Tensor-based Subspace Factorization for StyleGAN

Abstract

In this paper, we propose τ\tauGAN a tensor-based method for modeling the latent space of generative models. The objective is to identify semantic directions in latent space. To this end, we propose to fit a multilinear tensor model on a structured facial expression database, which is initially embedded into latent space. We validate our approach on StyleGAN trained on FFHQ using BU-3DFE as a structured facial expression database. We show how the parameters of the multilinear tensor model can be approximated by Alternating Least Squares. Further, we introduce a tacked style-separated tensor model, defined as an ensemble of style-specific models to integrate our approach with the extended latent space of StyleGAN. We show that taking the individual styles of the extended latent space into account leads to higher model flexibility and lower reconstruction error. Finally, we do several experiments comparing our approach to former work on both GANs and multilinear models. Concretely, we analyze the expression subspace and find that the expression trajectories meet at an apathetic face that is consistent with earlier work. We also show that by changing the pose of a person, the generated image from our approach is closer to the ground truth than results from two competing approaches.

Full text

Tensor-based Subspace Factorization for StyleGAN
Ren´
e Haas, Stella Graßhof and Sami S. Brandt
IT University of Copenhagen, Copenhagen, Denmark
Abstract— In this paper, we propose τGAN a tensor-based
method for modeling the latent space of generative models.
The objective is to identify semantic directions in latent space.
To this end, we propose to fit a multilinear tensor model
on a structured facial expression database, which is initially
embedded into latent space. We validate our approach on
StyleGAN trained on FFHQ using BU-3DFE as a structured
facial expression database. We show how the parameters of the
multilinear tensor model can be approximated by Alternating
Least Squares. Further, we introduce a stacked style-separated
tensor model, defined as an ensemble of style-specific models
to integrate our approach with the extended latent space of
StyleGAN. We show that taking the individual styles of the
extended latent space into account leads to higher model
flexibility and lower reconstruction error. Finally, we do several
experiments comparing our approach to former work on both
GANs and multilinear models. Concretely, we analyze the
expression subspace and find that the expression trajectories
meet at an apathetic face that is consistent with earlier work.
We also show that by changing the pose of a person, the
generated image from our approach is closer to the ground
truth than results from two competing approaches.
I. INTRODUCTION
In this paper, we propose a novel framework for finding
semantic directions in the latent space of Generative Ad-
versarial Networks (GANs) [10]. GANs have, since their
proposal, emerged as one of the most dominant approaches
for unsupervised representation learning in Computer Vision
and beyond [23].
Architecturally GANs refer to the simultaneous training
of two neural networks: a generator and a discriminator.
The generator produces images by sampling from its latent
space, while the discriminator, a binary classifier, tries to
discriminate the generated images from the training images.
The goal of training is to reach the equilibrium of the min-
max game between the two adversaries, such that neither can
improve by changing the parameter values. At equilibrium,
the discriminator can be discarded, and the generator can
then be used to produce new data by sampling from the latent
distribution. The new data points follow the same statistics as
the training data but are not contained in it. Modern state-
of-the-art GAN variations have borrowed from the Style-
transfer literature [14], [22] to disentangle the latent space
and synthesize high-quality face images. Work by [17], [18],
and most recently [16], showed how to a train state-of-the-art
StyleGAN model, even in cases of limited data.
A recent goal has been to find semantically interpretable
directions in GAN latent spaces, and several approaches
for semantic face editing have been proposed. Semantic
face editing refers to the ability to change various semantic
input reconstructed rotated
estimate
parameters
edit
parameters
Fig. 1: Overview of the proposed approach.
attributes, such as identity, expression, and rotation, gender,
of the generated images. Early work used an information
criterion (InfoGAN) [6] to determine semantic directions.
However, as pointed out in [8], there is no guarantee that
the latent codes produced by this method are semantically
meaningful. Additional unsupervised approaches for finding
semantic directions in StyleGAN include Principal Compo-
nent Analysis (PCA) on sampled latent codes [15] and the
closed-form factorization suggested by [25].
A recent approach for finding semantic directions in
StyleGAN in a supervised fashion is to train binary linear
classifiers (SVMs) to detect single binary semantic attributes
such as smile vs. no smile, male vs. female, glasses vs. no
glasses. For a given semantic attribute, the semantic direction
could then be defined as the normal to the supporting hyper-
planes of the trained SVM [24].
In the literature, a wide collection of multilinear meth-
ods have been proposed to model and analyze faces and
expressions. Early, PCA or dictionary-based 3D Morphable
Models (3DMM) [3], [9] capture the variation in shape and
texture of neutral 3D faces. Recently 3DMMs have also
been used to make semantic edits to images generated by
StyleGAN [27]. More recently, factorization methods, based
on higher-order data representations, were introduced with
the benefit of better disentanglement of dimensions, such
as person-specific shape and expression, when compared to
matrix methods [28], [30]. These models were built on the
Higher-Order Singular Value Decomposition (HOSVD) to
factorize the data, and have successfully been used to model
faces, their 3D reconstruction, as well as in transferring
expressions [4], [5]. Moreover, in [11], [12] a HOSVD
tensor model was constructed from the Binghamton 3D
facial expression database (BU-3DFE) [33], which revealed
a practically planar expression subspace, in which the six
basic emotions form one-dimensional affine subspaces [11].
These six lines intersect in a common vertex, the origin of
expressions, which surprisingly does not represent the neutral
face, but an extrapolated expression referred to as apathetic.
The main novelty of this work is to use a multilinear
face model to analyze the latent space of GANs. More
specifically, we propose to use the HOSVD to factorize the978-1-6654-3176-7/21/$31.00 ©2021 IEEE

z∈ Z
FC
FC
FC
. . .
Mapping Network
f:Z → W
w∈ W
conv
conv
conv
conv
Synthesis Network
g:W → X
x∈ X
.
.
.
Latent Space ZAuxiliary
Latent Space WImage Space X
Fig. 2: Architecture of the StyleGAN generator.
latent space into semantically meaningful linear subspaces
that yield a multilinear tensor model. Given an input image,
we estimate the model parameters to approximate the input,
and then change one attribute, such as rotation, as illustrated
in Fig. 1.
The main contributions of this paper are as follows:
•We propose a novel method for semantic face editing
with StyleGAN.
•We propose a method to estimate model parameters
and present reasonable regularization, enabling stable
parameter transfer.
•We show that expression trajectories intersect at a
unique point, corresponding to the origin of expressions,
which differs from the neutral face confirming the
earlier findings [11], [12] based on BU-3DFE.
•We propose an extended model, based on style separa-
tion, which leads to greater model flexibility and lower
reconstruction error for independent test images.
The paper is organized as follows: In Sec. II we will
review the architecture and outline the process on how to
embed reference images into the latent space of StyleGAN.
In Sec. III we present our Tensor-Based GAN model which
we build ”on top of” the StyleGAN latent space. Here we will
also elaborate on how we can approximate model parameters
for a given latent vector. Experiments and results of our
proposed approach are presented in Sec. IV, followed by
a summary and conclusion in Sec. V.
II. STY LEGAN
In this section, we will review the StyleGAN architecture
and explain how to embed reference images into the latent
space of the pre-trained models released by Nvidia [17], [18].
A. StyleGAN Architecture
The StyleGAN generator G:Z → X , where G=g◦f,
is composed of two networks, the mapping network f:Z →
Wand the synthesis network g:W → X , see Fig. 2. The
mapping network f, maps the latent vector z∈ Z onto the
auxiliary latent space Wto the vector w=f(z)while the
synthesis network g:W → X maps the vector w∈ W
to the final output image x∈ X in image space. The full
generator Gthus maps the latent vector zto an image x.
The notation used in this paper is summarized in Tab. I.
ResNet
input: x
b
w0
g(b
w0) = b
x0
VGG16 loss b
wi
g(b
wi) = b
xi
initial: b
x0
x
b
xi
repeat
Fig. 3: Diagram illustrating image embedding into the aux-
iliary latent space W.
B. Generator Inversion
GANs do not include an encoder as part of their archi-
tecture. Therefore, a goal in GAN research has been to
find a method for finding a latent code that produces an
image as close as possible to a given reference image, which
we refer to as embedding an image into the latent space.
The problem can be considered as inverting the synthesis
network g−1:X → W [1], [21] while inverting G, and
thereby embedding into Zspace, has been investigated in
[18]. Contemporary techniques for Wspace embedding, i.e.
finding g−1, use a VGG network [26]. Our approach for
embedding onto the auxiliary latent space Wis illustrated
in Fig. 3. The inverse generator G−1:X → Z yields the
latent vector z=G−1(x)with G−1=f−1◦g−1for the
input image x.
The initial estimate for the auxiliary latent vector for
a given reference image is computed as follows. We use
the pre-trained weights of StyleGAN [17] and the recently
revised StyleGAN2 [18] architecture. Then, as proposed in
[2], we train a ResNet [13] in a supervised setting using
synthetic StyleGAN data to approximate g−1that yields the
initial estimate ˆ
w0for the latent vector. The refinement for
the auxiliary latent vector is computed by first using the
VGG16 network [26], pre-trained on ImageNet database, and
then removing the classification layer, hence the truncated
network produces a high dimensional feature vector for a
given input image, as described in [34]. Since the trained
generator is fully differentiable, the loss can be calculated
in VGG space and gradients back-propagated through the
generator, hence we can iteratively update the latent code.
This approach is also used in [21]. We also found that
using the ResNet estimate as initialization for the VGG
optimization process, leads to faster convergence than not
using ResNet initialization.
III. MULTILINEAR MODEL
This section introduces τGAN, our latent space factor-
ization method for GANs that augments the StyleGAN
synthesis network gwith a multilinear tensor model. We
do this by embedding a facial expression database into the

X W Q
x
e
x
b
w
e
w
b
q
e
q
g−1τ−1
τ
g
e
d
i
t
p
a
r
a
m
e
t
e
r
s
Fig. 4: Overview of the different spaces and how the function
relate them, c.f. Tab. I. The blue line indicates a manual
change of one of the parameter vectors for transfer of person,
expression or rotation.
TABLE I: Overview of the notation used in this work.
Symbol Description
XImage Space
ZLatent Space
WAuxiliary Latent Space
QParameter Space
Operator Name
f:Z → W Mapping Network
g:W → X Synthesis Network
g−1:X → W StyleGAN Embedder
τ−1:W → Q Parameter Estimator
τ:Q→W Tensor Model
auxiliary latent space Wof StyleGAN. We then order the
embedded database into a tensor, which we factorize into
semantic subspaces. The resulting parameter space Qwill
thus be the Cartesian product of the semantic subspaces
Q=QP× QE× QR, where QPis the person space, QE
the expression space, and QRis the rotation subspace. An
overview of the different spaces and how the operators relate
them are displayed in Fig. 4and Tab I.
A. Tensor Factorization
The Higher-Order Singular Value Decomposition
(HOSVD) is a generalization of the matrix SVD to
higher-order tensors [7], [32], [11], [29], [28], [19].
The starting point for our analysis is a standardized data
tensor T∈RN×P×E×R, where Nrefers to the number of
elements in the latent vector, Pis the number of people, E
the number of expressions, and Rnumber of viewpoints or
rotations. Using the HOSVD Tcan then be factorized as
T≃C×1U1×2U2×3U3×4U4,(1)
where ×kdenotes the k-way product, C∈R˜
Nט
Pט
Eט
Ris
the core tensor, and U1∈RNט
N,U2∈RPט
P,U3∈
REט
E,U4∈RRט
Rare matrices with orthonormal columns
constructed from the singular vectors of the k-mode matrix
unfoldings of T. In general we have that ˜
N≤N,˜
P≤P,
˜
E≤E, and ˜
R≤R.
B. Multilinear Tensor Model for GANs
The HOSVD (1) factorizes the data tensor into a core
tensor, and a set of factor matrices Ui, one for each subspace.
By selecting appropriate rows from Ui,i= 2,3,4, one
normalized latent vector, i.e. a single mode-1 fiber of T,
can be recovered. For example, to recover the latent vector of
person pperforming expression ewith rotation r, the pth row
of U2,eth row of U3, and rth row of U4is selected. This can
be conveniently formulated by a canonical basis, where the
parameter vectors q′
2∈RP,q′
3∈REand q′
4∈RRpick a
weighted linear combination of the rows of the Uimatrices.
Therefore, a given latent code y′can be approximated by
the model prediction b
y′as
b
y′=C×1U1×2q′
2
TU2×3q′
3
TU3×4q′
4
TU4.(2)
This expression can be further simplified by defining qT
i≡
q′T
iUiand analogously b
y=UT
1b
y′. Now applying ×1UT
1to
both sides of (2) and recalling that the columns the respective
Umatrices are orthonormal we can write a more compact
model representation as
b
y=C×2qT
2×3qT
3×4qT
4,(3)
where the unprimed coordinates refer to the latent code in the
eigenspace spanned by the columns of the Uimatrices. In
this formulation, we have 3 individual parameter vectors and
use repeated n-mode products to relate these to the model
prediction.
We can rewrite (3) in a more general form to illustrate
the mathematical structure of our model. Let us define the
P×E×R, rank-1 parameter tensor Q=qT
2⊗qT
3⊗qT
4,
where ⊗refers to the tensor product. Then the components
of the rank-1 parameter tensor Q∈RP×E×Ris given by
Qνρλ =q(2)
νq(3)
ρq(4)
λwhere q(k)
νrefers to the νth component
of the subspace vector qk∈ Qkfor k={2,3,4}.
With this definition, we can write (3) in a more compact
and convenient representation using the Einstein summation
convention
ˆ
Yµ=Cµνρλ Qνρλ.(4)
This lets us write the latent code, in the auxiliary latent
space W, as an application of the multilinear map, defined
by the core tensor C, on the parameter tensor Q.
Our entire tensor model τcan thus be written as the
composite map of the core Cfollowed by the change-of-
basis transformation defined by U1:W → W, and the
inverse standardization operator Ω−1:W → W, where Ω−1
translates and scales a latent vector back to the original scale
of Wspace according to the mean and variance of the BU-
3DFE data.
C. Stacked Style-Separated Model
In addition to the previously presented model, we propose
an alternative approach, where styles are separated instead
of vectorizing the latent code. That is, we interpret the
Sstyles of was separate vectors of dimension L, which
is also indicated in Fig. 2. To separate the Sstyles, we

propose to order the latent codes into the data tensor Tstyle ∈
RS×L×P×E×R.
Then the shape dimension can be addressed separately by
defining the style-specific tensors
Ts∈RL×P×E×R, s = 1,2,· · · , S. (5)
We factorize each style-specific tensor Ts, and define style-
specific tensor model τs. The ensemble of these models is
referred to as the stacked style-separated model τS, which
has S(P+E+R)parameters. In conclusion, while the prior
vectorized model τ, based on T, has P+E+Rparameters,
this formulation τShas S(P+E+R)parameters since it
models the style separately.
D. Optimization
Our next aim is to estimate the model parameters by
constructing the estimator τ−1:W → Q. The estimator
is defined as the solution to the optimization problem
min
Q||b
y−y||2
2subject to
||qi||2
2≤c2and
||Uiqi||1≤c1for i= 2,3,4.
(6)
The form of (6) is inspired by [11], [12], and enforces
constraints on the model parameters to retrieve a stable
representation of new latent vectors by linear combinations
within the training data. We regularize the model using
Ridge and Lasso regression. Then the Lagrangian for the
constrained problem (6) can be written as
L(Q, λ1, λ2) = ||b
y−y||2
2
+
4
X
k=2
λ2,k ||qk||2
2+λ1,k ||q′
k||1
(7)
where λ1,k, λ2,k ≥0refer to regularization parameters,
i.e. Lasso and Ridge. Note that there is no prime on the
Ridge term since ||q′
i||2
2= (UT
iqi)T(UT
iqi) = ||qi||2
2since
UT
iUi=I. We will now continue to present a strategy
for solving the constrained optimization problem in (6) by
Alternating Least Squares.
As in [11], [12] the minimization can be solved by first
rewriting (3) as a matrix-vector multiplication separately for
each of the three model parameter vectors as
ˆ
y=A(k)qk, k = 2,3,4,(8)
where the matrices A(k)are given by
A(2) =C×3qT
3×4qT
4,(9)
A(3) =C×2qT
2×4qT
4,(10)
A(4) =C×2qT
2×3qT
3.(11)
Therefore, an unknown latent vector ycan be estimated
by alternating between the systems (8), while updating the
matrices A(k)in each step.
IV. EXP ERI MEN TS
In the following, we give some additional details for the
BU-3DFE database and continue to report on our experimen-
tal results.
Fig. 5: Validation of decomposition results. Energy of sin-
gular values for each mode of T.
A. Facial Expression Database
As mentioned in the introduction, we use the BU-3DFE
database [33]. The database contains 3D face scans and
images of 100 persons (56 female and 44 male), with varying
ages (18-70 years) and diverse ethnic/racial ancestries. Each
subject was asked to perform the six basic emotions: anger,
disgust, happiness, fear, sadness, and surprise, each with
four levels of intensity. Additionally, for each participant,
the neutral face was recorded. Hence, for each person, there
are a total of 25 facial expressions recorded from two pose
directions, left and right, resulting in 5000 face images.
B. Data Prepossessing
As a pre-processing step, we embedded each face im-
age from the BU-3DFE database, into the latent space of
StyleGAN, as described in Sec. II-B. We then collected the
resulting latent vectors into the 4-way data tensor T0∈
RN×P×E×R. We then calculated the mode-1unfolding
T(1)
0∈RN×P ER of T0containing all the P E R latent
vectors. We then standardized this matrix to zero mean and
unit variance for each latent variable and then finally folded
this standardized matrix into a N×P×E×Rdimensional
tensor Twhich we used for all subsequent experiments.
C. Subspace Analysis
The standardized tensor Twas factorized by the HOSVD,
as described in (1), yielding the four subspaces spanned by
the columns of Uk,k= 1,...,4. The distribution of the
energy of the subspaces is shown in Fig. 5, which illustrates
the compactness of the subspaces.
In Fig. 6we show a visualization of the expression
subspace. As an initial step, we truncated the expression
subspace from 25 dimensions to 3D. It can be seen that
for each emotion, the variation in expression strength forms
linear trajectories in expression space. These trajectories are
star-shaped and meet at an origin of expression which is
shared by all emotion trajectories. This is neither the neutral
nor the mean face, but the “apathetic” face, found in [11],
[12], see Fig. 7(a)-(c). In this case, the apathetic face in
Fig. 7(c) is closer to the mean face than in [11], [12],
displayed in Fig. 7(f) for comparison.
D. Vectorized vs. Stacked Style-Separated Model
In Sec. III we proposed to build two different versions of
tensor models. (1) The vectorized model flattens each latent
code of one image and then orders them into the tensor
T∈RN×P×E×R, and (2) the stacked style-separated model
Tstyle ∈RS×L×P×E×Rwhich considers the S= 18 styles of
StyleGAN separately. We estimated the parameters for the

Fig. 6: Projection of the expression subspace, defined by U3,
onto 3 dimensions.
(a) Mean (b) Neutral (c) Apathy
(d) Mean (e) Neutral (f) Apathy
Fig. 7: Synthesized faces for (a) the mean face, (b) neutral
face, and (c) apathetic face. Accordingly, (d), (e), (f) show
the 3D faces synthesized by the method in [11].
two models, using the ALS procedure (8). The results are
illustrated in Fig. 8. It can be seen, that the ground truth
(Fig. 8a), is visually closer to the stacked style-separated
model (Fig. 8c) than the vectorized model (Fig. 8b) for
test images from the BU-3DFE data set (top row), as well
as for arbitrary images (2nd and 3rd row). We conclude
that the proposed adaptation by the separate styles improves
performance.
E. Validation of Regularization Parameters
The optimization problem defined in (7) contains six
regularization parameters λ1,k and λ2,k,k= 2,3,4, two for
each of the three parameter vectors, which must be manually
set. In the following experiment we investigated how the
hyperparameters influenced the quality of the results, and
assume that they are the same for the three parameters, hence
λ1=λ1,k, and λ2=λ2,k . Here we used the vectorized
model on the basis of the standardized latent codes in (3).
Initially, we divided the data into a training, validation, and
test set by a randomized 90–5–5 split over the P= 100
person identities. The validation set thus had a total of
5ER = 250 samples. We estimated the tensor model based
(a) Reference (b) Vectorized (c) Style-Separated
Fig. 8: Reconstructions: (a) Ground truth images, and the
results from either (b) the vectorized model, and (c) the style-
separated model. The top row shows an example from the
BU-3DFE database, while the 2nd and 3rd rows illustrate
reconstruction of novel images which are not part of BU-
3DFE.
on the training set. For each latent vector in the validation
set we then estimated the subspace parameters qiby ALS
using (8).
We evaluated three kinds of errors for the validation set:
the approximation error, and the expression and rotation
transfer errors. The approximation error between the ground
truth yand estimated latent code b
yiis defined as ϵapprox =
||b
y−y||2
2. The transfer errors result from exchanging es-
timated parameters b
qkby known values e
qk. Hence using
e
yexpr ≡τ(b
qperson ⊗e
qexpr ⊗b
qrot)gives rise to an expression
transfer error which we define as ϵexpr =||e
yexpr −y||2
2.
Analogously, the rotation transfer error is defined as the
error arising from only changing the parameters associated
with the rotation subspace according to ϵrot =||e
yrot −y||2
2.
The three error metrics ϵapprox,ϵexpr , and ϵrot were then
calculated for each sample, with varying hyperparameter
values λ1and λ2. In this experiment, we investigate Lasso
and Ridge regression independently, i.e., we set λ1= 0 while
varying λ2, and vice versa. We restrict ourselves to only
consider cases where the regularization strength is equal for
all subspaces.
The results are illustrated in Fig. 9. In general, it can be
seen that the approximation error is more stable than the
other two errors. Fig. 9a suggests that high values of λ1
should be chosen for rotation transfer, while for expression
transfer λ1≈1seems to be a reasonable choice. Fig. 9b
reveals that for λ2≈1all error metrics are small, and hence
this interval is a good choice.

(a) Lasso (L1 penalty)
(b) Ridge (L2 penalty)
Fig. 9: Influence of the hyper parameters, λ1and λ2steering
the (a) Lasso and (b) Ridge constraints, on (from top to
bottom row) the approximation error, expression transfer
error, and rotation transfer error.
F. Regularization and Parameter Transfer
We used the regularization parameters above to perform
expression and rotation transfer on samples from the test set.
We then synthesized images from the estimated parameters
by applying the composite transformation b
x=g(τ(b
Q))
to the estimated subspace parameters b
Q. Additionally, we
performed expression and rotation transfer by replacing one
of the three estimated parameter vectors by known values,
as described before. We did this for the regularized model
(λ1>0, λ2>0) and the non-regularized model (λ1=
λ2= 0). Fig. 10 shows how well the ground truth, in W
space, (Fig. 10a) can be approximated by the non-regularized
solution (Fig. 10b) and the regularized solution (Fig. 10c). It
seems that the non-regularized solution matched the ground
truth slightly better with respect approximation expression
transfer. However, for rotation transfer (Fig. 10e) the reg-
ularized solution clearly outperformed the non-regularized
solution. Because in the non-regularized solution the result-
ing image is not recognizable as a face anymore at all, while
the regularized solution is not deformed and the rotation of
the depicted faces conform to ground truth. This experiment
thus showed that adding a small L2 regularization term yields
stable rotation transfer.
(a) Ground Truth (b) Non-regularized (c) Regularized
(d) Expression Transfer (e) Rotation Transfer
Fig. 10: Reconstruction and regularization results. (a) Ground
truth (b) approximation by the non-regularized model, and
(c) the regularized model. (d,e) Results from rotation and
expression transfer containing ground truth (top row), the
non-regularized solutions (middle row), and the regularized
solution (bottom row).
Fig. 11: To find the optimal interpolation strength αfor
rotation transfer for InterFaceGAN [24] and GANSpace [15]
we compare the images generated by shifting the latent code
corresponding to an image from the one rotation towards the
other and compare the result with the ground truth.
G. Quantitative Comparison
Finally, we compare τGAN to InterFaceGAN [24] and
GANSpace [15] for the application of semantic face editing
by using rotation transfer as one example.
Since the BU-3DFE database [33], see Sec. IV-A, contains
5000 faces images, 2500 from the left and from the 2500
right; we chose one of the two views as the reference image,
and then used InterFaceGAN, GANSpace and τGAN to
estimate a reconstruction of the image from the comple-
mentary rotation. The resulting image was then compared to
the Ground Truth (GT) by 1) Pearson correlation coefficient
(pcorr), 2) Structural Similarity Index Measure (SSIM) [31],
and 3) Learned Perceptual Image Patch Similarity (LPIPS)
[34]. For the LPIPS measure, we employed two versions:
one based on VGG [26], referred to as lpips-vgg, and the
other, lpips-alex, on AlexNet [20].
In InterFaceGAN [24] the authors find semantic directions

of StyleGAN by fitting SVMs to single semantic attributes
using an annotated data set. Using these directions, semantic
editing can be performed by interpolating in the direction
n∈RNdefined by the SVM hyper-plane normal vector for
a given latent code w∈ W, as
wedit =w+αn,(12)
where αis the strength of the shift in semantic direction
associated with n. To perform rotation transfer, we chose the
pose direction for the StyleGAN1 model trained on FFHQ
provided by [24] as n.
GANSpace finds semantic directions in an unsupervised
fashion using PCA. The semantic meaning of the found
principal components needs to be assigned by a one-time
manual labeling. In the paper the authors report that the
10th principal component applied only to the first 7 layers
produces a shift in rotation for the pretrained StyleGAN1
network. Using this definition, and the rotation direction, we
can perform semantic edits with GANSpace in a similar way
as in eq. 12.
To determine the optimal interpolation strength αfor
both methods, we design an experiment where we perform
rotation transfer with varying values for α. From the latent
code representing an image of one rotation, we edit the latent
code towards the complementary rotation resulting in a latent
vector wedit which is then used to synthesize an edited image.
We then compare the edited image to the ground truth using
the four metrics mentioned above. For each value of αwe
average the metrics and pick the minimum. The results are
presented in Fig. 11, where it can be seen that the best
performance for InterFaceGAN is reached at α= 2.77, and
for GANSpace at α= 1.66, respectively. These values are
used for the quantitative comparison presented in Fig. 13.
To perform rotation transfer with τGAN model, we first
estimated the model parameter vectors b
qk,k= 2,3,4for a
given input image as described in Sec. III-D. Then we used
the rotation subspace defined by U4in (1). For τGAN we
take the subspace direction m=u(4)
2−u(4)
1∈ QR, where
u(4)
1,u(4)
2are the first and second row of U4, respectively.
The rotation parameter was then changed as
e
q4=b
q4+γm,(13)
which then yields the edited latent code
wτ,edit =τ(b
q2⊗b
q3⊗e
q4).(14)
Fig. 12 shows synthesized images produced by InterFace-
GAN, GANSpace and τGAN, respectively. These are com-
pared against the reconstructions generated by latent codes
interpolated directly in Wspace by w=βwleft + (1 −
β)wright where wleft and wright refer to the left and right
rotation, respectively. The results show that τGAN provides
an alternative way for generating rotation in the StyleGAN
latent space. Compared to InterFaceGAN, our model seems
to create rotations which better preserve features like skin
tone and gaze direction, and compared to GANSpace the
face shape seems better preserved. However, for all methods
(a) Direct Interpolation
(b) InterFaceGAN
(c) GANSpace
(d) τGAN (Ours)
Fig. 12: Comparison of rotation transfer among varying
methods. The ground truth images in pixel space are shown
in the top row in the outermost columns. We use the latent
code corresponding to the left hand rotation (top left) and try
to recover the right hand rotation (top right). The provided
images have been created by: (a) direct interpolation, (b)
InterFaceGAN, (c) GANSpace, and (d) our proposed τGAN.
we note that the identity of the person slightly changes in
this example.
Additionally, we objectively compare the quality of ro-
tation transfer resulting from different methods as follows.
We apply the previously introduced three methods: Inter-
FaceGAN, GANSpace, and our proposed τGAN, to shift the
rotation of the 125 left-oriented images in the validation set
towards the right orientation. We then compare the edited
images to the known ground truth using the same four
metrics introduced at the beginning of this section. The
results in Fig. 13 show that τGAN has the lowest median
value for all metrics when compared with InterFaceGAN and
GANSpace.
V. C ONCLUSIONS
In this work, we proposed τGAN, a tensor-based model
for the auxiliary latent space of the StyleGAN. It is con-
structed by first embedding the images of the BU-3DFE
database into the latent space of StyleGAN. The latent codes
were stored into a tensor which is then factorized into
semantically meaningful subspaces by HOSVD. This con-
struction ensured that the semantic directions were directly
interpretable in contrast to unsupervised methods, where this

Fig. 13: Quantitative comparison of rotation transfer per-
formed by varying methods. We start with images from
the left rotation and shift the latent codes towards the right
rotations using τGAN, InterFaceGAN, and GANSpace. The
edited images are then compared to the GT based on the
previously used adapted metrics, redefined to be the lower
the better. We observe that the edited images produced by
τGAN are more similar to the GT across all four metrics.
is not always the case.
We were able to generalize previous results [11] of face
analysis by showing that the expression subspace has the
structure where the expression trajectories meet in a specific
apathetic expression, which is different from the mean or
neutral face. We evaluated our approach quantitatively and
qualitatively, and compared different versions of the pro-
posed tensor models on the basis of approximation of unseen
samples, and demonstrated the stability in the transfer of
expression and rotation. From the results, we conclude that
the proposed approach is a powerful way for characterizing
and parameterizing the latent space of StyleGAN.
The current setting assumes complete data that contains
measurements of all the people performing the same ex-
pressions from each rotation without any missing data. This
requirement could be relaxed by low-rank completion meth-
ods that is left for future work. To conclude we employed a
model trained on FFHQ, and received promising results on
the BU-3DFE data set.
REFERENCES
[1] R. Abdal, Y. Qin, and P. Wonka. Image2stylegan: How to embed
images into the stylegan latent space? In Proc. ICCV, pages 4431–
4440, 2019.
[2] P. Baylies. Stylegan encoder - converts real images to latent space.
https://github.com/pbaylies/stylegan-encoder/,
2019.
[3] V. Blanz and T. Vetter. A morphable model for the synthesis of 3d
faces. In Proc. SIGGRAPH, pages 187–194, 1999.
[4] A. Brunton, T. Bolkart, and S. Wuhrer. Multilinear Wavelets: A
Statistical Shape Space for Human Faces. In Proc. ECCV, pages 297–
312, Jan. 2014.
[5] C. Cao, Y. Weng, S. Zhou, Y. Tong, and K. Zhou. FaceWarehouse:
A 3D Facial Expression Database for Visual Computing. IEEE
Transactions on Visualization and Computer Graphics, 20(3):413–425,
Mar. 2014.
[6] X. Chen, Y. Duan, R. Houthooft, J. Schulman, I. Sutskever, and
P. Abbeel. Infogan: Interpretable representation learning by in-
formation maximizing generative adversarial nets. In D. D. Lee,
M. Sugiyama, U. von Luxburg, I. Guyon, and R. Garnett, editors,
NIPS, pages 2172–2180, 2016.
[7] L. De Lathauwer and B. De Moor. A multi-linear singular value
decomposition. Society for Industrial and Applied Mathematics,
21:1253–1278, 03 2000.
[8] Y. Deng, J. Yang, D. Chen, F. Wen, and X. Tong. Disentangled
and controllable face image generation via 3d imitative-contrastive
learning. In IEEE Computer Vision and Pattern Recognition, 2020.
[9] C. Ferrari, G. Lisanti, S. Berretti, and A. D. Bimbo. A dictionary
learning-based 3d morphable shape model. IEEE Transactions on
Multimedia, 19(12):2666–2679, 2017.
[10] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley,
S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In
Proceedings of the 27th International Conference on Neural Infor-
mation Processing Systems - Volume 2, NIPS’14, page 2672–2680,
Cambridge, MA, USA, 2014. MIT Press.
[11] S. Graßhof, H. Ackermann, S. Brandt, and J. Ostermann. Apathy is
the root of all expressions. 12th IEEE Conference on Automatic Face
and Gesture Recognition (FG2017), 2017.
[12] S. Graßhof, H. Ackermann, S. S. Brandt, and J. Ostermann. Mul-
tilinear Modelling of Faces and Expressions. IEEE Transactions on
Pattern Analysis and Machine Intelligence, 43(10):3540–3554, Oct.
2021. Conference Name: IEEE Transactions on Pattern Analysis and
Machine Intelligence.
[13] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image
recognition. Proc. CVPR, pages 770–778, 2016.
[14] X. Huang and S. Belongie. Arbitrary style transfer in real-time with
adaptive instance normalization. In Proc. ICCV, 2017.
[15] E. H¨
ark¨
onen, A. Hertzmann, J. Lehtinen, and S. Paris. Ganspace:
Discovering interpretable gan controls. In Proc. NeurIPS, 2020.
[16] T. Karras, M. Aittala, J. Hellsten, S. Laine, J. Lehtinen, and T. Aila.
Training generative adversarial networks with limited data. In Proc.
NeurIPS, 2020.
[17] T. Karras, S. Laine, and T. Aila. A style-based generator architecture
for generative adversarial networks. In Proc. CVPR, pages 4396–4405,
2019.
[18] T. Karras, S. Laine, M. Aittala, J. Hellsten, J. Lehtinen, and T. Aila.
Analyzing and improving the image quality of StyleGAN. In Proc.
CVPR, 2020.
[19] T. G. Kolda and B. W. Bader. Tensor decompositions and applications.
SIAM REVIEW, 51(3):455–500, 2009.
[20] A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification
with deep convolutional neural networks. In Proceedings of the 25th
International Conference on Neural Information Processing Systems -
Volume 1, NIPS’12, page 1097–1105. Curran Associates Inc., 2012.
[21] D. Nikitko. Stylegan – encoder for official tensorflow implementation.
https://github.com/puzer/stylegan-encoder/, 2019.
[22] D. Y. Park and K. H. Lee. Arbitrary style transfer with style-attentional
networks. In Proc. IEEE Computer Society Conference on Computer
Vision and Pattern Recognition, 2019-June:5873–5881, Dec 2018.
[23] S. Pidhorskyi, D. A. Adjeroh, and G. Doretto. Adversarial latent
autoencoders. In Proc. CVPR, June 2020.
[24] Y. Shen, J. Gu, X. Tang, and B. Zhou. Interpreting the latent space
of gans for semantic face editing. In CVPR, 2020.
[25] Y. Shen and B. Zhou. Closed-form factorization of latent semantics
in gans. In CVPR, 2021.
[26] K. Simonyan and A. Zisserman. Very deep convolutional networks
for large-scale image recognition, 2014.
[27] A. Tewari, M. Elgharib, G. Bharaj, F. Bernard, H.-P. Seidel, P. P´
erez,
M. Z¨
ollhofer, and C. Theobalt. Stylerig: Rigging stylegan for 3d
control over portrait images, cvpr 2020. In IEEE Conference on
Computer Vision and Pattern Recognition (CVPR). IEEE, june 2020.
[28] M. A. O. Vasilescu and D. Terzopoulos. Multilinear analysis of
image ensembles: Tensorfaces. In Proceedings of the 7th European
Conference on Computer Vision-Part I, ECCV ’02, page 447–460,
Berlin, Heidelberg, 2002. Springer-Verlag.
[29] M. A. O. Vasilescu and D. Terzopoulos. Multilinear subspace analysis
of image ensembles. In Proc. of the IEEE Conf. on Computer Vision
and Pattern Recognition, volume 2, pages II– 93, 07 2003.
[30] D. Vlasic, M. Brand, H. Pfister, and J. Popovi´
c. Face Transfer with
Multilinear Models. In ACM SIGGRAPH, pages 426–433, 2005.
[31] Z. Wang and A. Bovik. A universal image quality index. Signal
Processing Letters, IEEE, 9:81 – 84, 04 2002.
[32] K. Yano and K. Harada. Multilinear face model. In Visualization,
Imaging, and Image Processing (VIIP 2008), 2008.
[33] L. Yin, X. Wei, Y. Sun, J. Wang, and M. Rosato. A 3d facial expression
database for facial behavior research. In 7th Intern. Conf. on Automatic
Face and Gesture Recognition (FGR06), pages 211–216, 2006.
[34] R. Zhang, P. Isola, A. A. Efros, E. Shechtman, and O. Wang. The
unreasonable effectiveness of deep features as a perceptual metric. In
CVPR, 2018.