-spaces for ease
of decoding. To counteract this tendency we explicitly push the latent encodings of individual
responses to common stimuli together by penalizing Euclidean distances in the encoded space
between timepoints corresponding to common stimuli. Therefore, we implement the embedding
alignment penalty as L[alignment]f,g1,...,gn [(][Y][1][, . . ., Y][n][) =][ P]i=j,k
_̸_ _[∥][f]_ [(][Y][i,k][)][ −] _[f]_ [(][Y][j,k][)][∥][2][.]

**Manifold Regularization Penalty** In order to ensure a smooth embedding space that enables
faithful interpolation, inference, and extendibility to new data, a geometric regularization should
be applied to ensure the latent space learned by the autoencoder follows the data geometry.To this
end, within each decoder, we apply a regularization in the bottleneck layer using the manifold
embedding Pj that are learned with PHATE (Moon et al. (2019)) using the train data for each subject
_j. We note that the strong subject batch effects in fMRI data make the task of directly learning a_
subject-independent manifold with PHATE challenging. Indeed, without careful consideration, it
often yields a group fMRI data geometry with variance dominated by batch effects, which would then
submerge stimulus-related variance that is of actual interest. We therefore bypass this issue here by
only considering individual-subject manifolds and letting the combined autoencoder deduce a latent
representation that can easily be mapped to them. Thus, as shown in Figure 1,we split each decoder
intothe common latent representation to a subject-dependent one that captures well the intrinsic structure gj = gj[′] _[◦][g]j[′′][, where intuitively we expect the first part][ g]j[′]_ [to “implement batch effects” going from]
of their stimuli responses. It is to this first layer of gj[′] [we apply the manifold regularization. The]
second part gj[′′] [will then map this intrinsic (manifold) representation to the observable measurements]
to finish reconstruction. In this way, we obtain an intermediary representation after the first layer
of each decoder (where the single-layer limitation restricts the implemented “batch effects” to
be relatively simple) that should match the PHATE embeddings, implemented in training via the
manifold regularization as L[geometric]f,g1[′] _[,...,g]n[′]_ [(][Y][1][, . . ., Y][n][) =][ ∥][g]j[′] [(][f] [(][Y][j,k][))][ −P][j][(][Y][j,k][)][∥][2][.]

**Combined Loss** To combine three loss terms together, we introduce coefficients λ, µ controlling
the amount of alignment and geometric regularization (correspondingly) applied to the hidden layers
of the MRMD-AE, yielding the total loss

_L =_ _Yj,k_ _gj(f_ (Yj,k)) + λ _f_ (Yi,k) _f_ (Yj,k) + µ _gj[′]_ [(][f] [(][Y][j,k][))][ −P][j][(][Y][j,k][)][∥][2][ (1)]
Xj,k _∥_ _−_ _∥[2]_ _iX≠_ _j,k_ _∥_ _−_ _∥[2]_ _∥_

optimized over network functionsand organized together as illustrated in Fig. 1. f, gj = gj[′] _[◦]_ _[g]j[′′][,][ j][ = 1][, . . ., m][, parameterized via neural networks]_

**Additional Cross-Subject Training Penalty** The multiple decoder framework allows us to add
carefully designed loss penalties for specific tasks. For example, we can explicitly train for crosssubject translation. We achieve the purpose by adding the following regularization to encourage
translation,
_L[translation]f,g1,...,gn_ [(][Y][1][, . . ., Y][n][) =] Xj,k Xl≠ _j_ _∥Yj,k −_ _gj(f_ (Yl,k))∥[2]. (2)

In most of the following experiments this cross-subject penalty was not used, as we did not aim to
train for translation. In the last section of Results we explore the use of this penalty to facilitate
learning to translate between subjects’ fMRI activities.

5 RESULTS

To empirically validate MRDM-AE, we test it on tasks of manifold extension, stimulus classification,
and cross-subject translation. The manifold-extension tasks test the ability of MRDM-AE to place
points not seen during training appropriately within the learned data manifold in the embedded
space. The classification tasks analyze the ability of MRDM-AE to capture stimulus-relevant signal
from fMRI data, i.e., features of the movies from voxel activations. Cross-subject classification and
translation tasks test the ability of MRDM-AE to capture a shared latent space that can be used
for common encoding and individual decoding. These tasks are tested on two different datasets
comprising of subjects watching different movies. We compare MRDM-AE to several baselines
including dimensionality reduction methods (PCA, PHATE), the shared response model (SRM)
commonly used in fMRI data, as well as ablations of our autoencoder method.


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**Datasets** The Sherlock dataset is an open-access dataset of 16 participants watching a 48-minute
clip of the BBC television series “Sherlock.” Data was collected in two runs of 946 and 1030
[timepoints (repetition time; TRs) respectively. This data was downloaded from the DataSpace public](http://arks.princeton.edu/ark:/88435/dsp01nz8062179)
[repository. For full details on the stimulus, see the original publication of this dataset (Chen et al.](http://arks.princeton.edu/ark:/88435/dsp01nz8062179)
(2017)). Annotations for the Sherlock movie stimulus were released with Vodrahalli et al. (2018)
[and available freely on Github. These annotations were created by humans who viewed the film and](https://github.com/kiranvodrahalli/fMRI_Text_maps_NI)
labeled each timepoint on a variety of metrics. In our classification tasks, we used two sets of binary
labels that correspond with visual and auditory aspects of the stimulus: whether a given timepoint
featured an indoor or outdoor scene (Indoor/Outdoor) and whether or not the timepoint featured
music (Music Present).

As a second testbed, we used the StudyForrest open-access dataset downloaded from DataLad; full
details for this can be found at the original publication of the movie dataset (Hanke et al. (2014)) and
the localizer extension dataset (Sengupta et al. (2016)). Here we included data from 15 participants
who completed both the movie-viewing and functional localizer tasks, where participants viewed 24
unique grayscale images of objects from 6 categories (faces, bodies, houses, small objects, outdoor
scenes, and scrambled images) in a separate session to localize brain regions that reliably respond to
a specific category. In the movie task, participants watched a 2-hour version of the movie Forrest
_Gump. In a series of classification tasks, we use labels for each timepoint of the movie indicating the_
time of day (2 levels), and the flow of time in the narrative (4 levels), annotated in (Husler & Hanke
(2016)), and image categories for the localizer data (6 levels).
**Autoencoder Hyperparameters** All autoencoder based models have three fully connected layers
in the encoder and the decoder(s), with a bottleneck latent space layer in between. The hidden neurons
have the configurations of 256-128-64-20-64-128-256. Leaky ReLU activations are applied on all of
the layers. We used the Adam optimization algorithm with a learning rate of 0.001, and a batch size
of 64. The neural networks were trained for 4000 epochs. When regularization is used, λ and µ are
either 0.1 or 0.01 and the same set of values are used across subjects for a specific task.
**Model Comparison** We compare our proposed model with a standard autoencoder (AE) with the
same hidden dimensions as the proposed MRMD-AE, but with a single decoder for all subjects. We
refer a standard autoencoder with manifold regularization applied to the bottleneck layer as MR-AE.
**Dimensionality Reduction Methods** Non-linear dimensionality reduction methods such as
PHATE (and other methods such as t-SNE and UMAP) typically provides fixed latent representations
for the input data, and do not provide a natural embedding function that can project new data to
an existing manifold. PHATE preserves manifold structure while tSNE and UMAP usually shatter
data trajectories to create clusters even when they do not exist ( Moon et al. (2019), Kuchroo et al.
(2020), Rieck et al. (2020)). We therefore compare with an adaption of PHATE to interpolate new
datapoints via landmark approach used by Moon et al. (2019), a subsampling method that was used
and shown to be effective in improving scalability of PHATE. We also compare with linear PCA
approach. For within-subject analyses, the dimensionality reduction was learned for a subject on one
half of the dataset and applied to unseen timepoints. For cross-subject analyses, the dimensionality
reduction was learned on all but one subject and applied to the held-out subject, and repeated holding
out each subject once. For cross-subject analyses, we also compared to SRM by learning a reduceddimensional shared signal on data from all but one subject and evaluating the fit of the shared space
on the held-out subject, repeating for all subjects.
**Place Future fMRI Time Points on Learned Manifold** We first show that MRMD-AE achieves
efficient and accurate manifold extension by placing new fMRI time points at the right positions on
learned manifold. Consider the fMRI time series contain two parts, including the first mtrain TRs
that we receive as pilot data with which we learned individual manifolds _train, a mtrain_ _d matrix,_
with each row being the PHATE coordinates of one TR. We would like to faithfully extend these P _×_
learned manifolds to subsequently receive test data of future mtest fMRI TRs, so that the extended
manifold Ptest closely resemble the corresponding time points ground truth manifold P, that is
learned from all mtrain + mtest TRs.

Since the manifold coordinates are fixed to the input data, we cannot directly compare the Ptest with

[TRtest], the corresponding rows of . Instead, we first compute the manifold distance matrices
_P_ _P_
between each test and train time point pairs, where [ˆ](i, j) = _test(i, )_ _train(j, )_, and the
_D_ _||P_ _−P_ _||_
ground truth (i, j) = (i, ) (j, ) _, i_ _TRtest, j_ _TRtrain. Then we evaluate the similarity_
_D_ _||P_ _−P_ _||_ _∈_ _∈_
using the MSE between _D[ˆ] and D. Fig.2 shows that MRMD-AE achieves lower or matching MSE_
than the landmark approach, especially with lower percentages of train data. The extended manifold


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Figure 2: Comparison of extended manifold by MRMD-AE, MR-AE and PHATE Landmark on the
Sherlock early-visual ROI. The MSE between extended manifold coordinates and ground truth were
computed (lower is better). MRMD-AE extended manifold embeddings achieve highest classification
accuracy in predicting movie stimulus labels. Error bars are standard deviation across subjects.

generated by MRMD-AE more closely resemble the ground truth. MRMD-AE also outperforms
MR-AE, suggesting the advantage of per-subject decoders in reconstructing subject-specific signals.

We also use the extended manifold embedding to predict movie stimulus labels. An support vector
classifier (SVC) is trained on each subject’s extended manifold coordinates and the average accuracy
and standard deviation via a 5-fold cross validation are reported. We found that MRMD-AE extended
manifold achieves higher classification accuracies than the landmark approach, suggesting the
embeddings retain more meaningful signals that can be used to recover signals of the movie stimulus
(right panel of Fig. 2 and Table 1).
**Effect of Latent Space Alignment Penalty** We visualize the raw data and the embedding
_f_ (Yj), j = 1, 2, . . ., 15 of the Forrest image localizer data using PHATE, and color the plots by
subject ID in Figure 3. The raw data visualization exhibits clear batch effect. By increasing λ in
Eq. 1 and penalizing latent space misalignment, the batch effect in the latent space is eliminated. We
computed the average Earth Mover’s Distance (EMD) between subject embedding pairs and showed
improved alignment between subjects in the latent space comparing to the raw data and the improved
alignment by increasing alignment penalty.
**Stimuli Classification** Our goal is to test how well our MRMD-AE generalizes to unseen times
series of fMRI data. We evaluate the model by how well we can recover meaningful labels in the
latent space. After training on the train half of data from all subjects, we applied the trained model
on the test half of data and collected the latent space representation at the bottleneck layer. A SVC
is trained within each subject on the test embedding and the average classification accuracy and
standard deviation via a 5-fold cross validation are recorded. The average accuracy and standard
deviation across subjects are reported.

Results are reported in table 1 for Sherlock movie annotation classification and table 2 for Forrest
movie and image localizer classification. MRMD-AE effectively recovers the latent factors corresponding to the stimuli and achieve the best or second best performance in all combinations of ROI
and stimulus annotations. We showed previously that a learned PHATE manifold via landmark does
not generalize to out-of-sample data. Here we confirm that the classification accuracy of the test set
decreases significantly from the train set. The linear projection operator learned by PCA also fails to
generalize well to new data (Appendix Fig. 1 shows in-sample vs out-of-sample comparison for PCA
and PHATE Landmark). MRMD-AE also outperforms per-subject individual MR-AE, suggesting
the synergy between subjects offered by the shared encoder and multi-decoder framework benefits
generalization and recovering meaningful signal from data. Appendix Figure 2 shows that across
subjects, MDMR-AE consistently outperforms the other methods.
**Untrained Cross-Subject Translation** Our MRMD-AE framework allows integration of data
from multiple subjects, thereby enabling translation between subjects. Similar as before, we trained
our MRMD-AE with the train half of all subjects’ data. Then we translate the test half timepoints of
subject i to each of the other subjects by _Y[ˆ]i_ _j = gj(f_ (Yi)). We evaluate the quality of translation by
_→_
comparing the MSE and correlation between _Y[ˆ]i_ _j and Yj( 3). It should be noted that the model was_
_→_
not trained for translation, thus in training, the decoders did not learn to translate other subjects data.
Yet, the common encoder maps the neural activities to the same latent space and builds invariance to
individual noise in the encoded space, allowing the decoders to perform the translation task. SRM was


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Table 1: Sherlock within-subject manifold classification. Within each subject’s data, a latent space
was learned on one half of the data and extended to the held-out timepoints. A support vector classifier
was trained/tested on the latent representation of the unseen timepoints with 5-fold cross validation
to predict features of the movie stimulus. The mean and standard deviation of test accuracy across
subjects and folds were reported. MRMD-AE outperformed all comparison methods suggesting
efficient manifold extension to new time points and better preservation of factors from the stimulus.

|ROI|Method|IndoorOutdoor|Col4|MusicPresent|Col6|
|---|---|---|---|---|---|
|||Accuracy|Std. Dev.|Accuracy|Std. Dev.|
|early- visual|PCA PHATE-Landmark AE Individual MR-AE MRMD-AE|0.5923 0.5681 0.7379 0.7289 0.7760|0.0629 0.0333 0.0499 0.0196 0.0520|0.5215 0.5530 0.6840 0.6746 0.7170|0.0247 0.0302 0.0240 0.0259 0.0383|
|early- auditory|PCA PHATE-Landmark AE Individual MR-AE MRMD-AE|0.5948 0.5831 0.7285 0.7213 0.7523|0.0606 0.0290 0.0445 0.0315 0.0544|0.5544 0.5665 0.6950 0.6875 0.7128|0.0314 0.0296 0.0268 0.0309 0.0381|
|pmc|PCA PHATE-Landmark AE Individual MR-AE MRMD-AE|0.5622 0.5395 0.7080 0.7096 0.7318|0.0640 0.0368 0.0446 0.0207 0.0589|0.5216 0.5342 0.6585 0.6600 0.6845|0.0244 0.0316 0.0243 0.0265 0.0403|



Table 2: Forrest within-subject manifold classification. For Flow of Time and Time of Day
classifications, latent spaces were learned on one half of the movie timepoints and then extended to
the held-out timepoints. For the Localizer classification, latent spaces were trained on the entirety
of the movie watching dataset and then extended to the localizer dataset, which was collected in a
separate session. Results are reported as the average and standard deviation across subjects and cross
validation folds.

|ROI|Method|Flow of Time|Col4|Time of Day|Col6|Localizer|Col8|
|---|---|---|---|---|---|---|---|
|||Acc.|Std. Dev.|Acc.|Std. Dev.|Acc.|Std. Dev.|
|early- visual|PCA PHATE-Landmark AE Indiv. MR-AE MRMD-AE|0.4590 0.2434 0.5547 0.5771 0.5845|0.0186 0.0135 0.0210 0.0242 0.0192|0.6406 0.5266 0.6025 0.6695 0.6912|0.0322 0.0233 0.0298 0.0197 0.0216|0.3965 0.3863 0.4542 0.4321 0.4468|0.0464 0.0182 0.0561 0.0484 0.0683|
|late- visual|PCA PHATE-Landmark AE Indiv. MR-AE MRMD-AE|0.4690 0.2540 0.5613 0.5773 0.5837|0.0167 0.0219 0.0220 0.0227 0.0209|0.6450 0.5244 0.6278 0.6659 0.6816|0.0287 0.0236 0.0270 0.0229 0.0237|0.4442 0.3817 0.4153 0.4023 0.4255|0.0187 0.0104 0.0491 0.0392 0.0514|



Table 3: Cross-subject Translation Comparison when MRMD-AE is not trained for translation.
The translation accross subjects _Y[ˆ]i_ _j = gj(f_ (Yi)), i = j is compared with the target subject Yj.
_→_ _̸_
Lower MSE and higher correlation indicate better translation. MRMD-AE trained with latent space
alignment penalties (λ > 0 in Eq. 1) achieves better translation.

|ROI|Method|MSE|Col4|Correlation|Col6|
|---|---|---|---|---|---|
|||Reconstruction|Translation (untrained)|Reconstruction|Translation (untrained)|
|early- visual|PCA SRM MRMD-AE (λ = 0) MRMD-AE (λ > 0)|0.1548 0.1640 0.1797 0.1933|1.3903 1.3504 1.4321 1.3626|0.9193 0.9142 0.9058 0.8981|0.2275 0.2327 0.2235 0.2517|
|pmc|PCA SRM MRMD-AE (λ = 0) MRMD-AE (λ > 0)|0.0752 0.0766 0.0848 0.1647|1.5354 1.5130 1.5540 1.4795|0.9616 0.9609 0.9567 0.9139|0.1928 0.1951 0.1900 0.1977|


MSE Correlation

ROI Method Reconstruction Translation Reconstruction Translation

(untrained) (untrained)

PCA 0.1548 1.3903 0.9193 0.2275

early- SRM 0.1640 **1.3504** 0.9142 0.2327
visual MRMD-AE (λ = 0) 0.1797 1.4321 0.9058 0.2235

MRMD-AE (λ > 0) 0.1933 1.3626 0.8981 **0.2517**

PCA 0.0752 1.5354 0.9616 0.1928

SRM 0.0766 1.5130 0.9609 0.1951

pmc

MRMD-AE (λ = 0) 0.0848 1.5540 0.9567 0.1900

MRMD-AE (λ > 0) 0.1647 **1.4795** 0.9139 **0.1977**


-----

developed to achieve this goal by mapping all subjects to a shared response, then project to individual
spaces. We show that our model achieves best translation performance on various brain ROIs in at
least one metric. By incorporating latent space alignment penalty (λ > 0 in Eq. 1), MRMD-AE
learns a more aligned shared latent space and allows for better translation across subjects. More
experimental details can be found in the supplementary materials.

Figure 3: PHATE visualization of raw data and latent representation f (Yj), 1 _j_ 15, colored
_≤_ _≤_
by subject. The raw data exhibits clear batch effect while the latent space is better aligned between
subjects. The incorporation of higher levels of latent space alignment penalty by increasing λ in Eq. 1
better removes batch effect in the shared space, demonstrated by decreasing EMD.

**Trained Cross Subject Translation** Our MRMD-AE framework provides flexibility to specialize
in different tasks by incorporating carefully designed loss penalties. By adding a cross-subject
training penalty in Eq. 2 to the combined loss, we explicitly train our model for translating between
subjects. We show in Figure 4 the improved translation performance by using different combinations
of loss penalties. Combining the alignment penalty with translation penalty achieves the lowest
translation MSE.

|Col1|Alignment Penalty|Translation Penalty|
|---|---|---|
|MRMD-AE(1)|✗|✗|
|MRMD-AE(2)|✓|✗|
|MRMD-AE(3)|✓|✓|



Figure 4: Cross subject translation MSE comparison. Combinations of latent space alignment
penalty and translation penalty were used. MRMD-AE with better aligned latent space achieve better
translation MSE. We also trained the model explicitly for translation by incorporating the translation
loss, achieving the best translation performance with the lowest translation MSE.

6 DISCUSSION

Extracting meaningful brain mechanisms shared across subjects is a daunting task, chiefly due to
stimulus-independent noise and the challenge of comparing subjects’ activity pattern in a common
reference space.To address this, we proposed MRDM-AE, a multi-decoder manifold-regularized
autoencoder that takes fMRI data from a group of subjects as input, first projects it to a common
latent space, then to subject-specific decoders. We showed that the subject-specific decoders enabled
the autoencoder to decouple shared information from individual variations such that the shared latent
space offered higher-accuracy classifications of underlying stimuli, and better manifold extensions to
unseen data. Finally, we showed that the individual decoders can potentially be used for subject-tosubject translation, setting up fMRI response predictions of subjects to unseen data based on training
and calibration on common stimuli. Overall, we show the manifold learning and structural priors
bring in improve the state of the art in shared latent space learning in fMRI from many subjects.


-----

REPRODUCIBILITY STATEMENT

The results presented in the paper are reproducible. We will provide a link to an anonymous repository
to the reviewers and area chairs. The content of the link will contain complete instructions on how to
run the code to reproduce the results presented for the paper. In addition, all details for the models
and training procedures are described in the main paper under Sec. 4 and the Appendix.

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A SUPPLEMENTARY MATERIALS

**fMRI pre-processing and region identification** After all standard fMRI preprocessing steps,
functional images from both the Sherlock and Forrest datasets were aligned to the common MNI
template brain based on anatomical landmarks. In this anatomical template, we defined several
regions of interest (ROIs) to target in our analyses: the early visual cortex, late visual cortex, early
auditory cortex, and the posterior medial cortex (PMC). We selected these regions as they robustly
respond to audiovisual stimuli (early visual, late visual, and early auditory) and PMC was targeted
in the original sherlock analyses for its memory involvement (Chen et al. (2017)). We extracted
timeseries data from the voxels in each of these regions (early visual: 307 voxels, late visual: 1008
voxels, early auditory: 1018 voxels, PMC: 481 voxels) and collapsed the spatial structure within this
ROI into a [timepoints, voxels] matrix of activity for each voxel across time. Within this matrix, we
z-scored each voxel across its timeseries to account for differences across voxels in mean activation.

**Dimension reduction methods on out-of-sample data** Dimensionality reduction methods do not
generalize to out-of-sample data with decremented performance when extending trained models to
new data points. In Figure 1 we show that the stimuli classification accuracy decreases when PCA
and PHATE (via landmark approach) are applied to out-of-sample data. Experiment was conducted
on Sherlock early-visual ROI data.

Appendix Figure 1: Classification accuracy comparison. PCA and PHATE both fail to generalize to
out-of-sample data, shown as decremented accuracy. Landmark approach was used to extend learned
PHATE embedding to new data.

**Stimuli classification experimental details** For 5-fold cross validation, we proceed following
the standard procedures. We split the dataset into 5 groups at random, and for each group we take
as a hold out or test dataset and use the remaining groups as a training data set. Fit a classifier
on the training set and evaluate on the test set, retain the test accuracy and summarize after all
folds. For each training set, we balance the samples to reach an even split between classes during
training, so the classification tasks are entirely balanced. We repeat the procedure for each subject.
The mean accuracy from the 5-fold cross validation within each subject is reported as the skill
score of the subject. Then we report the mean and standard deviation across subjects. For SVM
classifiers, we used the C-support vector classification from Scikit-learn package, with an RBF kernel,
regularization=10, and gamma set to be ’scale’,1/(nfeatures _X.var()) ._
_×_

**Stimuli classification comparison across subjects** Since we perform classification within each
subject, we further look into how our model compare to the other methods for each subject. We
present the results on the Sherlock early visual ROI and show that higher classification accuracy was
achieved with MRMD-AE manifold embedding in 13 out of 16 subjects comparing to the next best
method that is a vanilla AE. Moreover, MRMD-AE outperformed individual MR-AE (third-best)
in all subjects. Although there are not enough samples to quantify subject-level statistics, we show
MRME-AE achieves better performance in almost all subjects( 2).

**Intrinsic dimensions of fMRI data** We analyzed the intrinsic dimensions of fMRI data by examining the spectral entropy of PHATE diffusion matrix of fMRI data. We refer the readers to Moon


-----

Appendix Figure 2: Classification accuracy comparison across subjects. Classification by indooroutdoor label on Sherlock early visual ROI using the top 3 performing models in Table 1. Left: The
mean accuracy from the 5-fold cross-validation within each subject was reported. Error bar was
standard deviation from the 5-fold cross-validation. MRMD-AE outperforms the next best method,
a vanilla AE, in 13/16 subjects and outperforms the third best method, individual MR-AE, in all
subjects. Right: Box plot of the same set of accuracy values show better performance of MRMD-AE
compared to the other methods.

et al. (2019) for how PHATE computes the diffusion operator and the relationship between diffusion
operator to the data geometry. For the purpose of current analysis, one can consider the diffusion
operator to represent the ambient data geometry and the spectral entropy analysis via Von Neumann
Entropy similar to the analysis of principle components that account for most variation in PCA. We
report the intrinsic data dimensions of various ROIs of the Sherlock and Forrest fMRI datasets that
we use in the experiments of this paper in Appendix Table 1.

|Dataset|Sherlock|Col3|Col4|Forrest|Col6|
|---|---|---|---|---|---|
|ROI|early-visual|pmc|early-auditory|late-visual|early-visual|
|Intrinsic Dimension|30|29|41|13|15|



Appendix Table 1: fMRI ROI data intrinsic dimensions


-----

