Original: F INDING THE GLOBAL SEMANTIC REPRESENTATION IN GAN THROUGH F R ´ ECHET M EAN
Revised: F INDING THE GLOBAL SEMANTIC REPRESENTATION IN GAN THROUGH F R ´ ECHET M EAN

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: The ideally disentangled latent space in
Revised: The ideally disentangled latent space in

Original: GAN involves the global representation of latent space with semantic attribute coordinates.
Revised: GAN involves the global representation of latent space with semantic attribute coordinates.

Original: In other words, considering that this disentangled latent space is a vector space, there exists the global semantic basis where each basis component describes one attribute of generated images.
Revised: In other words, considering that this disentangled latent space is a vector space, there exists the global semantic basis where each basis component describes one attribute of generated images.

Original: In this paper we propose an unsupervised method for finding this global semantic basis in the intermediate latent space in GANs.
Revised: In this paper we propose an unsupervised method for finding this global semantic basis in the intermediate latent space in GANs.

Original: This semantic basis represents sample-independent meaningful perturbations that change the same semantic attribute of an image on the entire latent space.
Revised: This semantic basis represents sample-independent meaningful perturbations that change the same semantic attribute of an image on the entire latent space.

Original: The proposed global basis, called
Revised: The proposed global basis, called

Original: Fr ´ echet basis is derived by introducing Fr ´ echet mean to the local semantic perturbations in a latent space.
Revised: Fr ´ echet basis is derived by introducing Fr ´ echet mean to the local semantic perturbations in a latent space.

Original: Fr ´ echet basis is discovered in two stages.
Revised: Fr ´ echet basis is discovered in two stages.

Original: First, the global semantic subspace is discovered by the Fr ´ echet mean in the Grassmannian manifold of the local semantic subspaces.
Revised: First, the global semantic subspace is discovered by the Fr ´ echet mean in the Grassmannian manifold of the local semantic subspaces.

Original: Second, Fr ´ echet basis is found by optimizing a basis of the semantic subspace via the Fr´echet mean in the Special Orthogonal
Revised: Second, Fr ´ echet basis is found by optimizing a basis of the semantic subspace via the Fr´echet mean in the Special Orthogonal

Original: Group.
Revised: Group.

Original: Experimental results demonstrate that Fr´echet basis provides better semantic factorization and robustness compared to the previous methods.
Revised: Experimental results demonstrate that Fr´echet basis provides better semantic factorization and robustness compared to the previous methods.

Original: Moreover, we suggest the basis refinement scheme for the previous methods.
Revised: Moreover, we suggest the basis refinement scheme for the previous methods.

Original: The quantitative experiments show that the refined basis achieves better semantic factorization while generating the same semantic subspace as the previous method
Revised: The quantitative experiments show that the refined basis achieves better semantic factorization while generating the same semantic subspace as the previous method

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Generative Adversarial Networks (GANs, (Goodfellow et al., 2014)) have achieved impressive success in high-fidelity image synthesis, such as ProGAN (Karras et al., 2018), BigGAN (Brock et al., 2018), and StyleGANs (Karras et al., 2019; 2020a;b; 2021).
Revised: Generative Adversarial Networks (GANs, (Goodfellow et al., 2014)) have achieved impressive success in high-fidelity image synthesis, such as ProGAN (Karras et al., 2018), BigGAN (Brock et al., 2018), and StyleGANs (Karras et al., 2019; 2020a;b; 2021).

Original: Interestingly, even when a GAN model is trained without any information about the semantics of data, its latent space often represents the semantic property of data (Radford et al., 2016; Karras et al., 2019).
Revised: Interestingly, even when a GAN model is trained without any information about the semantics of data, its latent space often represents the semantic property of data (Radford et al., 2016; Karras et al., 2019).

Original: To understand how GAN models represent the semantics, several studies investigated the disentanglement (Bengio et al., 2013) property of latent space in GANs (Goetschalckx et al., 2019; Jahanian et al., 2019; Plumerault et al., 2020;
Revised: To understand how GAN models represent the semantics, several studies investigated the disentanglement (Bengio et al., 2013) property of latent space in GANs (Goetschalckx et al., 2019; Jahanian et al., 2019; Plumerault et al., 2020;

Original: Shen et al., 2020).
Revised: Shen et al., 2020).

Original: Here, a latent space in GAN is called disentangled if there exists an optimal basis representation where each basis coefficient corresponds to one disentangled semantics (generative factor).
Revised: Here, a latent space in GAN is called disentangled if there exists an optimal basis representation where each basis coefficient corresponds to one disentangled semantics (generative factor).

Original: One approach to studying the disentanglement property is to find meaningful latent perturbations that induce the disentangled semantic variation on generated images (Ramesh et al., 2018;
Revised: One approach to studying the disentanglement property is to find meaningful latent perturbations that induce the disentangled semantic variation on generated images (Ramesh et al., 2018;

Original: H¨ark¨onen et al., 2020; Shen & Zhou, 2021; Choi et al., 2022b).
Revised: H¨ark¨onen et al., 2020; Shen & Zhou, 2021; Choi et al., 2022b).

Original: This approach can be interpreted as investigating how the semantics are represented around each latent variable.
Revised: This approach can be interpreted as investigating how the semantics are represented around each latent variable.

Original: We classify the previous works on meaningful latent perturbations into local and global methods depending on whether the proposed perturbation is sample-dependent or sample-ignorant.
Revised: We classify the previous works on meaningful latent perturbations into local and global methods depending on whether the proposed perturbation is sample-dependent or sample-ignorant.

Original: In this work, we focus on the global methods (H¨ark¨onen et al., 2020; Shen & Zhou, 2021).
Revised: In this work, we focus on the global methods (H¨ark¨onen et al., 2020; Shen & Zhou, 2021).

Original: If the latent space is ideally disentangled, the optimal semantic basis becomes the global semantic perturbation that represents a change in the same generative factor on the entire latent space.
Revised: If the latent space is ideally disentangled, the optimal semantic basis becomes the global semantic perturbation that represents a change in the same generative factor on the entire latent space.

Original: In this regard, these global methods are attempts to find the best-possible semantic basis on the target latent space.
Revised: In this regard, these global methods are attempts to find the best-possible semantic basis on the target latent space.

Original: Throughout this work, the semantic subspace represents the subspace generated by the corresponding semantic basis.
Revised: Throughout this work, the semantic subspace represents the subspace generated by the corresponding semantic basis.

Original: In this paper, we propose an unsupervised method for finding the global semantic perturbations in a latent space in GAN, called Fr´echet Basis .
Revised: In this paper, we propose an unsupervised method for finding the global semantic perturbations in a latent space in GAN, called Fr´echet Basis .

Original: Fr´echet Basis is based on the Fr´echet mean on the Riemannian manifold.
Revised: Fr´echet Basis is based on the Fr´echet mean on the Riemannian manifold.

Original: Fr ´ echet mean is a generalization of centroid to the general metric space
Revised: Fr ´ echet mean is a generalization of centroid to the general metric space

Original: SO(𝑑 𝒲 )
Revised: SO(𝑑 𝒲 )

Original: ℝ 𝑑 ෩𝒲 ×𝑑 𝒲 𝑃 𝑠
Revised: ℝ 𝑑 ෩𝒲 ×𝑑 𝒲 𝑃 𝑠

Original: 𝑠 𝐵 𝑠 ≔ 𝑀 𝑠 𝑂 𝑇 𝑤 𝑖 𝒲
Revised: 𝑠 𝐵 𝑠 ≔ 𝑀 𝑠 𝑂 𝑇 𝑤 𝑖 𝒲

Original: 𝑤 𝑖 𝑑 𝒲
Revised: 𝑤 𝑖 𝑑 𝒲

Original: 𝑤 𝑖 𝒲 𝑃 𝑠𝑜 𝑀 𝑠𝑇 𝑀
Revised: 𝑤 𝑖 𝒲 𝑃 𝑠𝑜 𝑀 𝑠𝑇 𝑀

Original: 𝑤 𝑖 𝑂 𝐺 𝑟 𝑑 𝒲 ,
Revised: 𝑤 𝑖 𝑂 𝐺 𝑟 𝑑 𝒲 ,

Original: ℝ 𝑑 ෩𝒲 𝑠 𝑠
Revised: ℝ 𝑑 ෩𝒲 𝑠 𝑠

Original: (Fr ´ echet, 1948) and the Riemannian manifold is the metric space (Lee, 2013).
Revised: (Fr ´ echet, 1948) and the Riemannian manifold is the metric space (Lee, 2013).

Original: In particular, Fr ´ echet
Revised: In particular, Fr ´ echet

Original: Basis is discovered in two steps (Fig 1).
Revised: Basis is discovered in two steps (Fig 1).

Original: First, we find the global semantic subspace S s of latent space as Fr ´ echet mean in the Grassmannian manifold (Boothby, 1986) of the intrinsic tangent spaces.
Revised: First, we find the global semantic subspace S s of latent space as Fr ´ echet mean in the Grassmannian manifold (Boothby, 1986) of the intrinsic tangent spaces.

Original: Here, the intrinsic tangent space represent the local semantic subspace (Choi et al., 2022b).
Revised: Here, the intrinsic tangent space represent the local semantic subspace (Choi et al., 2022b).

Original: Second,
Revised: Second,

Original: Fr´echet basis B s is discovered by selecting the optimal basis of S s via Fr´echet mean in the Special
Revised: Fr´echet basis B s is discovered by selecting the optimal basis of S s via Fr´echet mean in the Special

Original: Orthogonal Group (Lang, 2012).
Revised: Orthogonal Group (Lang, 2012).

Original: Our experiments show that Fr´echet basis provides better semantic factorization and robustness compared to the previous unsupervised global methods.
Revised: Our experiments show that Fr´echet basis provides better semantic factorization and robustness compared to the previous unsupervised global methods.

Original: Moreover, the second step in finding Fr ´ echet basis provides the basis refinement scheme to the previous global methods.
Revised: Moreover, the second step in finding Fr ´ echet basis provides the basis refinement scheme to the previous global methods.

Original: In our experiments, the basis refinement achieves better semantic factorization than the previous methods while keeping the same semantic subspace.
Revised: In our experiments, the basis refinement achieves better semantic factorization than the previous methods while keeping the same semantic subspace.

Original: Our contributions are as follows:
Revised: Our contributions are as follows:

Original: We propose unsupervised global semantic perturbations, called Fr ´ echet basis.
Revised: We propose unsupervised global semantic perturbations, called Fr ´ echet basis.

Original: Fr ´ echet basis is discovered by introducing Fr ´ echet mean to the local semantic perturbations.
Revised: Fr ´ echet basis is discovered by introducing Fr ´ echet mean to the local semantic perturbations.

Original: We show that Fr ´ echet basis achieves better semantic factorization and robustness compared to the previous global approaches.
Revised: We show that Fr ´ echet basis achieves better semantic factorization and robustness compared to the previous global approaches.

Original: We propose the basis refinement scheme, which optimizes the semantic basis on the given semantic subspace.
Revised: We propose the basis refinement scheme, which optimizes the semantic basis on the given semantic subspace.

Original: We can refine the previous global approaches by
Revised: We can refine the previous global approaches by

Original: applying the basis refinement on their semantic subspaces.
Revised: applying the basis refinement on their semantic subspaces.

Original:  R ELATED W ORKS AND B ACKGROUND
Revised:  R ELATED W ORKS AND B ACKGROUND

Original: Latent Perturbation for Image Manipulation
Revised: Latent Perturbation for Image Manipulation

Original: The latent space of GANs often represents the semantics of data even when the model is trained without the supervision of the semantic attributes.
Revised: The latent space of GANs often represents the semantics of data even when the model is trained without the supervision of the semantic attributes.

Original: Early approaches to understanding the semantic property of latent space showed that the vector arithmetic on latent space leads to the semantic arithmetic on the image space (Radford et al., 2016;
Revised: Early approaches to understanding the semantic property of latent space showed that the vector arithmetic on latent space leads to the semantic arithmetic on the image space (Radford et al., 2016;

Original: Upchurch et al., 2017).
Revised: Upchurch et al., 2017).

Original: In this regard, a line of research has been conducted to find meaningful latent perturbations that perform image manipulation in disentangled semantics.
Revised: In this regard, a line of research has been conducted to find meaningful latent perturbations that perform image manipulation in disentangled semantics.

Original: We categorize the previous works into local and global methods according to sample dependency.
Revised: We categorize the previous works into local and global methods according to sample dependency.

Original: The local method finds meaningful perturbations for each latent variable for one semantic attribute, e.g., Ramesh et al.
Revised: The local method finds meaningful perturbations for each latent variable for one semantic attribute, e.g., Ramesh et al.

Original: (2018), Latent Mapper in StyleCLIP (Patashnik et al., 2021), Attribute-conditioned normalizing flow in StyleFlow (Abdal et al., 2021), Local image editing in Zhu et al.
Revised: (2018), Latent Mapper in StyleCLIP (Patashnik et al., 2021), Attribute-conditioned normalizing flow in StyleFlow (Abdal et al., 2021), Local image editing in Zhu et al.

Original: (2021), and Local Basis (Choi et al., 2022b).
Revised: (2021), and Local Basis (Choi et al., 2022b).

Original: By contrast, the global methods offer the sample-independent meaningful perturbations for each latent space, e.g., Global directions in StyleCLIP (Patashnik et al., 2021),
Revised: By contrast, the global methods offer the sample-independent meaningful perturbations for each latent space, e.g., Global directions in StyleCLIP (Patashnik et al., 2021),

Original: GANSpace (H¨ark¨onen et al., 2020), and SeFa (Shen & Zhou, 2021).
Revised: GANSpace (H¨ark¨onen et al., 2020), and SeFa (Shen & Zhou, 2021).

Original: Among them, StyleCLIP is a supervised method requiring text descriptions of each generated image to train CLIP model (Radford et al., 2021).
Revised: Among them, StyleCLIP is a supervised method requiring text descriptions of each generated image to train CLIP model (Radford et al., 2021).

Original: Throughout this paper, we investigate unsupervised methods such as GANSpace and
Revised: Throughout this paper, we investigate unsupervised methods such as GANSpace and

Original: SeFa.
Revised: SeFa.

Original: GANSpace (H ¨ ark ¨ onen et al., 2020) suggested the principal components of latent space obtained by performing PCA as global meaningful perturbations.
Revised: GANSpace (H ¨ ark ¨ onen et al., 2020) suggested the principal components of latent space obtained by performing PCA as global meaningful perturbations.

Original: SeFa (Shen & Zhou, 2021) proposed the singular vectors of the first weight matrix as global disentangled perturbations.
Revised: SeFa (Shen & Zhou, 2021) proposed the singular vectors of the first weight matrix as global disentangled perturbations.

Original: Unsupervised global disentanglement score The disentanglement of latent space is expressed as the correspondence between the semantic attributes of data and the axes of latent space.
Revised: Unsupervised global disentanglement score The disentanglement of latent space is expressed as the correspondence between the semantic attributes of data and the axes of latent space.

Original: Because the definition of disentanglement depends on attributes, most of the existing disentanglement metrics for latent spaces are supervised ones, e.g., DCI score (Eastwood & Williams, 2018), β -VAE metric (Higgins et al., 2017), and FactorVAE metric (Kim & Mnih, 2018).
Revised: Because the definition of disentanglement depends on attributes, most of the existing disentanglement metrics for latent spaces are supervised ones, e.g., DCI score (Eastwood & Williams, 2018), β -VAE metric (Higgins et al., 2017), and FactorVAE metric (Kim & Mnih, 2018).

Original: They require the attribute annotations of generated images.
Revised: They require the attribute annotations of generated images.

Original: This requirement restricts the broad applicability of disentanglement evaluation on real datasets.
Revised: This requirement restricts the broad applicability of disentanglement evaluation on real datasets.

Original: To address this restriction, Choi et al.
Revised: To address this restriction, Choi et al.

Original: (2022a) proposed an unsupervised global disentanglement score, called Distortion .
Revised: (2022a) proposed an unsupervised global disentanglement score, called Distortion .

Original: Distortion metric measures the variation of tangent space on the learned latent manifold W .
Revised: Distortion metric measures the variation of tangent space on the learned latent manifold W .

Original: Hence, Distortion metric relies purely on the geometric property of the latent space and does not require attribute labels.
Revised: Hence, Distortion metric relies purely on the geometric property of the latent space and does not require attribute labels.

Original: Background We describe Distortion metric in more detail because our Fr´echet mean is closely related to it.
Revised: Background We describe Distortion metric in more detail because our Fr´echet mean is closely related to it.

Original: Here, the learned latent manifold W = f ( Z ) denotes an image of the subnetwork f from the input noise space Z to the target latent space.
Revised: Here, the learned latent manifold W = f ( Z ) denotes an image of the subnetwork f from the input noise space Z to the target latent space.

Original: We assume Z is the entire Euclidean space, i.e., Z = R d Z for some d Z .
Revised: We assume Z is the entire Euclidean space, i.e., Z = R d Z for some d Z .

Original: This is satisfied for the usual Gaussian prior p ( z ) =
Revised: This is satisfied for the usual Gaussian prior p ( z ) =

Original: N (0 , I d Z ) .
Revised: N (0 , I d Z ) .

Original: Choi et al.
Revised: Choi et al.

Original: (2022b) suggested a method for finding the k -dimensional local approximation W k w of W around w = f ( z ) ∈ W .
Revised: (2022b) suggested a method for finding the k -dimensional local approximation W k w of W around w = f ( z ) ∈ W .

Original: The local approximation W k w is discovered by the low-rank approximation problem of df z :
Revised: The local approximation W k w is discovered by the low-rank approximation problem of df z :

Original: For the i -th singular vector u z i ∈ R d Z , v w i ∈ R d W , and i -th singular value σ z i ∈ R of df z with σ z 1 ≥ · · · ≥ σ z m and m = min( d Z , d W ) , df
Revised: For the i -th singular vector u z i ∈ R d Z , v w i ∈ R d W , and i -th singular value σ z i ∈ R of df z with σ z 1 ≥ · · · ≥ σ z m and m = min( d Z , d W ) , df

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: for
Revised: for

Original: [Equation]
Revised: [Equation]

Original: Choi et al.
Revised: Choi et al.

Original: (2022b) showed that Local Basis (Eq 1) serves as the local semantic perturbations around w .
Revised: (2022b) showed that Local Basis (Eq 1) serves as the local semantic perturbations around w .

Original: In this respect, the tangent space at w represents the local semantic subspace because it is spanned the local semantic perturbations (Eq 3).
Revised: In this respect, the tangent space at w represents the local semantic subspace because it is spanned the local semantic perturbations (Eq 3).

Original: Upon this, Choi et al.
Revised: Upon this, Choi et al.

Original: (2022a) proposed the intrinsic local dimension estimation scheme for the latent manifold W through the robust rank estimate (Kritchman & Nadler, 2008) of df z .
Revised: (2022a) proposed the intrinsic local dimension estimation scheme for the latent manifold W through the robust rank estimate (Kritchman & Nadler, 2008) of df z .

Original: In local geometry, the intrinsic local dimension represents the number of dimensions required to properly approximate the denoised W .
Revised: In local geometry, the intrinsic local dimension represents the number of dimensions required to properly approximate the denoised W .

Original: Choi et al.
Revised: Choi et al.

Original: (2022a) showed that this local dimension becomes an upper-bound of the number of local semantic perturbations.
Revised: (2022a) showed that this local dimension becomes an upper-bound of the number of local semantic perturbations.

Original: Distortion metric is defined as the normalized variation of intrinsic tangent space on the latent manifold.
Revised: Distortion metric is defined as the normalized variation of intrinsic tangent space on the latent manifold.

Original: The normalized variation is expressed as the ratio of the distance between two tangent spaces at two random w ∈ W to the distance between two tangent spaces at two close w .
Revised: The normalized variation is expressed as the ratio of the distance between two tangent spaces at two random w ∈ W to the distance between two tangent spaces at two close w .

Original: Distortion metric uses the dimension-normalized Geodesic Metric d k geo (Ye &
Revised: Distortion metric uses the dimension-normalized Geodesic Metric d k geo (Ye &

Original: Lim, 2016; Choi et al., 2022a) in Grassmannian manifold (Boothby, 1986) to measure the distance between tangent spaces.
Revised: Lim, 2016; Choi et al., 2022a) in Grassmannian manifold (Boothby, 1986) to measure the distance between tangent spaces.

Original: Specifically, Distortion of W is defined as D W =
Revised: Specifically, Distortion of W is defined as D W =

Original: I rand /I
Revised: I rand /I

Original: local with
Revised: local with

Original: I rand =
Revised: I rand =

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: (cid:2)
Revised: (cid:2)

Original: d k
Revised: d k

Original: [Equation]
Revised: [Equation]

Original: (cid:0) T w
Revised: (cid:0) T w

Original: [Equation]
Revised: [Equation]

Original: k
Revised: k

Original: [Equation]
Revised: [Equation]

Original: 1 , T w 2 W k w 2 (cid:1) for k = min( k 1 , k 2 ) (cid:3) ,
Revised: 1 , T w 2 W k w 2 (cid:1) for k = min( k 1 , k 2 ) (cid:3) ,

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where k i denotes the local dimension estimate at w i =
Revised: where k i denotes the local dimension estimate at w i =

Original: f ( z i ) and d k geo refers to the dimensionnormalized Geodesic Metric between k -dimensional subspaces.
Revised: f ( z i ) and d k geo refers to the dimensionnormalized Geodesic Metric between k -dimensional subspaces.

Original: Interestingly, although Distortion metric does not exploit any semantic information, Distortion metric provides a strong correlation between the supervised disentanglement score and the global-basis-compatibility (Choi et al., 2022a).
Revised: Interestingly, although Distortion metric does not exploit any semantic information, Distortion metric provides a strong correlation between the supervised disentanglement score and the global-basis-compatibility (Choi et al., 2022a).

Original:  F R ´ ECHET M EAN G LOBAL B ASIS
Revised:  F R ´ ECHET M EAN G LOBAL B ASIS

Original: In this section, we propose an unsupervised method for finding global linear perturbation directions that make the same semantic manipulation on the entire latent space, called Fr´echet basis .
Revised: In this section, we propose an unsupervised method for finding global linear perturbation directions that make the same semantic manipulation on the entire latent space, called Fr´echet basis .

Original: If we have such global meaningful perturbations, the vector space representation of latent space along these semantic basis provides the global semantic representation of a model.
Revised: If we have such global meaningful perturbations, the vector space representation of latent space along these semantic basis provides the global semantic representation of a model.

Original: The proposed scheme is based on finding the Fr ´ echet mean (Fr ´ echet, 1948; Karcher, 1977) of the local disentangled perturbations.
Revised: The proposed scheme is based on finding the Fr ´ echet mean (Fr ´ echet, 1948; Karcher, 1977) of the local disentangled perturbations.

Original: The scheme is in two steps.
Revised: The scheme is in two steps.

Original: First, the optimal subspace representing the global semantics is discovered by the Fr´echet mean in the Grassmannian manifold (Boothby, 1986) of intrinsic tangent spaces (Choi et al., 2022b) on the target latent space.
Revised: First, the optimal subspace representing the global semantics is discovered by the Fr´echet mean in the Grassmannian manifold (Boothby, 1986) of intrinsic tangent spaces (Choi et al., 2022b) on the target latent space.

Original: Second, the optimal basis is obtained as the Fr´echet mean in the Special Orthogonal Group (Lang, 2012) of the projected local disentangled perturbations.
Revised: Second, the optimal basis is obtained as the Fr´echet mean in the Special Orthogonal Group (Lang, 2012) of the projected local disentangled perturbations.

Original: 3. M ETHOD
Revised: 3. M ETHOD

Original: Notation Throughout this work, we follow the notation presented in Sec 2.
Revised: Notation Throughout this work, we follow the notation presented in Sec 2.

Original: Let ˜ W = R d ˜ W be a ambient target latent space where we want to find global semantic perturbations.
Revised: Let ˜ W = R d ˜ W be a ambient target latent space where we want to find global semantic perturbations.

Original: We analyze the learned latent manifold W = f ( Z ) ⊂ ˜ W embedded in this latent space, which is given as an image of the subnetwork from the input noise Z to ˜ W .
Revised: We analyze the learned latent manifold W = f ( Z ) ⊂ ˜ W embedded in this latent space, which is given as an image of the subnetwork from the input noise Z to ˜ W .

Original: Motivation We investigate the problem of discovering global semantic perturbations through the local geometry of learned latent manifold W .
Revised: Motivation We investigate the problem of discovering global semantic perturbations through the local geometry of learned latent manifold W .

Original: Recently, Choi et al.
Revised: Recently, Choi et al.

Original: (2022a) discovered that the intrinsic tangent space T w W k w at each w ∈ W represents the local semantic variation of the generated image from w .
Revised: (2022a) discovered that the intrinsic tangent space T w W k w at each w ∈ W represents the local semantic variation of the generated image from w .

Original: Specifically, the intrinsic local dimension at w , denoted as k in T w W k w , corresponds to the number of local semantic perturbations.
Revised: Specifically, the intrinsic local dimension at w , denoted as k in T w W k w , corresponds to the number of local semantic perturbations.

Original: The top- k components of Local Basis
Revised: The top- k components of Local Basis

Original: Choi et al.
Revised: Choi et al.

Original: (2022b) are these local semantic perturbations and are the basis vectors of T w W k w (Eq 3).
Revised: (2022b) are these local semantic perturbations and are the basis vectors of T w W k w (Eq 3).

Original: Hence, the intrinsic tangent space T w W k w describes the local semantic variation of an image because it is spanned by the local meaningful perturbations.
Revised: Hence, the intrinsic tangent space T w W k w describes the local semantic variation of an image because it is spanned by the local meaningful perturbations.

Original: In this regard, we interpret the global semantic variation as the mean of these local semantic variations .
Revised: In this regard, we interpret the global semantic variation as the mean of these local semantic variations .

Original: One of the most popular methods for defining the mean of subspaces is through Fr´echet mean (Marrinan et al., 2014).
Revised: One of the most popular methods for defining the mean of subspaces is through Fr´echet mean (Marrinan et al., 2014).

Original: Fr´echet mean is a generalization of the mean in vector space to the general metric space.
Revised: Fr´echet mean is a generalization of the mean in vector space to the general metric space.

Original: The mean of vectors µ is the minimizer of the sum of squared distances to each vector.
Revised: The mean of vectors µ is the minimizer of the sum of squared distances to each vector.

Original: Similarly, Fr ´ echet mean µ fr in the metric space X with metric d is defined as the minimizer of squared metrics, i.e., for x 1 , x 2 , . .
Revised: Similarly, Fr ´ echet mean µ fr in the metric space X with metric d is defined as the minimizer of squared metrics, i.e., for x 1 , x 2 , . .

Original: , x n ∈ X ,
Revised: , x n ∈ X ,

Original: [Equation]
Revised: [Equation]

Original: In particular, a Riemannian manifold is an example of metric space where we can introduce the
Revised: In particular, a Riemannian manifold is an example of metric space where we can introduce the

Original: Fr ´ echet mean (Lou et al., 2020).
Revised: Fr ´ echet mean (Lou et al., 2020).

Original: In this work, we utilize the Grassmannian manifold (Boothby, 1986) to find the subspace of latent space for the global semantic representation and the Special Orthogonal
Revised: In this work, we utilize the Grassmannian manifold (Boothby, 1986) to find the subspace of latent space for the global semantic representation and the Special Orthogonal

Original: Group (Lang, 2012) to choose the optimal basis on it.
Revised: Group (Lang, 2012) to choose the optimal basis on it.

Original: 3.1. G LOBAL S EMANTIC S UBSPACE
Revised: 3.1. G LOBAL S EMANTIC S UBSPACE

Original: Our goal is to find a Riemannian manifold where we can embed these intrinsic tangent spaces { T w i W k
Revised: Our goal is to find a Riemannian manifold where we can embed these intrinsic tangent spaces { T w i W k

Original: i w i } 1 ≤ i ≤ n describing local semantic variations at each w i .
Revised: i w i } 1 ≤ i ≤ n describing local semantic variations at each w i .

Original: The Grassmannian manifold Gr ( k, V ) denotes the set of k -dimensional linear subspaces of vector space V (Boothby, 1986).
Revised: The Grassmannian manifold Gr ( k, V ) denotes the set of k -dimensional linear subspaces of vector space V (Boothby, 1986).

Original: Hence, these tangent spaces can be embedded to one Grassmannian manifold Gr (cid:0) d W , R d ˜ W (cid:1) by matching their dimensions to the dimension of learned latent manifold d W .
Revised: Hence, these tangent spaces can be embedded to one Grassmannian manifold Gr (cid:0) d W , R d ˜ W (cid:1) by matching their dimensions to the dimension of learned latent manifold d W .

Original: Specifically, we match the dimensions of tangent spaces by refining or extending them to the subspaces spanned by the top- d W components of Local Basis (Eq 3).
Revised: Specifically, we match the dimensions of tangent spaces by refining or extending them to the subspaces spanned by the top- d W components of Local Basis (Eq 3).

Original: This is equivalent to approximating W with the d W -dimensional local estimate W d W w at all w ∈ W
Revised: This is equivalent to approximating W with the d W -dimensional local estimate W d W w at all w ∈ W

Original: (Eq 2).
Revised: (Eq 2).

Original: We estimate the layer-wise dimension d W of learned latent manifold W by averaging local dimensions of n samples { k i } 1 ≤ i ≤ n ,
Revised: We estimate the layer-wise dimension d W of learned latent manifold W by averaging local dimensions of n samples { k i } 1 ≤ i ≤ n ,

Original: [Equation]
Revised: [Equation]

Original: (7) where v w
Revised: (7) where v w

Original: i
Revised: i

Original: i denotes Local Basis at w i
Revised: i denotes Local Basis at w i

Original: (Eq 1).
Revised: (Eq 1).

Original: Then, we define the global semantic subspace S s of
Revised: Then, we define the global semantic subspace S s of

Original: W as the Fr´echet mean on Gr (cid:0) d W , R d ˜ W (cid:1) with the geodesic metric d geo (Ye & Lim, 2016)
Revised: W as the Fr´echet mean on Gr (cid:0) d W , R d ˜ W (cid:1) with the geodesic metric d geo (Ye & Lim, 2016)

Original: [Equation]
Revised: [Equation]

Original: Here, the geodesic metric d geo is defined as d geo ( W, W ′ )
Revised: Here, the geodesic metric d geo is defined as d geo ( W, W ′ )

Original: = (cid:16)(cid:80)
Revised: = (cid:16)(cid:80)

Original: ki
Revised: ki

Original: =1 θ 2 i (cid:17) 1 / 2 for W, W ′ ∈
Revised: =1 θ 2 i (cid:17) 1 / 2 for W, W ′ ∈

Original: Gr ( k, R n ) where θ i denotes the i -th principal angle between W and W ′ .
Revised: Gr ( k, R n ) where θ i denotes the i -th principal angle between W and W ′ .

Original: That is, θ i = cos − 1 ( σ i ( M ⊤ W M W ′ )) where M W ∈ R n × k denotes the column-wise concatenation of orthonormal basis for W .
Revised: That is, θ i = cos − 1 ( σ i ( M ⊤ W M W ′ )) where M W ∈ R n × k denotes the column-wise concatenation of orthonormal basis for W .

Original: For optimization, we used the gradient descent algorithm in the Pymanopt (Townsend et al., 2016).
Revised: For optimization, we used the gradient descent algorithm in the Pymanopt (Townsend et al., 2016).

Original: 3.1. G LOBAL S EMANTIC B ASIS
Revised: 3.1. G LOBAL S EMANTIC B ASIS

Original: The aim of this work is to find global meaningful perturbations that represent the disentangled semantics.
Revised: The aim of this work is to find global meaningful perturbations that represent the disentangled semantics.

Original: However, the global semantic subspace S s is discovered by solving an optimization problem in the Grassmannian manifold, the set of subspaces.
Revised: However, the global semantic subspace S s is discovered by solving an optimization problem in the Grassmannian manifold, the set of subspaces.

Original: We need an additional step to find a specific basis on S s .
Revised: We need an additional step to find a specific basis on S s .

Original: In particular, we utilize the Fr´echet mean on the Special Orthogonal Group.
Revised: In particular, we utilize the Fr´echet mean on the Special Orthogonal Group.

Original: Why Special Orthogonal Group Let the columns of M S , M ′S ∈ R d ˜ W
Revised: Why Special Orthogonal Group Let the columns of M S , M ′S ∈ R d ˜ W

Original: ×
Revised: ×

Original: d W be the two distinct orthonormal basis of S s .
Revised: d W be the two distinct orthonormal basis of S s .

Original: Then, there exists an orthogonal matrix O ∈ R d W × d W , i.e., O ⊤ O = O ⊤ =
Revised: Then, there exists an orthogonal matrix O ∈ R d W × d W , i.e., O ⊤ O = O ⊤ =

Original: I , which satisfies M ′S = M S O .
Revised: I , which satisfies M ′S = M S O .

Original: Therefore, finding the optimal basis of S s is equivalent to finding the orthogonal matrix O given the initial M S .
Revised: Therefore, finding the optimal basis of S s is equivalent to finding the orthogonal matrix O given the initial M S .

Original: The Special Orthogonal Group SO ( n ) consists of n × n orthogonal matrices with determinant +

Revised: The Special Orthogonal Group SO ( n ) consists of n × n orthogonal matrices with determinant +

Original: (Lang, 2012).
Revised: (Lang, 2012).

Original: Note that the determinant of an arbitrary orthogonal matrix is +1 or − 1 .
Revised: Note that the determinant of an arbitrary orthogonal matrix is +1 or − 1 .

Original: We consider SO ( n ) instead of the orthogonal matrices for two reasons.
Revised: We consider SO ( n ) instead of the orthogonal matrices for two reasons.

Original: First, our task is independent of the flipping ( ( − 1) -multiplication) of each basis component.
Revised: First, our task is independent of the flipping ( ( − 1) -multiplication) of each basis component.

Original: The positive perturbation along v is identical to the negative perturbation along − v .
Revised: The positive perturbation along v is identical to the negative perturbation along − v .

Original: The flipping of a basis component in M ′S leads to the flipping of the corresponding column in O .
Revised: The flipping of a basis component in M ′S leads to the flipping of the corresponding column in O .

Original: This results in the ( − 1) -multiplication at the determinant of O .
Revised: This results in the ( − 1) -multiplication at the determinant of O .

Original: Therefore, without loss of generality, we may assume that O is a special orthogonal matrix.
Revised: Therefore, without loss of generality, we may assume that O is a special orthogonal matrix.

Original: Second, the Orthogonal Group is disconnected while the Special Orthogonal Group is connected.
Revised: Second, the Orthogonal Group is disconnected while the Special Orthogonal Group is connected.

Original: Hence, the Orthogonal Group is inadequate for finding the Fr ´ echet mean, which is optimized by the gradient descent algorithm.
Revised: Hence, the Orthogonal Group is inadequate for finding the Fr ´ echet mean, which is optimized by the gradient descent algorithm.

Original: Basis Refinement We propose the optimization scheme for finding the global semantic basis from the global semantic subspace S s .
Revised: Basis Refinement We propose the optimization scheme for finding the global semantic basis from the global semantic subspace S s .

Original: Here, we denote the column-wise concatenation of local semantic basis at each w i as M w i ∈ R d ˜ W × d W , i.e., each column is the top- d W Local Basis at w ∈ W .
Revised: Here, we denote the column-wise concatenation of local semantic basis at each w i as M w i ∈ R d ˜ W × d W , i.e., each column is the top- d W Local Basis at w ∈ W .

Original: Note that the column space of M w i is the local semantic subspace T w i
Revised: Note that the column space of M w i is the local semantic subspace T w i

Original: W
Revised: W

Original: d W w i .
Revised: d W w i .

Original: Likewise, M S refers to an initial orthonormal basis of S s .
Revised: Likewise, M S refers to an initial orthonormal basis of S s .

Original: As an overview, the proposed scheme is as follows:
Revised: As an overview, the proposed scheme is as follows:

Original: (i) Project each local semantic basis to the d W -dimensional global semantic subspace S s .
Revised: (i) Project each local semantic basis to the d W -dimensional global semantic subspace S s .

Original: (ii) Project these projected local semantic basis to SO ( d W ) .
Revised: (ii) Project these projected local semantic basis to SO ( d W ) .

Original: (iii) Find the Fr´echet mean O in SO ( d W ) and embed O back to the ambient space R d ˜ W .
Revised: (iii) Find the Fr´echet mean O in SO ( d W ) and embed O back to the ambient space R d ˜ W .

Original: Before the above optimization, we preprocess each local semantic basis at w i , i.e., the columns of
Revised: Before the above optimization, we preprocess each local semantic basis at w i , i.e., the columns of

Original: M w i , to be positively aligned to each column of M S , i.e., ⟨ M w i
Revised: M w i , to be positively aligned to each column of M S , i.e., ⟨ M w i

Original: [: , i ] , M S
Revised: [: , i ] , M S

Original: [: , i ] ⟩ > 0 for all i .
Revised: [: , i ] ⟩ > 0 for all i .

Original: (i)
Revised: (i)

Original: As a first step, we project each local semantic basis at w i onto the global semantic subspace S s , i.e.,
Revised: As a first step, we project each local semantic basis at w i onto the global semantic subspace S s , i.e.,

Original: M ⊤S M w i .
Revised: M ⊤S M w i .

Original: (ii)
Revised: (ii)

Original: Then, the matrix of projected local semantic basis M ⊤S M w i ∈ R d W × d W is projected to SO ( d W ) 1 .
Revised: Then, the matrix of projected local semantic basis M ⊤S M w i ∈ R d W × d W is projected to SO ( d W ) 1 .

Original: The projection on the orthogonal group P o and the proposed projection on the Special
Revised: The projection on the orthogonal group P o and the proposed projection on the Special

Original: Orthogonal Group P so can be obtained via SVD.
Revised: Orthogonal Group P so can be obtained via SVD.

Original: (See the appendix for proof.):
Revised: (See the appendix for proof.):

Original: Let X = U Σ V ⊤ be a SVD of X ∈ R d W × d W ,
Revised: Let X = U Σ V ⊤ be a SVD of X ∈ R d W × d W ,

Original: [Equation]
Revised: [Equation]

Original: where P o ( X ) = UV ⊤ .
Revised: where P o ( X ) = UV ⊤ .

Original: (9) (iii) Finally, we find the optimal orthogonal matrix O , that transforms the initial basis M S to the global semantic basis B s , via Fr ´ echet mean of projected local semantic basis { P so (cid:0) M ⊤ S M w i (cid:1) } i ⊂
Revised: (9) (iii) Finally, we find the optimal orthogonal matrix O , that transforms the initial basis M S to the global semantic basis B s , via Fr ´ echet mean of projected local semantic basis { P so (cid:0) M ⊤ S M w i (cid:1) } i ⊂

Original: SO ( d W ) .
Revised: SO ( d W ) .

Original: B s = M S
Revised: B s = M S

Original: O where O = arg min
Revised: O where O = arg min

Original: µ ∈ SO ( d W ) (cid:88)
Revised: µ ∈ SO ( d W ) (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Here, the Riemannian metric on SO ( d W ) is defined as d ( X 1 , X 2 ) =
Revised: Here, the Riemannian metric on SO ( d W ) is defined as d ( X 1 , X 2 ) =

Original: ∥
Revised: ∥

Original: Skew (cid:0) log( X ⊤ 1 X 2 ) (cid:1) ∥ F for
Revised: Skew (cid:0) log( X ⊤ 1 X 2 ) (cid:1) ∥ F for

Original: X 1 , X 2 ∈ SO ( d W ) where log denotes a matrix logarithm and Skew refers to the skew-symmetric component of a matrix, i.e., Skew( X 1 ) =
Revised: X 1 , X 2 ∈ SO ( d W ) where log denotes a matrix logarithm and Skew refers to the skew-symmetric component of a matrix, i.e., Skew( X 1 ) =

Original: 12 (cid:0) X 1 − X ⊤ 1 (cid:1) .
Revised: 12 (cid:0) X 1 − X ⊤ 1 (cid:1) .

Original: As in the Grassmannian manifold, we utilized the Pymanopt (Townsend et al., 2016) implementation for finding the Fr´echet mean on
Revised: As in the Grassmannian manifold, we utilized the Pymanopt (Townsend et al., 2016) implementation for finding the Fr´echet mean on

Original: SO ( d W ) .
Revised: SO ( d W ) .

Original: Then, the global semantic basis B s is obtained as the embedding of Fr´echet mean O to R d ˜ W , i.e., B s = M S O .
Revised: Then, the global semantic basis B s is obtained as the embedding of Fr´echet mean O to R d ˜ W , i.e., B s = M S O .

Original: We call this global semantic basis B s Fr´echet Basis .
Revised: We call this global semantic basis B s Fr´echet Basis .

Original: 3. F R ´ ECHET B ASIS AS L 2 -D ISTORTION MINIMIZER
Revised: 3. F R ´ ECHET B ASIS AS L 2 -D ISTORTION MINIMIZER

Original: The Fr´echet basis can be interpreted as the minimizer of the unsupervised disentanglement metric,
Revised: The Fr´echet basis can be interpreted as the minimizer of the unsupervised disentanglement metric,

Original: Distortion (Choi et al., 2022a).
Revised: Distortion (Choi et al., 2022a).

Original: Distortion metric is based on the inconsistency of intrinsic tangent
Revised: Distortion metric is based on the inconsistency of intrinsic tangent

Original: [Figure]
Revised: [Figure]

Original:   
Revised:   

Original: [Figure]
Revised: [Figure]

Original:    
Revised:    

Original: [Figure]
Revised: [Figure]

Original:      
Revised:      

Original: '& ,      
Revised: '& ,      

Original: [Figure]
Revised: [Figure]

Original: spaces.
Revised: spaces.

Original: Specifically, Distortion metric D W is the ratio between the inconsistency at two random w ∈ W and at two close w ∈ W , i.e., D W =
Revised: Specifically, Distortion metric D W is the ratio between the inconsistency at two random w ∈ W and at two close w ∈ W , i.e., D W =

Original: I rand /I
Revised: I rand /I

Original: local (Eq 4 and 5).
Revised: local (Eq 4 and 5).

Original: Here, the intrinsic tangent space represents the local semantic variations.
Revised: Here, the intrinsic tangent space represents the local semantic variations.

Original: From this point of view, the Distortion-based global semantic subspace S D would be a representative of these tangent spaces that minimize the inconsistency to each tangent space.
Revised: From this point of view, the Distortion-based global semantic subspace S D would be a representative of these tangent spaces that minimize the inconsistency to each tangent space.

Original: [Equation]
Revised: [Equation]

Original: with I
Revised: with I

Original: [Equation]
Revised: [Equation]

Original: (11) where µ is a subspace of R d ˜ W , k refers to the local dimension at w and µ k denotes the k -dimensional refinement of µ .
Revised: (11) where µ is a subspace of R d ˜ W , k refers to the local dimension at w and µ k denotes the k -dimensional refinement of µ .

Original: The Fr ´ echet basis assumes that the entire latent manifold W is approximated with d W -dimensional local estimate at all w ∈ W .
Revised: The Fr ´ echet basis assumes that the entire latent manifold W is approximated with d W -dimensional local estimate at all w ∈ W .

Original: Under this assumption, I global ( µ ) becomes
Revised: Under this assumption, I global ( µ ) becomes

Original: [Equation]
Revised: [Equation]

Original: p ( z ) , w = f (
Revised: p ( z ) , w = f (

Original: [Equation]
Revised: [Equation]

Original: ) (cid:2) d geo (cid:0) µ, T w W d W w (cid:1) (cid:3) for µ ∈ Gr ( d W , R d ˜ W ) .
Revised: ) (cid:2) d geo (cid:0) µ, T w W d W w (cid:1) (cid:3) for µ ∈ Gr ( d W , R d ˜ W ) .

Original: Therefore, the comparison with Eq 8 shows that the global semantic subspace by Fr ´ echet mean S s can be interpreted as L 2 -Distortion minimizer.
Revised: Therefore, the comparison with Eq 8 shows that the global semantic subspace by Fr ´ echet mean S s can be interpreted as L 2 -Distortion minimizer.

Original: Although the original L 1 -Distortion was proven to provide high correlations with the global-basis-compatibility and the supervised disentanglement score, L 2Distortion was not tested (Choi et al., 2022a).
Revised: Although the original L 1 -Distortion was proven to provide high correlations with the global-basis-compatibility and the supervised disentanglement score, L 2Distortion was not tested (Choi et al., 2022a).

Original: Therefore, we evaluated whether L 2 -Distortion is also a meaningful metric to verify the validity of minimizing it.
Revised: Therefore, we evaluated whether L 2 -Distortion is also a meaningful metric to verify the validity of minimizing it.

Original: Following the experiments in Choi et al.
Revised: Following the experiments in Choi et al.

Original: (2022a), we assessed the global-basis-compatibility by the FID (Heusel et al., 2017)
Revised: (2022a), we assessed the global-basis-compatibility by the FID (Heusel et al., 2017)

Original: Gap between
Revised: Gap between

Original: Local Basis and GANSpace (H¨ark¨onen et al., 2020) under the same perturbation intensity.
Revised: Local Basis and GANSpace (H¨ark¨onen et al., 2020) under the same perturbation intensity.

Original: Also,
Revised: Also,

Original: DCI score (Eastwood & Williams, 2018) is adopted as the supervised disentanglement score.
Revised: DCI score (Eastwood & Williams, 2018) is adopted as the supervised disentanglement score.

Original: We utilized 40 binary attribute classifiers pre-trained on CelebA (Liu et al., 2015) to annotate the 10k generated images.
Revised: We utilized 40 binary attribute classifiers pre-trained on CelebA (Liu et al., 2015) to annotate the 10k generated images.

Original: Figure 2 demonstrates that L 2 -Distortion achieves high correlations comparable to the original Distortion score in the global-based compatibility and DCI score.
Revised: Figure 2 demonstrates that L 2 -Distortion achieves high correlations comparable to the original Distortion score in the global-based compatibility and DCI score.

Original: These results prove that our framework of minimizing L 2 -Distortion by Fr´echet mean is also valid.
Revised: These results prove that our framework of minimizing L 2 -Distortion by Fr´echet mean is also valid.

Original:  E XPERIMENTS 4. F R ´ ECHET B ASIS AS G LOBAL S EMANTIC P ERTURBATIONS
Revised:  E XPERIMENTS 4. F R ´ ECHET B ASIS AS G LOBAL S EMANTIC P ERTURBATIONS

Original: We evaluate the Fr´echet basis as the global semantic perturbations on the intermediate layers of the mapping network in various StyleGAN models (Karras et al., 2019; 2020b).
Revised: We evaluate the Fr´echet basis as the global semantic perturbations on the intermediate layers of the mapping network in various StyleGAN models (Karras et al., 2019; 2020b).

Original: For each StyleGAN model, we used the layers from 3 rd to 8 th because the local dimension estimate is rather unstable for the 1 st and 2 nd layers depending on the preprocessing hyperparameter θ pre (Choi et al., 2022a).
Revised: For each StyleGAN model, we used the layers from 3 rd to 8 th because the local dimension estimate is rather unstable for the 1 st and 2 nd layers depending on the preprocessing hyperparameter θ pre (Choi et al., 2022a).

Original: We chose these intermediate layers for evaluation because they are diverse and properly disentangled latent spaces.
Revised: We chose these intermediate layers for evaluation because they are diverse and properly disentangled latent spaces.

Original: In this manner, the Fr´echet basis can be tested on six latent spaces for each pre-trained
Revised: In this manner, the Fr´echet basis can be tested on six latent spaces for each pre-trained

Original: StyleGAN model.
Revised: StyleGAN model.

Original: In this section, all Fr´echet basis are discovered using 1,000 samples of Local Basis with θ pre = 0 . 01 .
Revised: In this section, all Fr´echet basis are discovered using 1,000 samples of Local Basis with θ pre = 0 . 01 .

Original: The max iteration is set to 200 when optimizing Fr´echet mean with Pymanopt.
Revised: The max iteration is set to 200 when optimizing Fr´echet mean with Pymanopt.

Original: The evaluation is performed on two properties: Semantic Factorization and Robustness.
Revised: The evaluation is performed on two properties: Semantic Factorization and Robustness.

Original: Fr ´ echet basis
Revised: Fr ´ echet basis

Original: [Figure]
Revised: [Figure]

Original: denotes the layer-wise edit the i -

Original: GANSpace traversalsare (Fig 3a Fig is compared with GANSpace (H ¨ ark ¨ onen et al., 2020) and SeFa (Shen & Zhou, 2021) because these two methods are also unsupervised global basis (Sec 2).
Revised: denotes the the traversals is compared with GANSpace (H¨ark¨onen et al., 2020) and SeFa (Shen & Zhou, 2021) because these two methods are also unsupervised global basis (Sec 2).

Original: Note that GANSpace and Fr ´ echet basis can be applied to arbitrary latent space, but SeFa is only applicable to W -space (Karras et al., 2019), i.e., the last layer of the mapping network in StyleGANs.
Revised: Note that GANSpace and Fr´echet basis can be applied to arbitrary latent space, but SeFa is only applicable to W -space (Karras et al., 2019), i.e., the last layer of the mapping network in StyleGANs.

Original: Semantic Factorization In Fig 3 and 4, we evaluated the semantic factorization of Fr ´ echet basis.
Revised: Semantic Factorization In Fig 3 and 4, we evaluated the semantic factorization of Fr ´ echet basis.

Original: (See the appendix for additional examples.)

Original: Figure 3 shows how the image changes as we perturb the latent variable along each global basis.
Revised: Figure 3 shows how the image changes as we perturb the latent variable along each global basis.

Original: For a fair comparison, we took the annotated basis in GANSpace (H ¨ ark ¨ onen et al., 2020) and compared those with Fr ´ echet basis on W -space of three StyleGAN models.
Revised: For a fair comparison, we took the annotated basis in GANSpace (H ¨ ark ¨ onen et al., 2020) and compared those with Fr´echet basis on W -space of three StyleGAN models.

Original: Because GANSpace performs layer-wise edits, we matched the set of layers, where the perturbed latent variable is fed, in the synthesis network as annotated.
Revised: Because GANSpace performs layerwise edits, we matched the set of layers, where the perturbed latent variable is fed, in the synthesis network as annotated.

Original: The corresponding Fr ´ echet basis component is selected by the cosine-similarity.
Revised: The corresponding Fr´echet basis component is selected by the cosine-similarity.

Original: Each subfigure shows the three images traversed with the same global basis, perturbation intensity, and the set of perturbed layers.
Revised: Each subfigure shows the three images traversed with the same global basis, perturbation intensity, and the set of perturbed layers.

Original: The original image is placed at the center.
Revised: The original image is placed at the center.

Original: Hence, these subfigures also show the semantic consistency of the global basis.
Revised: Hence, these subfigures also show the semantic consistency of the global basis.

Original: In StyleGAN2 trained on FFHQ, GANSpace shows image failure on the left side and semantic inconsistency on the third row (not representing hairy on the left) (Fig 3b).
Revised: In StyleGAN2 trained on FFHQ, GANSpace shows image failure on the left side and semantic inconsistency on the third row (not representing hairy on the left) (Fig 3b).

Original: In StyleGAN trained on LSUN-cat (Yu et al., 2015), GANSpace presents entangled semantic manipulation (Fig 3d).
Revised: In StyleGAN2 trained on LSUN-cat (Yu et al., 2015), GANSpace presents entangled semantic manipulation (Fig 3d).

Original: The latent traversal along GANSpace changes the light position as annotated, but also darkens the striped pattern of cats.
Revised: The latent traversal along GANSpace changes the light position as annotated, but also darkens the striped pattern of cats.

Original: On the other hand, Fr´echet basis achieves better semantic factorization without showing those problems (Fig 3a and 3c).
Revised: On the other hand, Fr´echet basis achieves better semantic factorization without showing those problems (Fig 3a and 3c).

Revised: (See the appendix G for additional examples of other attributes and datasets.

Revised: Also, since Fr´echet basis is an average of Local Basis, we compared these two methods and GANSpace in the appendix F. )

Original: For the quantitative comparison of semantic factorization, we compared DCI as in Sec 3.2.
Revised: For the quantitative comparison of semantic factorization, we compared DCI as in Sec 3.2.

Original: DCI is a supervised disentanglement metric that assesses the axis-wise alignment of semantics.
Revised: DCI is a supervised disentanglement metric that assesses the axis-wise alignment of semantics.

Original: Hence, we measured DCIs of the latent space representations with two global basis, GANSpace and Fr´echet basis.
Revised: Hence, we measured DCIs of the latent space representations with two global basis, GANSpace and Fr´echet basis.

Original: Specifically, we converted a latent variable w ∈ R d ˜ W into the d W -dimensional representation with each global basis: For a d W -dimensional global basis { v globali } 1 ≤ [Equation]
Revised: Specifically, we converted a latent variable w ∈ R d ˜ W into the d W -dimensional representation with each global basis: For a d W -dimensional global basis { v globali } 1 ≤ i ≤ d W ,

Original: ≤ [Equation] Representation of w with { v globali } 
Revised: Representation of w with { v globali } 1 ≤ i ≤ d W = (cid:16) w · v global

Original: [Equation]
Revised: [Equation]

Original: Figure 4 presents the DCI results.
Revised: Figure 4 presents the DCI results.

Original: StyleGAN2-e denotes the config-e model of StyleGAN2 (Karras et al., 2020b), and all three models are trained on FFHQ (Karras et al., 2019).
Revised: StyleGAN2-e denotes the config-e model of StyleGAN2 (Karras et al., 2020b), and all three models are trained on FFHQ (Karras et al., 2019).

Original: The intrinsic dimension of each latent space is estimated with θ pre = 0 .
Revised: The intrinsic dimension of each latent space is estimated with θ pre = 0 .

Original: (Choi et al., 2022a).
Revised: (Choi et al., 2022a).

Original: In all latent spaces except for the 5 -th layer in StyleGAN2-e, the latent space achieves a higher DCI score when represented with
Revised: In all latent spaces except for the 5 -th layer in StyleGAN2-e, the latent space achieves a higher DCI score when represented with

Original: [Figure]
Revised: [Figure]

Original:        ) U p F KH W % D V L V  '& ,     
Revised:        ) U p F KH W % D V L V  '& ,    

Original: [Figure]
Revised: [Figure]

Original:      

Original: [Figure]

Original:      *$16SDFH),'      ) U p F KH W
Revised:      *$16SDFH),'      ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '     
Revised:  ) , '    

Original: [Figure]
Revised: [Figure]

Original: 
Revised: 

Original: [Figure]
Revised: [Figure]

Original:     [Figure] *$16SDFH),'      ) U p F KH W
Revised:   *$16SDFH),'      ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       [Figure]
Revised:  ) , '       (c) StyleGAN2-FFHQ

Original: Fr ´ echet basis.
Revised: Fr ´ echet basis.

Original: This quantitative result shows that Fr ´echet basis provides a better semantic factorization along each basis component at the same latent space .
Revised: ´

Original: Robustness We tested the robustness of Fr´echet basis by comparing the image fidelity under the latent perturbation.
Revised: Robustness We tested the robustness of Fr´echet basis by comparing the image fidelity under the latent perturbation.

Original: For each global basis, we evaluated the FID (Heusel et al., 2017) of 50k latent perturbed images.
Revised: For each global basis, we evaluated the FID (Heusel et al., 2017) of 50k latent perturbed images.

Original: The perturbation direction is selected to be the 1st component for the GANSpace.
Revised: The perturbation direction is selected to be the 1st component for the GANSpace.

Original: In Fr ´ echet basis, we chose the component with the highest cosine-similarity to the 1st component of
Revised: In Fr ´ echet basis, we chose the component with the highest cosine-similarity to the 1st component of

Original: GANSpace.
Revised: GANSpace.

Original: The perturbation intensity is 2 in StyleGAN1, and 5 in StyleGAN2-e and StyleGAN2.
Revised: The perturbation intensity is 2 in StyleGAN1, and 5 in StyleGAN2-e and StyleGAN2.

Original: The FID scores of the latent spaces in three models are provided in Fig 5.
Revised: The FID scores of the latent spaces in three models are provided in Fig 5.

Original: (See the appendix for FID scores under various perturbations intensity.)
Revised: (See the appendix for FID scores under various perturbations intensity.)

Original: We think this higher robustness is because the global semantic subspace S s is the Fr ´ echet mean of the intrinsic tangent spaces of learned latent manifold.
Revised: We think this higher robustness is because the global semantic subspace S s is the Fr ´ echet mean of the intrinsic tangent spaces of learned latent manifold.

Original: The strong robustness of traversing along the tangent space at each latent variable was observed in
Revised: The strong robustness of traversing along the tangent space at each latent variable was observed in

Original: Choi et al.
Revised: Choi et al.

Original: (2022b).
Revised: (2022b).

Original: Therefore, traversing along Fr´echet basis can be interpreted as traversing along the mean of these locally robust perturbation directions, because Fr´echet basis is a basis of S s .
Revised: Therefore, traversing along Fr´echet basis can be interpreted as traversing along the mean of these locally robust perturbation directions, because Fr´echet basis is a basis of S s .

Original: 4. B ASIS REFINEMENT BY F R ´ ECHET MEAN
Revised: 4. B ASIS REFINEMENT BY F R ´ ECHET MEAN

Original: [Table]
Revised: [Table]

Original: [Table]

Original: Fr´echet basis consists of two parts: finding the global semantic subspace S s and selecting the global semantic basis B s from the discovered semantic subspace.
Revised: Fr´echet basis consists of two parts: finding the global semantic subspace S s and selecting the global semantic basis B s from the discovered semantic subspace.

Original: In particular, the second step can be utilized to refine a given global basis discovered by previous methods.
Revised: In particular, the second step can be utilized to refine a given global basis discovered by previous methods.

Original: We ran this basis refinement on the subspace generated by the existing global methods, GANSpace and SeFa.
Revised: We ran this basis refinement on the subspace generated by the existing global methods, GANSpace and SeFa.

Original: This experiment reveals the contribution of the second step separated from the first step.
Revised: This experiment reveals the contribution of the second step separated from the first step.

Original: Let S global be the d W -dimensional semantic subspace generated by the global basis { v globali } 1 ≤ i ≤ d W .
Revised: Let S global be the d W -dimensional semantic subspace generated by the global basis { v globali } 1 ≤ i ≤ d W .

Original: Then, we optimized the constrained-optimal global basis B global in S global via Sec 3.1.2.
Revised: Then, we optimized the constrained-optimal global basis B global in S global via Sec 3.1.2.

Original: Table
Revised: Table

Original: [Figure]
Revised: [Figure]

Original: 
Revised: 

Original: [Figure]
Revised: [Figure]

Original: shows FID and DCI scores of Fr´echet basis, GANSpace, GANSpace refinement, SeFA, and SeFa refinement in W -space of StyleGAN2 trained on FFHQ.
Revised: shows FID and DCI scores of Fr´echet basis, GANSpace, GANSpace refinement, SeFA, and SeFa refinement in W -space of StyleGAN2 trained on FFHQ.

Original: These two scores are evaluated in the same manner as in Sec 4.1 with the perturbation intensity 2 for FID.
Revised: These two scores are evaluated in the same manner as in Sec 4.1 with the perturbation intensity 2 for FID.

Original: Most importantly, Fr´echet basis achieves the best FID and DCI.
Revised: Most importantly, Fr´echet basis achieves the best FID and DCI.

Original: Also, the basis refinement monotonically improves the two scores of the two previous methods, GANSpace and SeFa.
Revised: Also, the basis refinement monotonically improves the two scores of the two previous methods, GANSpace and SeFa.

Original: The comparison with the previous global method and its refinement proves the contribution of basis optimization.
Revised: The comparison with the previous global method and its refinement proves the contribution of basis optimization.

Original: Also, the superior performance of
Revised: Also, the superior performance of

Original: Fr ´ echet basis over the two refinements shows the contribution of subspace optimization.
Revised: Fr ´ echet basis over the two refinements shows the contribution of subspace optimization.

Original: 4. G EODESIC I NTERPOLATION ON G RASSMANNIAN M ANIFOLD
Revised: 4. G EODESIC I NTERPOLATION ON G RASSMANNIAN M ANIFOLD

Original: We further investigate the optimality of global semantic subspace S s by analyzing the interpolation from S s to GANSpace subspace S GS .
Revised: We further investigate the optimality of global semantic subspace S s by analyzing the interpolation from S s to GANSpace subspace S GS .

Original: Similar to Sec 4.2, this experiment examines the contribution of the first step in Fr ´ echet basis.
Revised: Similar to Sec 4.2, this experiment examines the contribution of the first step in Fr ´ echet basis.

Original: In the Grassmannian manifold Gr ( k, R n ) , there exists at least one length-minimizing geodesic between any two subspaces in Gr ( k, R n ) .
Revised: In the Grassmannian manifold Gr ( k, R n ) , there exists at least one length-minimizing geodesic between any two subspaces in Gr ( k, R n ) .

Original: Moreover, there is an explicit parametrization of this length-minimizing geodesic (Eq 14) (Chakraborty & Vemuri, 2015).
Revised: Moreover, there is an explicit parametrization of this length-minimizing geodesic (Eq 14) (Chakraborty & Vemuri, 2015).

Original: For
Revised: For

Original: X , Y ∈
Revised: X , Y ∈

Original: Gr ( k, R n ) , let X, Y ∈ R n × k be the column-wise concatenation of their orthonormal basis.
Revised: Gr ( k, R n ) , let X, Y ∈ R n × k be the column-wise concatenation of their orthonormal basis.

Original: Then, the length-minimizing geodesic Γ( X , Y , t ) from X to Y is defined as:
Revised: Then, the length-minimizing geodesic Γ( X , Y , t ) from X to Y is defined as:

Original: [Equation]
Revised: [Equation]

Original: where X ⊤ Y is non-singular, ( Y − XX ⊤ Y )
Revised: where X ⊤ Y is non-singular, ( Y − XX ⊤ Y )

Original: ( X ⊤ Y ) − 1 = U Σ V ⊤ is the thin SVD, and Θ = arctan Σ .
Revised: ( X ⊤ Y ) − 1 = U Σ V ⊤ is the thin SVD, and Θ = arctan Σ .

Original: Note that the last V ⊤ multiplication is unnecessary as the subspace.
Revised: Note that the last V ⊤ multiplication is unnecessary as the subspace.

Original: However, we added it to match the basis to X at t = 0 .
Revised: However, we added it to match the basis to X at t = 0 .

Original: We used this geodesic to perform the interpolation and extrapolation from S s to S GS :
Revised: We used this geodesic to perform the interpolation and extrapolation from S s to S GS :

Original: [Equation]
Revised: [Equation]

Original: with n = 6 .
Revised: with n = 6 .

Original: Note that i = 0 , n + 2 represent the extrapolations of t = ( − 1 /n ) , 1 + (1 /n ) .
Revised: Note that i = 0 , n + 2 represent the extrapolations of t = ( − 1 /n ) , 1 + (1 /n ) .

Original: Because the above interpolation is performed on a subspace-scale, we conducted the basis refinement (Sec 4.2) for each interpolation subspace S i to find the interpolation basis B i .
Revised: Because the above interpolation is performed on a subspace-scale, we conducted the basis refinement (Sec 4.2) for each interpolation subspace S i to find the interpolation basis B i .

Original: Figure 6 presents the geodesic metric d geo in the Grassmannian manifold for each interpolation subspace to GANSpace and Fr´echet basis (Fig 6a), and the DCI score evaluated at each interpolation basis
Revised: Figure 6 presents the geodesic metric d geo in the Grassmannian manifold for each interpolation subspace to GANSpace and Fr´echet basis (Fig 6a), and the DCI score evaluated at each interpolation basis

Original: B i (Fig 6b) on W -space of
Revised: B i (Fig 6b) on W -space of

Original: StyleGAN2-FFHQ.
Revised: StyleGAN2-FFHQ.

Original: First, Fig 6a shows that S i performs interpolation from Fr´echet basis at i = 

Revised: First, Fig 6a shows that S i performs interpolation from Fr´echet basis at i =

Original: to
Revised: to

Original: GANSpace at i = 7 .
Revised: GANSpace at i = 7 .

Original: This interpolation is linear in the Grassmannian metric d geo .
Revised: This interpolation is linear in the Grassmannian metric d geo .

Original: Second, the DCI score at each interpolation basis B i are presented in Fig 6b.
Revised: Second, the DCI score at each interpolation basis B i are presented in Fig 6b.

Original: Fr´echet basis at i = 

Revised: Fr´echet basis at i =

Original: achieves the best
Revised: achieves the best

Original: DCI score among the interpolation basis.
Revised: DCI score among the interpolation basis.

Original: Note that the DCI score of original GANSpace without refinement is 0.312 (Tab 1), which is lower than the score after refinement at i = 7 as in Sec 4.2.
Revised: Note that the DCI score of original GANSpace without refinement is 0.312 (Tab 1), which is lower than the score after refinement at i = 7 as in Sec 4.2.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we proposed the unsupervised global semantic basis on the intermediate latent space in a GAN, called Fr´echet basis.
Revised: In this paper, we proposed the unsupervised global semantic basis on the intermediate latent space in a GAN, called Fr´echet basis.

Original: Fr´echet basis is discovered by utilizing the Fr´echet mean on the
Revised: Fr´echet basis is discovered by utilizing the Fr´echet mean on the

Original: Grassmannian manifold and the Special Orthogonal Group.
Revised: Grassmannian manifold and the Special Orthogonal Group.

Original: Our experiments demonstrate that Fr´echet basis achieves better semantic factorization and robustness than the previous unsupervised global methods.
Revised: Our experiments demonstrate that Fr´echet basis achieves better semantic factorization and robustness than the previous unsupervised global methods.

Original: In addition, we suggest the basis refinement scheme using the Fr´echet mean.
Revised: In addition, we suggest the basis refinement scheme using the Fr´echet mean.

Original: Given the same semantic subspace generated by the previous global methods, the refined basis attains better semantic factorization and robustness.
Revised: Given the same semantic subspace generated by the previous global methods, the refined basis attains better semantic factorization and robustness.

Original: A P ROJECTION ONTO S PECIAL O RTHOGONAL G ROUP
Revised: A P ROJECTION ONTO S PECIAL O RTHOGONAL G ROUP

Original: In this section, we provide proof for the projection of an invertible matrix A ∈ GL ( n ) onto the special orthogonal group SO ( n ) .
Revised: In this section, we provide proof for the projection of an invertible matrix A ∈ GL ( n ) onto the special orthogonal group SO ( n ) .

Original: Formally, the set of invertible matrices GL ( n ) , orthogonal group
Revised: Formally, the set of invertible matrices GL ( n ) , orthogonal group

Original: O ( n ) , and special orthogonal group SO ( n ) are defined as follows:
Revised: O ( n ) , and special orthogonal group SO ( n ) are defined as follows:

Original: [Equation]
Revised: [Equation]

Original: For a non-invertible matrix, the projection onto SO ( n ) is not uniquely defined because of the subspace generated by singular vectors with σ = 0.
Revised: For a non-invertible matrix, the projection onto SO ( n ) is not uniquely defined because of the subspace generated by singular vectors with σ = 0.

Original: However, the set of non-invertible matrices has measure-zero in the set of n × n matrices R n × n .
Revised: However, the set of non-invertible matrices has measure-zero in the set of n × n matrices R n × n .

Original: Thus, this did not happen in practice during the
Revised: Thus, this did not happen in practice during the

Original: Fr´echet basis optimization.
Revised: Fr´echet basis optimization.

Original: Theorem 1.
Revised: Theorem 1.

Original: The following optimization problem, i.e., the projection of A ∈ GL ( n ) onto the Special
Revised: The following optimization problem, i.e., the projection of A ∈ GL ( n ) onto the Special

Original: Orthogonal Group SO ( n ) ,
Revised: Orthogonal Group SO ( n ) ,

Original: [Equation]
Revised: [Equation]

Original: where A ∈ GL ( n ) , (19) has a solution
Revised: where A ∈ GL ( n ) , (19) has a solution

Original: [Equation]
Revised: [Equation]

Original: (20) where A = U Σ V ⊤ is the Singular Value Decomposition (SVD) of A with the projection onto orthogonal group P o ( A ) = UV ⊤ .
Revised: (20) where A = U Σ V ⊤ is the Singular Value Decomposition (SVD) of A with the projection onto orthogonal group P o ( A ) = UV ⊤ .

Original: (The projection is unique if we assume σ 1 > σ 2 > . . .
Revised: (The projection is unique if we assume σ 1 > σ 2 > . . .

Original: > σ n >where { σ i } 1 ≤ i ≤ n denote the singular values of A.)
Revised: > σ n >where { σ i } 1 ≤ i ≤ n denote the singular values of A.)

Original: Proof.
Revised: Proof.

Original: If P o ( A ) = UV ⊤ ∈ SO ( n ) , done.
Revised: If P o ( A ) = UV ⊤ ∈ SO ( n ) , done.

Original: ( ∵ SO ( n ) ⊂ O ( n ) and det ( P o ( A ))) = 1) .
Revised: ( ∵ SO ( n ) ⊂ O ( n ) and det ( P o ( A ))) = 1) .

Original: If P o ( A ) = UV ⊤ / ∈ SO ( n ) , i.e., det( A ) < 0 ,
Revised: If P o ( A ) = UV ⊤ / ∈ SO ( n ) , i.e., det( A ) < 0 ,

Original: [Equation]
Revised: [Equation]

Original: For any skew-symmetric matrix K , define f ( t ) as
Revised: For any skew-symmetric matrix K , define f ( t ) as

Original: [Equation]
Revised: [Equation]

Original: Note that Xe tK ∈ SO ( n ) and tr( A ⊤ A ) is given.
Revised: Note that Xe tK ∈ SO ( n ) and tr( A ⊤ A ) is given.

Original: Therefore, if X is the minimizer of ∥ X − A ∥ F , we have f ′ (0)
Revised: Therefore, if X is the minimizer of ∥ X − A ∥ F , we have f ′ (0)

Original: = − 2 tr
Revised: = − 2 tr

Original: ( A ⊤ XK )
Revised: ( A ⊤ XK )

Original: = 

Revised: =

Original: for all skew-symmetric K .
Revised: for all skew-symmetric K .

Original: Thus, A ⊤ X is symmetric.
Revised: Thus, A ⊤ X is symmetric.

Original: Without loss of generality, we may assume that A = U 0 Σ 0 V ⊤ 0 , X
Revised: Without loss of generality, we may assume that A = U 0 Σ 0 V ⊤ 0 , X

Original: = U 

Revised: = U

Original: X ′ V ⊤ 0 for some U 0 , V 0 ∈
Revised: X ′ V ⊤ 0 for some U 0 , V 0 ∈

Original: SO ( n ) where Σ 0 is the diagonal matrix as Σ in SVD but Σ 0 might have the negative elements.
Revised: SO ( n ) where Σ 0 is the diagonal matrix as Σ in SVD but Σ 0 might have the negative elements.

Original: This decomposition can be obtained by flipping the singular vectors in U, V of SVD to make
Revised: This decomposition can be obtained by flipping the singular vectors in U, V of SVD to make

Original: U 0 , V 0 ∈ SO ( n ) and letting X ′ =
Revised: U 0 , V 0 ∈ SO ( n ) and letting X ′ =

Original: U ⊤ 0 XV 0 .
Revised: U ⊤ 0 XV 0 .

Original: Explicitly, from det( A ) < 0 ,
Revised: Explicitly, from det( A ) < 0 ,

Original: [Equation]
Revised: [Equation]

Original: det
Revised: det

Original: [Equation]
Revised: [Equation]

Original: Then, since X ⊤ A = (cid:0) A ⊤ X (cid:1) ⊤ ,
Revised: Then, since X ⊤ A = (cid:0) A ⊤ X (cid:1) ⊤ ,

Original: X ⊤ A
Revised: X ⊤ A

Original: = V
Revised: = V

Original: [Equation]
Revised: [Equation]

Original: X ′ )
Revised: X ′ )

Original: ⊤ Σ 0 V ⊤ 0 is symmetric and has negative determinant.
Revised: ⊤ Σ 0 V ⊤ 0 is symmetric and has negative determinant.

Original: Therefore, ( X ′ )
Revised: Therefore, ( X ′ )

Original: ⊤ Σ 0 is also symmetric and thus diagonalizable.
Revised: ⊤ Σ 0 is also symmetric and thus diagonalizable.

Original: [Equation]
Revised: [Equation]

Original: where λ i denotes the i -th eigenvalue of ( X ′ )
Revised: where λ i denotes the i -th eigenvalue of ( X ′ )

Original: ⊤ Σ 0 with | λ 1 | ≥ | λ 2 | ≥ . .
Revised: ⊤ Σ 0 with | λ 1 | ≥ | λ 2 | ≥ . .

Original: ≥ | λ n | .
Revised: ≥ | λ n | .

Original: Note that since
Revised: Note that since

Original: X ′ =
Revised: X ′ =

Original: U ⊤ 0 XV 0 ∈ SO ( n ) , the i -th singular value σ i of A satisfies σ i = | λ i | .
Revised: U ⊤ 0 XV 0 ∈ SO ( n ) , the i -th singular value σ i of A satisfies σ i = | λ i | .

Original: Moreover, an odd number of signed singular values in Σ 0 is negative because det( A ) < 0 .
Revised: Moreover, an odd number of signed singular values in Σ 0 is negative because det( A ) < 0 .

Original: Also, det(( X ′ )
Revised: Also, det(( X ′ )

Original: ⊤ Σ 0 )
Revised: ⊤ Σ 0 )

Original: = det( X ⊤ A ) < 0 implies that an odd number of eigenvalues is negative.
Revised: = det( X ⊤ A ) < 0 implies that an odd number of eigenvalues is negative.

Original: Hence, ∥ X − A ∥ F is minimized when λ i =
Revised: Hence, ∥ X − A ∥ F is minimized when λ i =

Original: σ i for 1 ≤ i ≤ n − 1 and λ n = − σ n .
Revised: σ i for 1 ≤ i ≤ n − 1 and λ n = − σ n .

Original: If X ′ is the diagonal matrix satisfying this condition, X = U diag (1 , 1 , . . .
Revised: If X ′ is the diagonal matrix satisfying this condition, X = U diag (1 , 1 , . . .

Original: , 1 , − 1) V ⊤ .
Revised: , 1 , − 1) V ⊤ .

Original: Lemma 1 proves that when σ 1 > σ 2 > . . .
Revised: Lemma 1 proves that when σ 1 > σ 2 > . . .

Original: >
Revised: >

Original: σ n is satisfied, the uniqueness of X ′ is guaranteed.
Revised: σ n is satisfied, the uniqueness of X ′ is guaranteed.

Original: Lemma 1.
Revised: Lemma 1.

Original: Let X ′ ∈
Revised: Let X ′ ∈

Original: SO ( n ) and Σ 0 = diag ( σ 1 , . . .
Revised: SO ( n ) and Σ 0 = diag ( σ 1 , . . .

Original: , σ n − 1 , − σ n ) with σ 1 > . . .
Revised: , σ n − 1 , − σ n ) with σ 1 > . . .

Original: >
Revised: >

Original: σ n − 1 > σ n > 0 .
Revised: σ n − 1 > σ n > 0 .

Original: If Y = ( X ′ )
Revised: If Y = ( X ′ )

Original: ⊤ Σ 0 is symmetric, X ′ is a diagonal matrix.
Revised: ⊤ Σ 0 is symmetric, X ′ is a diagonal matrix.

Original: Proof.
Revised: Proof.

Original: Since Y is symmetric, it is diagonalizable with the orthogonal matrix
Revised: Since Y is symmetric, it is diagonalizable with the orthogonal matrix

Original: P ∈
Revised: P ∈

Original: O
Revised: O

Original: ( n ) .
Revised: ( n ) .

Original: [Equation]
Revised: [Equation]

Original: We interpret these two decompositions Y =
Revised: We interpret these two decompositions Y =

Original: ( X ′ )
Revised: ( X ′ )

Original: ⊤ Σ 

Revised: ⊤ Σ

Original: I = PDP t as two SVD-like representations of Y because ( X ′ )
Revised: I = PDP t as two SVD-like representations of Y because ( X ′ )

Original: ⊤ ∈ SO ( n ) and I ⊤
Revised: ⊤ ∈ SO ( n ) and I ⊤

Original: =
Revised: =

Original: I .
Revised: I .

Original: The ordered singular vectors in the domain of Y are uniquely determined as the basis for each eigenspace of Y ⊤ Y .
Revised: The ordered singular vectors in the domain of Y are uniquely determined as the basis for each eigenspace of Y ⊤ Y .

Original: Note that if the dimension of eigenspace is bigger than 1, there is a freedom of choosing a basis in it.
Revised: Note that if the dimension of eigenspace is bigger than 1, there is a freedom of choosing a basis in it.

Original: From Σ 0 , the possible eigenvalues of Y are {± σ i } 1 ≤ i ≤ n and the eigenvalues of Y ⊤ Y are { σ 2 i } 1 ≤ i ≤ n .
Revised: From Σ 0 , the possible eigenvalues of Y are {± σ i } 1 ≤ i ≤ n and the eigenvalues of Y ⊤ Y are { σ 2 i } 1 ≤ i ≤ n .

Original: Therefore, since the standard basis { e i } 1 ≤ i ≤ n of R n are the domain singular vectors of Y from
Revised: Therefore, since the standard basis { e i } 1 ≤ i ≤ n of R n are the domain singular vectors of Y from

Original: Y =
Revised: Y =

Original: ( X ′ )
Revised: ( X ′ )

Original: ⊤ Σ 

Revised: ⊤ Σ

Original: I and Y is diagonalizable,
Revised: I and Y is diagonalizable,

Original: Y ( e i )
Revised: Y ( e i )

Original: [Equation]
Revised: [Equation]

Original: e i
Revised: e i

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: n
Revised: n

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Hence, the standard basis are also the codomain singular vectors of Y , which implies that X ′ is diagonal.
Revised: Hence, the standard basis are also the codomain singular vectors of Y , which implies that X ′ is diagonal.

Original: B I MPLEMENTATION D ETAIL
Revised: B I MPLEMENTATION D ETAIL

Original: In this section, we summarize the hyperparameters for Frechet basis presented in the experimental results in Section 4.
Revised: In this section, we summarize the hyperparameters for Frechet basis presented in the experimental results in Section 4.

Original: • Preprocossing hyperparameter θ pre for local dimension estimation Choi et al.
Revised: • Preprocossing hyperparameter θ pre for local dimension estimation Choi et al.

Original: (2022a): θ pre = 0 .
Revised: (2022a): θ pre = 0 .

Original: • Global Semantic Subspace Optimization – Number of samples n = 1000 .
Revised: • Global Semantic Subspace Optimization – Number of samples n = 1000 .

Original: – Max iteration in Frechet mean Optimization using Pymanopt Townsend et al.
Revised: – Max iteration in Frechet mean Optimization using Pymanopt Townsend et al.

Original: = 1 , 000 .
Revised: = 1 , 000 .

Original: – Max time in Frechet mean Optimization using Pymanopt = 2 , 000 .
Revised: – Max time in Frechet mean Optimization using Pymanopt = 2 , 000 .

Original: • Global Semantic Basis Optimization – Number of samples n = 1000 .
Revised: • Global Semantic Basis Optimization – Number of samples n = 1000 .

Original: – Max iteration in Frechet mean Optimization using Pymanopt = 200 .
Revised: – Max iteration in Frechet mean Optimization using Pymanopt = 200 .

Original: – Max time in Frechet mean Optimization using Pymanopt = 10 , 000 .
Revised: – Max time in Frechet mean Optimization using Pymanopt = 10 , 000 .

Original: C F ULL CORRELATION RESULTS OF L 2 -D ISTORTION
Revised: C F ULL CORRELATION RESULTS OF L 2 -D ISTORTION

Original: [Figure]
Revised: [Figure]

Original:     ) , '
Revised:     ) , '

Original:  * DS      
Revised:  * DS      

Original: [Figure]
Revised: [Figure]

Original:     
Revised:     

Original: [Figure]
Revised: [Figure]

Original:  
Revised:  

Original: [Figure]
Revised: [Figure]

Original:  
Revised:  

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: D Q UANTITATIVE COMPARISON OF ROBUSTNESS (FID) FOR VARIOUS PERTURBATION INTENSITY
Revised: D Q UANTITATIVE COMPARISON OF ROBUSTNESS (FID) FOR VARIOUS PERTURBATION INTENSITY

Original:        *$16SDFH),'        ) U p F KH W
Revised:        *$16SDFH),'        ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (a) I = 1 .       *$16SDFH),'        ) U p F KH W
Revised:  ) , '       (a) I = 1 .       *$16SDFH),'        ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (b) I = 1 .     *$16SDFH),'      ) U p F KH W
Revised:  ) , '       (b) I = 1 .     *$16SDFH),'      ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (c) I = 2 .       *$16SDFH),'        ) U p F KH W
Revised:  ) , '       (c) I = 2 .       *$16SDFH),'        ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (d) I = 2 .
Revised:  ) , '       (d) I = 2 .

Original:       *$16SDFH),'       ) U p F KH W
Revised:       *$16SDFH),'       ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (a) I = 3 .        *$16SDFH),'         ) U
Revised:  ) , '       (a) I = 3 .        *$16SDFH),'         ) U

Original: p F KH W
Revised: p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (b) I = 4 .
Revised:  ) , '       (b) I = 4 .

Original:      *$16SDFH),'       ) U p F KH W
Revised:      *$16SDFH),'       ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (c) I = 5 .        *$16SDFH),'         ) U p F KH W
Revised:  ) , '       (c) I = 5 .        *$16SDFH),'         ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (d) I = 6 .
Revised:  ) , '       (d) I = 6 .

Original:       *$16SDFH),'       ) U p F KH W
Revised:       *$16SDFH),'       ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (a) I = 2 .      *$16SDFH),'       ) U p F KH W
Revised:  ) , '       (a) I = 2 .      *$16SDFH),'       ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (b) I = 3 .         *$16SDFH),'          ) U p F KH W
Revised:  ) , '       (b) I = 3 .         *$16SDFH),'          ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (c) I = 4 .
Revised:  ) , '       (c) I = 4 .

Original:     *$16SDFH),'      ) U p F KH W
Revised:     *$16SDFH),'      ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '       (d) I = 5 .
Revised:  ) , '       (d) I = 5 .

Original: E A BLATION STUDY ON THE OTHER MEANS OF G RASSMANNIAN MANIFOLD
Revised: E A BLATION STUDY ON THE OTHER MEANS OF G RASSMANNIAN MANIFOLD

Original: In this section, we conducted an ablation study on defining the global semantic subspace in the
Revised: In this section, we conducted an ablation study on defining the global semantic subspace in the

Original: Grassmanifold.
Revised: Grassmanifold.

Original: In particular, we compared the Fr´echet mean Fr´echet (1948); Marrinan et al.
Revised: In particular, we compared the Fr´echet mean Fr´echet (1948); Marrinan et al.

Original: and extrinsic mean Srivastava & Klassen (2002) of the Grassmannian manifold.
Revised: and extrinsic mean Srivastava & Klassen (2002) of the Grassmannian manifold.

Original: The extrinsic mean µ E is defined as the minimizer of squared extrinsic metrics d E , i.e., for x 1 , . .
Revised: The extrinsic mean µ E is defined as the minimizer of squared extrinsic metrics d E , i.e., for x 1 , . .

Original: , x n ∈ Gr ( k, R n ) ,
Revised: , x n ∈ Gr ( k, R n ) ,

Original: [Equation]
Revised: [Equation]

Original: where d E ( µ, x i ) =
Revised: where d E ( µ, x i ) =

Original: d Φ (Φ( µ ) , Φ( x i )) ,
Revised: d Φ (Φ( µ ) , Φ( x i )) ,

Original: [Equation]
Revised: [Equation]

Original: 29) where Φ denotes an appropriate embedding of Gr ( k, R n ) .
Revised: 29) where Φ denotes an appropriate embedding of Gr ( k, R n ) .

Original: Following Marrinan et al.
Revised: Following Marrinan et al.

Original: (2014), we set the embedding Φ to be the corresponding projection P x i , i.e., Φ( x i ) =
Revised: (2014), we set the embedding Φ to be the corresponding projection P x i , i.e., Φ( x i ) =

Original: P x i =
Revised: P x i =

Original: M ⊤ x i M x i where
Revised: M ⊤ x i M x i where

Original: M x i ∈ R n × k indicates the column-wise concatenation of an orthonormal basis of x i .
Revised: M x i ∈ R n × k indicates the column-wise concatenation of an orthonormal basis of x i .

Original: Also, the
Revised: Also, the

Original: Frobenius norm is adopted for d Φ .
Revised: Frobenius norm is adopted for d Φ .

Original:        
Revised:        

Original:         
Revised:         

Original:                    ([WULQVLF%DVLV'&,       
Revised:                    ([WULQVLF%DVLV'&,       

Original:   ) U p F KH W
Revised:   ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ' & ,       (a) DCI ( ↑ )         ([WULQVLF%DVLV),'         ) U p F KH W
Revised:  ' & ,       (a) DCI ( ↑ )         ([WULQVLF%DVLV),'         ) U p F KH W

Original: % D V L V
Revised: % D V L V

Original:  ) , '      
Revised:  ) , '      

Original: [Figure]
Revised: [Figure]

Original: Extrinsic basis is a variant of Fr´echet basis where the global semantic basis is discovered by the extrinsic mean of the Grassmannian manifold.
Revised: Extrinsic basis is a variant of Fr´echet basis where the global semantic basis is discovered by the extrinsic mean of the Grassmannian manifold.

Original: In both scores, our Fr´echet basis outperforms Extrinsic basis in 5 out 6 intermediate layers in the mapping network.
Revised: In both scores, our Fr´echet basis outperforms Extrinsic basis in 5 out 6 intermediate layers in the mapping network.

Original: F A DDITIONAL S EMANTIC F ACTORIZATION COMPARISON
Revised: F C OMPARISON TO L OCAL B ASIS F. Q UANTITATIVE C OMPARISON

Original: [Figure]
Revised: In this section, we introduced a new experiment to quantitatively compare semantic factorization between Fr´echet basis and Local Basis Choi et al.

Original: [Figure]
Revised: (2022b).

Original: [Figure]
Revised: Intuitively, this experiment measures the average of local DCI scores.

Original: [Figure]
Revised: To be more specific, consider n -samples { z i } 1 ≤ i ≤ n ⊂ Z of input

Original: [Figure]
Revised: Gaussian noise.

Original: [Figure]
Revised: (We set n = 100 )

Revised: Then, take m -samples from the neighborhood of each z i and map them to the target latent space W = f

Revised: ( Z ) , where f denotes the subnetwork from Z to W : w i , j = f

Revised: ( z i + ϵ i,j ) where ϵ i,j ∼ N (0 , σ 2 I ) and w i = f

Revised: ( z i ) , (30) for sufficiently small σ > 0 .

Revised: (We set m = 1 , 000 and σ = 0 . 5 )

Revised: Then, we measure DCI score for each neighborhood of w i , i.e., { w i,j } 1 ≤ j ≤ m and take an average of these local DCI scores, i.e., DCI ( { w i,j } 1 ≤ j ≤ m ) .

Revised: Below, we included the evaluated scores on W -space of StyleGAN2, e, and

Revised: FFHQ.

Revised: As

Revised: , Fr´echet outperforms Local Basis in the quantitative assessment.

Revised: [Table]

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: G A DDITIONAL S EMANTIC F ACTORIZATION C OMPARISON

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]