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# Equivariance Allows Handling Multiple Nuisance Variables When Analyzing Pooled Neuroimaging Datasets

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## Abstract and Figures

Pooling multiple neuroimaging datasets across institutions often enables improvements in statistical power when evaluating associations (e.g., between risk factors and disease outcomes) that may otherwise be too weak to detect. When there is only a {\em single} source of variability (e.g., different scanners), domain adaptation and matching the distributions of representations may suffice in many scenarios. But in the presence of {\em more than one} nuisance variable which concurrently influence the measurements, pooling datasets poses unique challenges, e.g., variations in the data can come from both the acquisition method as well as the demographics of participants (gender, age). Invariant representation learning, by itself, is ill-suited to fully model the data generation process. In this paper, we show how bringing recent results on equivariant representation learning (for studying symmetries in neural networks) instantiated on structured spaces together with simple use of classical results on causal inference provides an effective practical solution. In particular, we demonstrate how our model allows dealing with more than one nuisance variable under some assumptions and can enable analysis of pooled scientific datasets in scenarios that would otherwise entail removing a large portion of the samples.
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Equivariance Allows Handling Multiple Nuisance Variables
When Analyzing Pooled Neuroimaging Datasets
Vishnu Suresh Lokhande
lokhande@cs.wisc.edu
Rudrasis Chakraborty
rudrasischa@gmail.com
Sathya N. Ravi
sathya@uic.edu
Vikas Singh
vsingh@biostat.wisc.edu
Abstract
Pooling multiple neuroimaging datasets across institu-
tions often enables improvements in statistical power when
evaluating associations (e.g., between risk factors and dis-
ease outcomes) that may otherwise be too weak to detect.
When there is only a single source of variability (e.g., dif-
ferent scanners), domain adaptation and matching the dis-
tributions of representations may sufﬁce in many scenar-
ios. But in the presence of more than one nuisance vari-
able which concurrently inﬂuence the measurements, pool-
ing datasets poses unique challenges, e.g., variations in the
data can come from both the acquisition method as well
as the demographics of participants (gender, age). Invari-
ant representation learning, by itself, is ill-suited to fully
model the data generation process. In this paper, we show
how bringing recent results on equivariant representation
learning (for studying symmetries in neural networks) in-
stantiated on structured spaces together with simple use
of classical results on causal inference provides an effec-
tive practical solution. In particular, we demonstrate how
our model allows dealing with more than one nuisance
variable under some assumptions and can enable analy-
sis of pooled scientiﬁc datasets in scenarios that would
otherwise entail removing a large portion of the samples.
Our code is available on https:// github. com/ vsingh-
group/DatasetPooling
1. Introduction
Observational studies in many disciplines acquire cross-
sectional/longitudinal clinical and imaging data to under-
stand diseases such as neurodegeneration and dementia
[44]. Typically, these studies are sufﬁciently powered for
the primary scientiﬁc hypotheses of interest. However, sec-
ondary analyses to investigate weaker but potentially inter-
esting associations between risk factors (such as genetics)
and disease outcomes are often difﬁcult when using com-
mon statistical signiﬁcance thresholds, due to the small-
/medium sample sizes.
Over the last decade, there are coordinated large scale
Figure 1. Learning Invariant Representations. In our framework,
input images Xare pooled together from multiple sites. An encoder Φ
maps Xto the latent representations Φ(X)that corresponds to high-level
causal features XCthat inﬂuences the label prediction. Unlike the input
images X,Φ(X)is robust to nuisance attributes like site (scanner) and
covariates (age). Φis trained alongside predictor hand decoder Ψ.
multi-institutional imaging studies (e.g., ADNI [26], NIH
All of Us and HCP [19]) but the types of data collected
or the project’s scope (e.g., demographic pool of partici-
pants) may not be suited for studying speciﬁc secondary
scientiﬁc questions. A “pooled” imaging dataset obtained
from combining roughly similar studies across different in-
stitutions/sites, when possible, is an attractive alternative.
The pooled datasets provide much larger sample sizes and
improved statistical power to identify early disease bio-
markers – analyses which would not otherwise be possi-
ble [16,30]. But even when study participants are consistent
across sites, pooling poses challenges. This is true even for
linear regression [56] – improvement in statistical power is
not always guaranteed. Partly due to these as well as other
reasons, high visibility projects such as ENIGMA [47] have
reported ﬁndings using meta-analysis methods.
Data pooling and fairness. Even under ideal conditions,
pooling imaging datasets across sites requires care. As-
sume that the participants across two sites, say site1and
site2, are perfectly gender matched with the same propor-
tion of male/female and the age distribution (as well as the
proportion of diseased/health controls) is also identical. In
this idealized setting, the only difference between sites may
come from variations in scanners or acquisition (e.g., pulse
sequences). When training modern neural networks for a
regression/classiﬁcation task with imaging data obtained in
this scenario, we may ask that the representations learned by
1
arXiv:2203.15234v1 [cs.LG] 29 Mar 2022
(a) Causal Diagram (b) Variation due to site (scanner) for particular age group. (c) Variation due to covariates (age) in Siemens scanner.
Figure 2. (a) ACausal Diagram listing variable of interest and their relationship for multi-site pooling problem. Nodes Dpopul ,Dacqui and Dpreval denote
the population, acquisition and prevalence biases that vary across sites. C’s are covariates (like age or gender). XCdenotes the high-level causal features
of an image Xthat inﬂuences the labels Y. Nodes in red d-separate the nodes in blue and green. (b) MRI images on control subjects from the ADNI [26]
dataset for different scanners in the age group 70-80.(c) Images obtained from the Siemens scanner (i.e., ﬁxing site) on control subjects for three extreme
age groups. The gantt chart on top of the image indicates the respective age range in the Phillips and GE scanners. As observed, different scanner groups
do not share a common support on “age” covariates, resulting in samples outside of the common support to be discarded in na¨
ıve pooling approaches.
the model be invariant to the categorical variable denoting
“site”. While this is not a “solved” problem, this strategy
has been successfully deployed based on results in invariant
representation learning [3,5,34] (see Fig. 1). One may al-
ternatively view this task via the lens of fairness – we want
the model’s performance to be fair with respect to the site
variable. This approach is effective, via constraints [52] or
using adversarial modules [17,53]. This setting also permits
re-purposing tools from domain adaptation [35,50,55] or
transfer learning [12] as a pre-processing step, before anal-
ysis of the pooled data proceeds.
Nuisance variables/confounds. Data pooling problems
one often encounters in scientiﬁc research typically vio-
lates many of the conditions in the aforementioned exam-
ple. The measured data Xat each site is inﬂuenced not
only by the scanner properties but also by a number of other
covariates / nuisance variables. For instance, if the age dis-
tribution of participants is not identical across sites, com-
parison of the site-wise distributions is challenging because
it is inﬂuenced both by age and the scanner. An example
of the differences introduced due to age and scanner biases
is shown in Figures 2b,2c. With multiple nuisance vari-
ables, even effective tools for invariant representation learn-
ing, when used directly, can provide limited help. The data
generation process, and the role of covariates/nuisance vari-
ables, available via a causal diagram (Figure 2a), can inform
how the formulation is designed [6,45]. Indeed, concepts
from causality have beneﬁted various deep learning mod-
els [37,41]. Specially, recent work [31] has shown the value
of integrating structural causal models for domain general-
ization, which is related to dataset pooling.
Causal Diagram. Dataset pooling under completely arbi-
trary settings is challenging to study systematically. So, we
assume that the site-speciﬁc imaging datasets are not sig-
niﬁcantly different to begin with, although the distributions
for covariates such as age/disease prevalence may not be
perfectly matched and each of these factors will inﬂuence
how these variables inﬂuence the measurements. We show
how the distribution matching criteria provided by a causal
diagram can be nicely handled for some ordinal covariates
that are not perfectly matched across sites by adapting ideas
from equivariant representation learning.
Contributions. We propose a method to pool multiple neu-
roimaging datasets by learning representations that are ro-
bust to site (scanner) and covariate (age) values (see Fig. 1
for visualization). We show that continuous nuisance co-
variates which do not have the same support and are not
identically distributed across sites, can be effectively han-
dled when learning invariant representations. We do not
require ﬁnding “closest match” participants across sites –
a strategy loosely based on covariate matching [39] from
statistics which is less feasible if the distributions for a co-
variate (e.g., age) do not closely overlap. Our model is
based on adapting recent results on equivariance together
with known concepts from group theory. When tied with
common invariant representation learners, our formulation
allows far improved analysis of pooled imaging datasets.
We ﬁrst perform evaluations on common fairness datasets
and then show its applicability on two separate neuroimag-
ing tasks with multiple nuisance variables.
2. Reducing Multi-site Pooling to Inﬁnite Di-
mensional Optimization
Let Xdenote an image of a participant and let Ybe the
corresponding (continuous or discrete) response variable or
target label (such as cognitive score or disease status). For
simplicity, consider only two sites – site1and site2. Let
Drepresent the site-speciﬁc shifts, biases or covariates that
we want to take into account. One possible data generation
process relating these variables is shown in Figure 2a.
Site-speciﬁc biases/confounds. Observe that Yis, in
fact, inﬂuenced by high-level (or latent) features XCspe-
ciﬁc to the participant. The images (or image-based disease
biomarkers) Xare simply our (lossy) measurement of the
participant’s brain XC[14]. Further, Xalso includes an
(unknown) confound: contribution from the scanner (or ac-
quisition protocol). Figure 2a also lists covariates C, such
as age and other factors which impact XC(and therefore,
X). A few common site-speciﬁc biases Dare shown in
2
Fig. 2a. These include (i) population bias Dpopul that
leads to differences in age or gender distributions of the
cohort [9]; (ii) we must also account for acquisition shift
Dacqui resulting from different scanners or imaging pro-
tocols – this affects Xbut not XC;(iii) data are also in-
ﬂuenced by a class prevalence bias Dpreval, e.g., healthier
individuals over-represented in site2will impact the distri-
bution of cognitive scores across sites.
For imaging data, in principle, site-invariance can be
achieved via an encoder-decoder style architecture to map
the images Xinto a “site invariant” latent space Φ(X).
Here, Φ(X)in the idealized setting, corresponds to the true
“causal” features XCthat is comparable across sites. In
practice, we know that images cannot fully capture the dis-
ease – so, Φ(X)is simply a surrogate, limited by the mea-
surements we have on hand. Given these caveats, an archi-
tecture is shown in Fig. 1. Ideally, the encoder will mini-
mize Maximum Mean Discrepancy (MMD) [20] or another
discrepancy between the distributions of latent representa-
tions Φ(·)of the data from site1and site2.
The site-speciﬁc attributes Dare often unobserved or
otherwise unavailable. For instance, we may not have full
To tackle these issues, we use a causal diagram, see Fig. 2a,
similar to existing works [31,55] with minimal changes. For
dealing with unobserved D’s, some standard approaches are
known [22]. Let us see how it can help here. Applying d-
separation (see [22,36] ) on Fig. 2a, we see that the nodes
(Dpopul, C, XC)form a so-called “head-to-tail” branch and
the nodes (Dacqui, X, XC),(Dpreval , Y, XC)form a “head-
to-head” branch. This implies that XCD|C. This
is exactly an invariance condition: XCshould not change
across different sites for samples with the same value of C.
To enforce this using Φ(·), we must optimize a discrepancy
between site-wise Φ(X)’s at a given value of C,
min
ΦMMDPsite1Φ(X)|C, Psite2Φ(X)|C(1)
Provably solving (1)?A brief comment on the difﬁculty
of the distributional optimization in (1) is useful. Generic
tools for (worst case) convergence rates for such problems
are actively being developed [51]. For the average case,
[38] presents an online method for a speciﬁc class of (ﬁ-
nite dimensional) distributionally robust optimization prob-
lems that can be deﬁned using standard divergence mea-
sures. Observe that even these convergence guarantees are
local in nature, i.e., they output a point that satisﬁes neces-
sary conditions and may not be sufﬁcient.
In practice, the outlook is a little better. Intuitively, an
optimal matching of the conditional distributions P(Φ(X)|
C)across the two sites corresponds to a (globally) optimal
solution to the probabilistic optimization task in (1). Ex-
isting works show that it is indeed possible to approach this
computationally via sub-sampling methods [55] or by learn-
ing elaborate matching functions to identify image or object
pairs across sites that are “similar” [31] or have the same
value for C. Sub-sampling, by deﬁnition, reduces the num-
ber of samples from the two sites by discarding samples
outside of the common support. This impacts the quality
of the estimator – for instance, [55] must restrict the analy-
sis only to that age range of Cwhich overlaps or is shared
across sites. Such discarding of samples is clearly undesir-
able when each image acquisition is expensive. Matching
functions also do not work if the support of Cis not identi-
cal across the sites, as brieﬂy described next.
Example 2.1. Let Cdenote an observed covariate, e.g.,
age. Consider Xiat site1with C=c1and Xjat site2
with C=c2. If c1c2, a matching will seek Φ(Xi)
Φ(Xj)in XCspace. If c1falls outside the support of c’s
acquired at site2, one must not only estimate Φ(·)but also a
transport expression Γc2c1(·)on XCsuch that Φ(Xi)
Γc2c1(Φ(Xj)). The “transportation” involves estimating
what a latent image acquired at age c2would look like at
age c1. This means that matching would need a solution to
the key difﬁculty, obtained further upstream.
2.1. Improved Distribution Matching via Equivari-
ant Mappings may be possible
Ignoring Yfor the moment, recall that matching here
corresponds to a bijection between unlabeled (ﬁnite) con-
ditional distributions. Indeed, if the conditional distribu-
tions take speciﬁc forms such as a Poisson process, it is in-
deed possible to use simple matching algorithms that only
responding empirical distributions [43], for example, the
well-known Gale-Shapley algorithm [46]. Unfortunately,
in applications that we consider here, such distributional as-
sumptions may not be fully faithful with respect to site spe-
ciﬁc covariates C. In essence, we want representations Φ
(when viewed as a function of C) that vary in a predictable
(or say deterministic) manner – if so, we can avoid match-
ing altogether and instead match a suitable property of the
site-wise distributions of the representation Φ(X). We can
make this criterion more speciﬁc. We want the site-wise
distributions to vary in a manner where the “trend” is con-
sistent across the sites. Assume that this were not true, say
P(Φ(X)|C)is continuous and monotonically increases
with Cfor site1but monotonically decreases for site2. A
match of P(Φ(X)|C)across the sites at a particular value
of C=cimplies at least one C=c0where P(Φ(X)|C)
do not match. The monotonicity argument is weak for high
dimensional Φ. Plus, we have multiple nuisance variables.
It turns out that our requirement for P(Φ(X)|C)to vary
in a predictable manner across sites can be handled using
the idea of equivariant mappings, i.e., P(Φ(X)|C)must
3
Figure 3. Visualization of Stage one. First, an image pair Xi, Xjare mapped onto a hypersphere using an encoder E. The resulting pair i,j
are passed through τnetwork to map them into the space of rotation matrices (which is the quotient group denoted by G/H). Fact 3ensure that τis a
G=SO(n)equivariant map. G(i, j)/G(i, j )1is the group action of transforming τ(i)to τ(j)(j)to τ(i)respectively.
be equivariant with respect to Cfor both sites. In addition,
we will also seek invariance to scanner attributes.
While we are familiar with the well-studied notion of
invariance through the MMD criterion [29], we will brieﬂy
formalize our idea behind an equivariant mapping which is
less common in this setting.
Deﬁnition 1. A mapping f:X → Y deﬁned over mea-
surable Borel spaces Xand Yis said to be Gequivariant
under the action of group Giff
f(g·x) = g·f(x), g G
We refer the reader to two recent surveys, Section 2.1of
[7] and Section 3.1of [8], which provide a detailed review.
Equivariance is often understood in the context of a
group action (say, a matrix group) [24,28]. While the co-
variates Cis a vector (and every vector space is an abelian
group), since this group will eventually act on the latent
space of our images, imposing additional structure will be
beneﬁcial. To do so, we will utilize a mapping between C’s
and a group suitable for our setting. Once this is accom-
plished, we will derive an equivariant encoder. We discuss
these steps next.
3. Methods
The dual goals of (i) equivariance to covariates (such as
age) and (ii) invariance to site, involves learning multiple
mappings. For simplicity, and to keep the computational
effort manageable, we divide our method into two stages.
Brieﬂy, our stages are (a) Stage one: Equivariance to Co-
variates. We learn a mapping to a space that provides the
essential ﬂexibility to characterize changes in covariate C
as a group action. This enables us to construct a space sat-
isfying the equivariance condition as per Def. 1(b) Stage
two: Invariance to Site. We learn a second encoding to
a generic vector space by apriori ensuring that the equiv-
ariance properties from Stage one are preserved. Such an
encoding is then tuned to optimize the MMD criterion, thus
generating a latent space that is invariant to site while re-
maining equivariant to covariates. We describe these stages
one by one in the following sections.
3.1. Stage one: Equivariance to Covariates
Given the space of images, X, with the covariates C,
ﬁrst, we want to characterize the effect of Con Xas a
group action for some group G. Here, an element gG
characterizes the change from covariate ciCto cjC
(for short, we will use iand j). The change in Ccorre-
sponds to a translation action which is difﬁcult to instanti-
ate in Xwithout invoking expensive conditional generative
models. Instead, we propose to learn a mapping to a latent
space Lsuch that the change in Ccan be characterized by
a group action pertaining to Gin the space L(the latent
space of X). As an example, let us say Xigoes to Xjin X
as (XiXj). This means that (XiXj)is caused due
to the covariate change (cicj)in C. Let Ebe a mapping
between the image space Xand the latent space L. In the la-
tent space L, for the moment, we want that (EXiEXj)
should correspond to the change in covariate (cicj).
Remark 2. We are mostly interested in normalized covari-
ates for example, in pnorm, while other volume based de-
terministic normalization functions may also be applicable.
In the simplest case of p= 2 norm, the corresponding group
action is naturally induced by the matrix group of rotations.
Based on this choice of group, we will learn an autoen-
coder (E,D)with an encoder E:X L and a decoder
D:L→X, here Lis the encoding space. Due to Re-
mark 2, we can choose Lto be a hypersphere, Sn1, and
(E,D)as a hyperspherical autoencoder [54]. Then, we can
characterize the “action of Con X” as the action of Gon
Sn1. That is to say that a covariate change (translation in
C) is a change in angles on L. This corresponds to a rotation
due to the choice of our group G. Note that for L=Sn1,
Gis the space of n×nrotation matrices, denoted by SO(n),
and the action of Gis well-deﬁned. What remains is to en-
courage the latent space Lto be G-equivariant. We start
with some group theoretic properties that will be useful.
3.1.1 Review: Group theoretic properties of SO(n)
Let SO(n) = XRn×n|XTX=In,det(X)=1be
the group of n×nspecial orthogonal matrices. The group
4
SO(n)acts on Sn1with the group action “·” given by
g·7→ g, for gSO(n)and Sn1. Here we use g
to denote the multiplication of matrix gwith . Under this
group action, we can identify Sn1with the quotient space
G/H with G=SO(n)and H=SO(n1) (see Ch. 3
of [13] for more details). Let τ:Sn1G/H be such
an identiﬁcation, i.e., τ() = gH for some gG. The
identiﬁcation τis equivariant to Gin the following sense.
Fact 3. Given τ:Sn1G/H as deﬁned above, τis
equivariant with the action of G, i.e., τ(g·) = ().
Next, we see that given two points i,jon Sn1there
is a unique group element in Gto move from τ(i)to τ(j).
Lemma 4. Given two latent space representations i,j
Sn1, and the corresponding cosets giH=τ(i)and
gjH=τ(j),!gij =gjg1
iGsuch that j=gij ·i.
Thanks to Fact 3and Lemma A.1, simply identifying a
suitable τwill provide us the necessary equivariance prop-
erty. To do so, next, we parameterize τby a neural network
and describe a loss function to learn such a τand (E,D).
3.1.2 Learning a G-equivariant τwith DNNs
Now that we established the key components: (a) an au-
toencoder (E,D)to map from Xto the latent space Sn1
(b) a mapping τ:Sn1SO(n)which is G=SO(n)-
equivariant, see Figure 3, we discuss how to learn such a
(E,D)and a G-equivariant τ.
Let Xi, Xj∈ X be two images with the corresponding
covariates i, j Cwith i6=j. Let i=E(Xi),j=
E(Xj). Using Lemma A.1, we can see that a gij Gto
move from ito jdoes exist and is unique. Now, to learn
aτthat satisﬁes the equivariance property (Fact 3), we will
need τto satisfy two conditions, τ(gij ·i) = gij τ(i)and
τ(gji ·j) = gj iτ(j)gG. The two conditions are
captured in the following loss function,
i=E(Xi)j=E(Xj)(2)
Lstage1=X
{(Xi,i),(Xj,j)}
⊂X ×C
kG(i, j )·τ(i)τ(j)k2+
kG1(i, j )·τ(j)τ(i)k2(3)
Here, G:C×CGwill be a table lookup given by
(i, j)7→ gij is the function that takes two values for the
covariate c, say, i, j corresponding to Xi, Xj∈ X and
simply returns the group element (rotation) gij needed to
move from E(Xi)to E(Xj).Choice of G:In general,
learning Gis difﬁcult since Cmay not be continuous. In
this work, we ﬁx Gand learn τby minimizing (3). We
will simplify the choice of Gas follows: assuming that
Cis a numerical/ordinal random variable, we deﬁne Gby
(i, j)7→ expm((ij)1m). Here m=n
2is the di-
mension of Gand expm is the matrix exponential, i.e.,
Algorithm 1 Learning representations that are Equivariant
to Covariates and Invariant to Site
Input: Training Sets from multiple sites (X, Y )site1 ,
(X, Y )site2. Nuisance covariates C.
Stage one: Equivariance to Covariates
1 : Parameterize Encoder-Decoder pairs (E,D)and τ
mapping with neural networks
2 : Optimize over (E,D)and τto minimize,
Lstage1+PikXiD(E(Xi))k2
Output: First latent space mapping Eand a supporting
mapping function τ. Here, τis G-equivariant to the co-
variates C(see Lemma (A.1) and (3)).
Stage two: Invariance to Site
1 : Parameterize encoder b, predictor hand decoder Ψ
with neural networks
2 : Preserve equivariance from stage one with an equiv-
ariant mapping Φ, (see Lemma (A.1))
3 : Optimize Φ, b, h and Ψto minimize Lstage2+MMD
Output: Second latent space mapping Φ. Here, Φis
equivariant to the covariates and invariant to site.
expm :so(n)SO(n), where so(n)is the Lie alge-
bra [21] of SO(n). Since so(n)is a vector space, hence
(ij)1mso(n). To reduce the runtime of expm,
we replace expm by a Cayley map [32,42] deﬁned by:
so(n)3A7→ (IA)(I+A)1SO(n). Here we used
expm for parameterization (other choices also suitable).
Finally, we learn the encoder-decoder (E,D)by using
a reconstruction loss constraint with Lstage1in (3). This
can also be thought of as a combined loss for this stage as
Lstage1+PikXiD(E(Xi))k2where the second term is
the reconstruction loss. The loss balances two terms and re-
quires a scaling factor (see appendix § A.7). A ﬂowchart of
all steps in this stage can be seen in Fig 3.
3.2. Stage two: Invariance to Site
Having constructed a latent space Lthat is equivariant to
changes in the covariates C, we must now handle the site
attribute, i.e., invariance with respect to site. Here, it will
be convenient to project Lonto a space that simultaneously
preserves the equivariant structure from Land offers the
ﬂexibility to enforce site-invariance. The following lemma,
inspired from the functional representations of probabilistic
symmetries (§4.2of [7]), provides us strategies to achieve
this goal. Here, consider Φ : L→Zto be the projection.
Lemma 5. For a τ:L → G/H as deﬁned above, and for
any arbitrary mapping b:L→Z, the function Φ : L→Z
deﬁned by
Φ() = τ()·bτ()1·(4)
is G-equivariant, i.e., Φ(g·) = gΦ().
5
Figure 4. t-SNE plots of latent representations τ().For ADNI (left) and Adult (right), an equivariant encoder ensures that the latent features are
evenly distributed and bear a monotonic trend with respect to the changes in the age covariate value. The non-equivariant space is generated from the Na¨
ıve
pooling baseline. Each color denotes a discretized age group. Age was discretized only for the ﬁgure to highlight the density of samples in each age group.
Proof is available in the appendix § A.1. Note that Φre-
mains equivariant for any mapping b. This provides us the
option to parameterize bas a neural network and train the
entirety of Φfor the desired site invariance where equivari-
ance will be preserved due to (9). In this work, we learn
such a Φ : L → Z with the help of a decoder Ψ : Z → L
by minimizing the following loss,
Lstage2=X
=E(X)∈L
X∈X ,Y ∈Y
Reconstruction loss
z }| {
kΨ(Φ())k2+
Prediction loss
z }| {
kYh(Φ())k2(5)
subject to Φ() = τ()·bτ()1·
| {z }
G-equivariant map
(6)
Minimizing the loss (5) with the constraint (6) allows learn-
ing the network b:L→Zand the decoder Ψ : Z → L.
We are now left with asking that Z∈ Z be such that the rep-
resentations are invariant across the sites. We simply use the
following MMD criterion although other statistical distance
measures can also be utilized.
MMD =kE
Z1
P(Φ())site1
K(Z1,·)E
Z2
P(Φ())site2
K(Z2,·)kH(7)
The criterion is deﬁned using a Reproducing Kernel Hilbert
Space with norm k·kHand kernel K. We combine (5), (6)
and (7) as the objective function to ensure site invariance.
Thus, the combined loss function Lstage2+MMD is min-
imized to learn ,Ψ). Scaling factor details are available
in the appendix § A.7.
Summary of the two stages. Our overall method com-
prises of two stages. The ﬁrst stage, Section 3.1, involves
learning the τfunction. The function learned in this stage is
G-equivariant by the choice of the loss Lstage1, see (3). Our
next stage, Section 3.2, employs the learned τfunction and
a trainable mapping bto generate invariant representations.
This stage preserves G-equivariance due to the Φmapping
in (9). The loss for the second step is Lstage2+MMD , see
(5). Our method is summarized in Algorithm 1. Conver-
gence behavior of the proposed optimization (of τ, Φ) still
seems challenging to characterize exactly, but recent papers
provide some hope, and opportunities. For example, if the
networks are linear, then results from [18] maybe applicable
which explain the our superior empirical performance.
4. Experiments
We evaluate our proposed encoder for site-invariance
and robustness to changes in the covariate values C. Eval-
uations are performed on two multi-site neuroimaging
datasets, where algorithmic developments are likely to be
most impactful. Prior to neuroimaging datasets, we also
conduct experiments on two standard fairness datasets, Ger-
man and Adult. The inclusion of fairness datasets in our
analysis, provides us a means for sanity tests and optimiza-
tion feasibility on an established problem. Here, the goal
of achieving fair representations is treated as pooling multi-
ple subsets of data indexed by separate sensitive attributes.
We begin our analysis by ﬁrst describing our measures of
evaluation and then reporting baselines for comparisons.
Measures of Evaluation. Recall that our method in-
volves learning τas in (3) to satisfy the equivariance prop-
erty. Moreover, we need to learn Φas in (9)–(5) to achieve
site invariance. Our measures assess the structure of the la-
tent space τ()and Φ(). The measures are: (a) Eq :This
metric evaluates the 2distance between τ(i)and τ(j)for
all pairs i, j. Formally, it is computed as
Eq =X
{(Xi,i),(Xj,j)}⊂X ×C
i=Ee(Xi),j=Ee(Xj)
|ij|kτ(i)τ(j)k2(8)
A higher value of this metric indicates that τ(i)and
τ(j)are related by the group action gij . Additionally,
we use t-SNE [48] to qualitatively visualize the effect of
τ.(b) Adv :This metric quantiﬁes the site-invariance
achieved by the encoder Φ. We evaluate if Φ()for a
learned ∈ L has any information about the site. A three
layered fully network (see appendix § A.6) is trained as an
adversary to predict site from Φ(), similar to [49]. A lower
value of Adv, that is close to random chance, is desirable.
(c) M:Here, we compute the MMD measure, as in (7),
on the test set. A smaller value of Mindicates better invari-
ance to site. Lastly, (d) ACC :This metric notes the test set
6
Figure 5. Statistical Analysis on the reconstructed outputs. The vox-
els that are signiﬁcantly associated with Alzheimer’s disease (p <0.001)
ferroni correction. A high density of signiﬁcant voxels indicates that our
method preserves disease related signal after pooling across scanners.
accuracy in predicting the target variable Y.
Baselines for Comparison. We contrast our method’s
performance with respect to a few well-known baselines.
(i) Na¨
ıve: This method indicates a na¨
ıve approach of pool-
ing data from multiple sites without any scheme to handle
nuisance variables. (ii) MMD [29]: This method mini-
mizes the distribution differences across the sites without
any requirements for equivariance to the covariates. The la-
tent representations being devoid of the equivariance prop-
erty result in lower accuracy values as we will see shortly.
(iii) CAI [49]: This method introduces a discriminator to
train the encoder in a minimax adversarial fashion. The
training routine directly optimizes the Adv measure above.
While being a powerful implicit data model, adversarial
methods are known to have unstable training and lack con-
vergence guarantees [40]. (iv) SS [55]: This method adopts
a Sub-sampling (SS) framework to divide the images across
the sites by the covariate values C. An MMD criterion is
minimized individually for each of the sub-sampled groups
and an average estimate is computed. Lastly, (v) RM [33]:
Also used in [31], RandMatch (RM) learns invariant repre-
sentations on samples across sites that ”match” in terms of
the class label (we match based on both Yand Cvalues) .
Below, we summarize each method and nuisance attribute
Correction Na¨
ıve MMD [29] CAI [49] SS [55] RM [33] Ours
Site 7 3 3 3 3 3
Covariates 7 7 7 3 3 3
Table 1. Baselines in the paper and their nuisance attribute correction.
We evaluate methods on the test partition provided with
the datasets. The mean of the metrics over three random
seeds is reported. The hyper-parameter selection is done on
a validation split from the training set, such that the predic-
tion accuracy falls within 5% window relative to the best
performing model [10] (more details in appendix § A.2).
4.1. Obtaining Fair Representations
We approach the problem of learning fair representations
through our multi-site pooling formulation. Speciﬁcally, we
consider each sensitive attribute value as a separate site. Re-
sults on two benchmark datasets, German and Adult [11],
are described below.
German Dataset. This dataset is a classiﬁcation prob-
lem used to predict defaults on the consumer loans in the
German market. Among the several features in the dataset,
the attribute foreigner is chosen as a sensitive attribute. We
train our encoder while maintaining equivariance with re-
spect to the continuous valued age feature. Table 2provides
a summary of the results in comparison to the baselines.
Our equivariant encoder maximizes the Eq metric indicat-
ing the the latent space τ()is well separated for different
values of age. Further, the invariance constraint improves
the Adv metric signifying a better elimination of sensitive
attribute information from the representations. The Mmet-
ric is higher relative to the other baselines. The ACC for all
the methods are within a 2% range.
if a person has an income higher (or lower) than $50Kper year. The dataset is biased with respect to gender, roughly, 1-in-5women (in contrast to 1-in-3men) are reported to make over$50K. Thus, the female/male genders are con-
sidered as two separate sites with age as a nuisance covari-
ate feature. As shown in Table 2, our equivariant encoder
improves on metrics Eq and Adv relative to all the base-
lines similar to the German dataset. In addition to the quan-
titative metrics, we visualize the t-SNE plots of the repre-
sentations τ()in Fig. 4(right). It is clear from the ﬁg-
ure that an equivariant encoder imposes a certain monotonic
trend as the Age values as varied.
4.2. Pooling Brain Images across Scanners
For our motivating application, we focus on pool-
ing tasks for two different brain imaging datasets where
the problem is to classify individuals diagnosed with
Alzheimer’s disease (AD) and healthy control (CN).
Setup. Images are pre-processed by ﬁrst normaliz-
ing and then skull-stripping using Freesurfer [15]. A linear
(afﬁne) registration is performed to register each image to
MNI template space. Images are trained using 3D convo-
lutions with ResNet [23] backbone (details in the appendix
§A.6). Since the datasets are small, we report results over
ﬁve random training-validation splits.
ADNI Dataset. The data for this experiment has been
Figure 6. Distribution of age covariate in the ADNI dataset. Two set-
tings are considered – (left) the intersection of the support is large, and
(right) with a smaller common support. Despite the mismatch of support
across scanner attributes, our approach minimizes the MMD measure (de-
sirable) on the test set relative to the na¨
ıve pooling method.
7
Eq :Equivariance Gap, Adv :Adversarial Test Accuracy, M:Test MMD measure, ACC :Test prediction accuracy
: Higher Value is preferred, : Lower Value is preferred
Eq Adv M ↓ ACC ↑ Eq Adv M ↓ ACC Eq Adv M ↓ AC C ↑ Eq Adv M ↓ ACC
Na¨
ıve 4.6(0.7) 0.62(0.03) 7.7(0.8) 74(0.9) 3.4(0.7) 83(0.1) 9.8(0.3) 84(0.1) 3.1(1.0) 59(2.9) 27(1.6) 80(2.6) 4.1(0.9) 49(8.4) 90(8.7) 83(4.4)
MMD [29]4.5(1.0) 0.66(0.04) 1.5(0.3) 73(1.5) 3.4(0.9) 83(0.1) 3.1(0.3) 84(0.1) 3.1(1.0) 59(3.3) 27(1.7) 80(2.6) 3.6(1.0) 49(11.9) 86(11.0) 84(6.5)
CAI [49]1.9(0.6) 0.65(0.01) 1.2(0.2) 76(1.3) 0.1(0.0) 81(0.7) 4.2(2.4) 84(0.04) 2.4(0.7) 61(2.1) 27(1.5) 74(3.6) 2.8(1.6) 56(6.9) 85(12.3) 82(5.1)
SS [55]3.8(0.5) 0.70(0.07) 1.5(0.6) 76(0.9) 2.8(0.5) 83(0.2) 1.5(0.2) 84(0.1) 3.7(0.5) 57(2.1) 26(1.6) 81(3.7) 3.4(1.3) 51(6.7) 88(14.6) 82(3.5)
RM [33]3.4(0.4) 0.66(0.04) 7.5(0.9) 74(2.1) 0.8(0.1) 82(0.4) 4.8(0.7) 84(0.3) 0.8(0.9) 52(5.4) 22(0.6) 78(3.8) 0.4(0.5) 40(4.7) 77(13.8) 84(5.3)
Ours 6.4(0.6) 0.54(0.01) 2.7(0.6) 75(3.3) 5.3(0.9) 75(1.4) 7.1(0.6) 83(0.1) 5.1(1.2) 50(4.2) 16(7.2) 77(4.8) 7.5(1.2) 49(7.3) 70(22.3) 81(1.8)
Table 2. Quantitative Results. We show Mean(Std) results over multiple run. For our baselines, we consider a Na¨
ıve encoder-decoder model, learning
representations via minimizing the MMD criterion [29] and Adversarial training [49], termed as CAI. We also compare against Sub-sampling (SS) [55]
that minimizes the MMD criterion separately for every age group, and the RandMatch (RM) [33] baseline that generates matching input pairs based on the
Age and target label values. The SS and RM baselines discard subset of samples if a match across sites is not available. The measure Adv represents the
adversarial test accuracy except for the German dataset where ROC-AUC is used due to high degree of skew in the data.
three scanner types in the dataset, namely, GE, Siemens
and Phillips. Similar to the fairness experiments, equivari-
ance is sought relative to the covariate Age. The values of
Age are in the range 50-95 as indicated in density plot of
Fig. 6(left). The Age distribution is observed to vary across
different scanners, albeit minimally, in the full dataset. In
the t-SNE plot, Fig. 4(left), we see that the latent space
has an equivariant structure. Closer inspection of the plot
shows that the representations vary in the same order as that
of Age. Different colors indicate different Age sub-groups.
Next, in Fig. 5, we present the t-statistics in the template
space on the reconstructed images after pooling. Here, the
t-statistics measure the association with AD/CN target la-
bels. As seen in the ﬁgure, the voxels signiﬁcantly associ-
ated with the Alzheimer’s disease (p < 0.001) are consid-
erable in number. This result supports our goal to combine
datasets to increase sample size and obtain a high power
in statistical analysis. Next, in Fig. 6(right), we increase
the difﬁculty of our problem by randomly sub-sampling for
each scanner group such that the intersection of support is
minimized. In such an extreme case, our method attains a
better Mmetric relative to the Na¨
ıve method, thus justify-
ing the applicability to situations where there is a mismatch
of support across the sites. Lastly, we inspect the perfor-
mance on the quantitative metrics on the entire dataset in
Table 2. All metrics Eq ,Adv and Mimprove relative to
the baselines with a small drop in the ACC.
ADCP dataset. This experiment’s data was collected
as part of the NIH-sponsored Alzheimer’s Disease Connec-
tome Project (ADCP) [1,25]. It is a two-center MRI, PET,
and behavioral study of brain connectivity in AD. Study in-
clusion criteria for AD / MCI (Mild Cognitive Impairment)
patients consisted of age 5590 years who retain decisional
capacity at initial visit, and meet criteria for probable AD or
MCI. MRI images were acquired at three sites. The three
sites differ primarily in terms of the patient demograph-
ics. We inspect the quantitative results of this experiment
in Tab. 2and place the qualitative results in the appendix
§A.4,A.5. The table reveals considerable improvements in
all our metrics relative to the Na¨
ıve method.
Limitations. Currently, our formulation assumes that the
to-be-pooled imaging datasets are roughly similar – there is
deﬁnitely a role for new developments in domain alignment
to facilitate deployment in a broader range of applications.
Secondly, larger latent space dimensions may cause com-
pute overhead due to matrix exponential parameterization.
Finally, algorithmic improvements can potentially simplify
the overhead of the two-stage training.
5. Conclusions
Retrospective analysis of data pooled from previous /
ongoing studies can have a sizable inﬂuence on identifying
early disease processes, not otherwise possible to glean
from analysis of small neuroimaging datasets. Our devel-
opment based on recent results in equivariant representation
learning offers a strategy to perform such analysis when
covariates/nuisance attributes are not identically distributed
across sites. Our current work is limited to a few such
variables but suggests that this direction is promising and
can potentially lead to more powerful algorithms.
Acknowledgments The authors are grateful to Vib-
hav Vineet (Microsoft Research) for discussions on
the causal diagram used in the paper. Thanks to Amit
Sharma (Microsoft Research) for the conversation on
their MatchDG project. Special thanks to Veena Nair
and Vivek Prabhakaran from UW Health for helping with
the ADCP dataset. Research supported by NIH grants to
UW CPCP (U54AI117924), RF1AG059312, Alzheimer’s
Disease Connectome Project (ADCP) U01 AG051216,
and RF1AG059869, as well as NSF award CCF 1918211.
Sathya Ravi was also supported by UIC-ICR start-up funds.
8
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A. Appendix
A.1. Proofs of theoretical results
In this section, we will provide the proofs of Lemma 4
and Lemma 5discussed in the main paper.
Lemma. Given two latent space representations i,j
Sn1, and the corresponding cosets giH=τ(i)and
gjH=τ(j),!gij =gjg1
iGsuch that j=gij ·i.
Proof. Given giH=τ(i)and gjH=τ(j), we use gij =
gig1
jGsuch that, gjH=gij giH.
Now using the equivariance fact (3) , we get,
gjH=gij giH
=τ(j) = gij τ(i)
=τ(j) = τ(gij ·i)
Now as τis an identiﬁcation, i.e., a diffeomorphism, we
get j=gij i. Note that Sn1is a Riemannian homoge-
neous space and the group Gacts transitively on Sn1, i.e.,
given x,ySn1,gGsuch that, y=g·x. Hence
from j=gij iand the transitivity property we can con-
clude that gij is unique.
Lemma. For a τ:L → G/H as deﬁned above, and a
mapping b:L→Z, the function Φ : L→Zdeﬁned by
Φ() = τ()·bτ()1·(9)
is G-equivariant, i.e., Φ(g·) = gΦ().
Proof. Let ∈ L. Consider the Φmapping of g·, that is
Φ(g·) = τ(g·)·bτ(g·)1·.
Using the fact (3) from the main paper, we have τ(g·
) = ()and τ(g·)1=τ()1g1. Substituting
these in Φ(g·), we get
Φ(g·) = ()·bτ()1g1g·
= ()bτ()1·
Thus, Φ(g·) = gΦ()
A.2. Details on Evaluation Metrics
Recall from Section 4of the paper, our discussion on
three metrics – Eq ,Adv and M. While Eq and M
are variants of distance measure on the latent space, Adv
assesses the ability to predict the nuisance attributes from
the latent representation (and is therefore probabilistic in
nature). Observe that Eq and Mare (euclidean) distance
measures and could be very different depending on the nor-
malization of the vectors. For our purposes of evaluating
these latent vectors/features in downstream tasks, we per-
form a simple feature normalization in order to obtain 01
latent vectors given by,
˜zi=zimin(zi)
max(zi)min(zi).(10)
Our feature normalization is composed of two steps: (i) cen-
tering – the numerator in (10) ensures that the mean of z
(along its coordinates) is 0; and (ii) scale – the denominator
projects the features zon the sphere at origin with radius
kzik
= max(zi)min(zi)0. Note that our scaling
step can be thought of as the usual projection in a special
case: when ziis guaranteed to be nonnegative (for exam-
ple, when zirepresent activations), then kzik
simply cor-
responds to a lower bound of the usual inﬁnity norm, kzk
(hence projection on a scaled ball). We adopt this nor-
malization only to compute Eq and Mmeasures, and not
for model training.
an adversarial neural network predicting the nuisance at-
tributes. We use a three-layered fully connected network
with batch normalization and train for 150 epochs. [34] uses
similar architecture for the adversaries with different hidden
layers of 0,1,2,3. We found that a three-layer adversary
is powerful enough to predict the nuisance attributes and
hence we use it to report the Adv measure.
as a public-private partnership, led by Principal Investi-
gator Michael W. Weiner, MD. ADNI was set up with an
objective to measure the progression of mild cognitive
impairment (MCI) and early Alzheimers disease (AD)
using serial magnetic resonance imaging (MRI), positron
emission tomography (PET), other biological markers. We
have three imaging protocol (scanner) types in the dataset,
namely, GE, Siemens and Phillips. The count of samples
AD/CN in each of these imaging protocols are provided in
Table 3. An example illustration (borrowed from [2]) of
using different scanner on the images is shown in Figure 8.
Preprocessing. All images were ﬁrst normalized and skull-
stripped using Freesurfer [15]. A linear (afﬁne) registra-
11
(a) Variation due to scanner for particular age group. (b) Variation due to covariates (age) in scanner 3.
Figure 7. Sample Images from ADCP dataset. (a) MRI images on control subjects from the ADCP dataset for different sites in the age group 70-80.
(b) Images obtained from Site 3for three extreme age groups. The gantt chart on top of the image indicates the respective age range in the other sites.
(a) GE (b) Siemens
Figure 8. Scanner effects on images. Two imaging protocols are shown:
(a) Siemens, (b) GE. The yellow region is the cortical ribbon segmenta-
tion, and the green circle shows that the imaging protocol from different
manufacturers have an effect on the scan. Image borrowed from [2].
tion was performed to register each image to MNI template
space.
Participants. The data for ADCP was collected through an
(ADCP) U01 AG051216. The study inclusion criteria for
AD (Alzheimer’s disease) / MCI (Mild Cognitive Impair-
ment) patients consisted of age between 55-90 years, will-
ing and able to undergo all procedures, retains decisional
capacity at initial visit, meets criteria for probable AD or
meets criteria for MCI.
Scanners. MRI images were acquired at three distinct sites
on GE scanners. T1-weighted structural images were ac-
quired using a 3D gradient-echo pulse sequence (repetition
time (TR) = 604 ms, echo time (TE) = 2.516 ms, inversion
time = 1060 ms, ﬂip angle = 8o, ﬁeld of view (FOV) = 25.6
cm, 0.8mm isotropic). T2-weighted structural images were
acquired using a 3D fast spin-echo sequence (TR = 2500
ms, TE = 94.398 ms, ﬂip angle = 90o, FOV = 25.6cm, 0.8
Table 3. Sample counts for ADNI dataset
Manufacturer=GE Medical Systems 44 78
Manufacturer=Philips Medical Systems 32 50
Manufacturer=Siemens 83 162
mm isotropic).
Preprocessing. The Human Connectome Project (HCP)
minimal preprocessing pipeline version 3.4.0[19] was fol-
lowed for data processing. This pipeline is based on FM-
RIB Software Library [27]. Next, the T1w and T2w images
are aligned, a B1 (bias ﬁeld) correction is performed, and
the subject’s image in native structural volume space is reg-
istered to MNI space using FSL’s FNIRT [4]. Only T1w
images in the MNI space were used for further analysis and
experiments.
Data Statistics. We plot the distributions of several at-
tributes in this dataset conditioned on the site. In Figure 10,
we show that the values of age and cognitive scores differ
across the three sites in this dataset. Cognitive scores are
computed based on an test assigned to the patients. Higher
scores indicate higher cognitive operation in the patient. Ta-
ble 4shows the sample counts for target variable of predic-
tion AD (Alzheimer’s disease) and Control group.
A.5. Visualizing the latent space
In the paper Figure 4, we have seen the latent space
τ()for the samples in the ADNI and the Adult datasets.
Here, we will see similar qualitative results for the
German and the ADCP dataset in Figure 9of the sup-
plement. In the plots, the latent representations for a
non-equivariant encoder are stretched thoughout the latent
space. In contrast, the representations of an equivariant
encoder, for a discretized value of Age, are localized to
speciﬁc regions. Further, these representations have a
monotonic behaviour with respect to the values of Age.
Table 4. Sample counts for ADCP dataset
site 1 10 39 29 20
site 2 10 33 30 13
site 3 5 19 14 10
12
(a) ADCP Dataset (b) German Dataset
Figure 9. t-SNE plots of latent representations of τ(). For both ADCP (left) and German (right), the the latent vectors of the equivariant encoder are
evenly distributed with respect to the age covariate value. The non-equivariant space is generated from the na¨
ıve pooling model. Different colors denote the
discretized set of age covariate value present in the data.
(a) Age (b) Cognitive Score
Figure 10. Distribution of attributes in the ADCP dataset. On the left
we observe the distribution of age for the three different sites present in the
ADCP dataset. On the right, we see the distribution of the cognitive scores.
The cognitive scores are computed based on a test that assesses executive
function. Higher scores indicate higher level of cognitive ﬂexibility. Both
age and cognitive scores are observed to vary across the sites.
Listing 1. Residual Block
BatchNorm3d
Swi s h
Conv3d
BatchNorm3d
Swi s h
Conv3d
Listing 2. Fully Connected Block
Flatten
Dropout
L i n e a r
BatchNorm1d
Swi s h
Dropout
L i n e a r
A.6. Hyper-parameters and NN Architectures
For tabular datasets such as German and Adult, our en-
coders and decoders comprise of fully connected networks
and a hidden layer of 64 nodes. The dimension of the quo-
tient latent space τ(i)is 30. Adam is used as a default
optimizer and the learning rate is adjusted based on the val-
idation set.
volutions and a ResNet architecture as the backbone. The
last layer is used to describe the quotient space τ(i). We
present the residual and the fully connected block below.
Detailed architectures can be viewed in the code.
A.7. Scaling factors
Recall from the Algorithm 1of the main paper that our
loss function for each stage comprises of reconstruction
and prediction losses in addition to the objectives concern-
ing equivariance and invariance. These multi-objective loss
functions require scaling factors that upweight one objec-
tive over the other. These scaling factors group up as hyper-
parameters for the Algorithm. In our experiments, it was
observed that the results were robust to a range of scaling
factor choices. For the results reported in Table 1of the
paper, they were identiﬁed through cross-validation. Here
we provide an example for the scaling factors used for the
the code for the scaling factors of other datasets.
Stage one: Equivariance to Covariates
Equivariance Loss Lstage1
Scaling Factor : 1.0
Reconstruction Loss PikXiD(E(Xi))k
Scaling Factor : 0.02
Stage two: Invariance to Site
Invariance Loss MMD
Scaling Factor : 0.1
Prediction Loss kYh(Φ())k2
Scaling Factor : 1.0
Reconstruction Loss kΨ(Φ(`))k2
Scaling Factor : 0.1
We refer the reader to Algorithm 1and Section 3of the main
paper for the details on the notations used above.
13
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