Grassmann Manifold Flows for Stable Shape Generation
Authors: Ryoma Yataka
Kazuki Hirashima
Masashi Shiraishi
Venue: NeurIPS 2023 poster (2023)
Keywords: Generative Models
Geometric Deep Learning
Normalizing Flows
Shape Analysis
Grassmann Manifold
TL;DR: This paper establishes the theoretical foundations for learning distributions on the Grassmann manifold via continuous normalizing flows, with the explicit goal of generating stable shapes.

Abstract: Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias.
Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling stable shape analysis. 
In this paper, we present a novel approach in which we establish the theoretical foundations for learning distributions on the Grassmann manifold via continuous normalization flows, with the explicit goal of generating stable shapes.
Our approach facilitates more robust generation by effectively eliminating the influence of extraneous transformations, such as rotations and inversions, through learning and generating within a Grassmann manifold designed to accommodate the essential shape information of the object.
The experimental results indicated that the proposed method could generate high-quality samples by capturing the data structure.
Furthermore, the proposed method significantly outperformed state-of-the-art methods in terms of the log-likelihood or evidence lower bound.
The results obtained are expected to stimulate further research in this field, leading to advances for stable shape generation and analysis.

Grassmann Manifold Flows for Stable Shape Generation
Ryoma Yataka 1.2 Kazuki Hirashima', Masashi Shiraishi' Information Technology R&D Center (ITC), Mitsubishi Electric Corporation,
Kamakura, Kanagawa, 247-8501, Japan
Mitsubishi Electric Research Laboratories (MERL), Cambridge, MA 02139, USA (Yataka. .wweaaada.yomddd....ommmm htw../ommmee..tttt...eeeeee.ee.. MitsubbsshiEeetriiee co. jjp, yataka@merl, )omm
Abstract
Recently, studies on machine learning have focused on methods that use symmetry implicit in specific manifold inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling stable shape analysis. In this paper, we present novel approach in which we establish the theoretical foundations for learning distributions on the Grassmann manifold via continuous normalization flows, with the explicit goal of generating stable shapes. Our approach facilitates more robust generation by effectively eliminating the influence of extraneous transformations, such rotations and inversions, through learning and generating within a Grassmann manifold designed accommodate the essential shape information of the object. The experimental results indicated that the proposed method could generate high-quality samples by capturing the data structure. Furthermore, the proposed method significantly outperformed state-of-the-art methods in terms of the log- log-likelihood or evidence lower bound. The results obtained are expected to stimulate further research in this field, leading to advances for stable shape generation and analysis.
Introduction
Many machine learning algorithms are designed to automatically learn and extract latent factors that explain a specific dataset. Symmetry known to be inductive bias (that is, prior knowledge other than training data that can contribute significantly to learning results) for learning latent factors (Cohen & Welling (2016) 2017); Weiler & Cesa (2019); [Storras al. (202T;; Bronstein al. (202T): Puny et al. (202)) ). Moreover, they exist in many phenomena in the natural sciences. If target M is invariant when the operation g e S designated by S applied to M, M has symmetry S. For example. a sphere remains a sphere even if a rotation R is applied; a symmetric shape remains symmetric even a left-right reversal is applied.
Recent studies have focused on methods incorporating symmetry into models based on equivariance and invariance, when the data space forms mon-Euclidean space (a sphere S or special unitary group SU())) (Cohen et al. (2018) 2019b); Graham et al. (2020); Haan et al. (2021) Boyda et al. (2021a)). Among them, discriminative and generative models for subspace data (e.g., shape matrices of point cloud data such as firee-dimensionl molecules and general object shapes) have been proposed viable approaches using Grassmann manifolds, which are quotient manifolds obtained by introducing function that induces invariance with respect to orthogonal transformations into
'Invariance implics the transformation æ satisfics (a)) = π (x') ग (g(x)) for x' (x) obtained by applying the operation the input Transformation is considered invariant with respect operation g. transformation и considered equivariant with respect operation p я satisfies g (5 (x)) = n (g(z)).
Invariance is special case equivariance where (g) an identity transformation.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

---

set of orthonormal basis matrices namely Stiefel Manifold. Numerous studies have confirmed
their effectiveness (Liu al. (2003); Hamm Lee (2008); : Turaga et al.] (2008); Harandi et al.
(2011): Fan et al. (2011 1 ; Lui (2012); : Huang et al. 2015 2018); Souza et al. (2020): Haitman et al.
(2021): Doronina et al. (2022) Souza et al. (022)) ). Shape theory studies the equivalent class of all configurations that can be obtained by a specific class of transformation (e.g. linear, affine, projective) on a single basis shape [Patrangenaru & Mardia (003);; (Seiiasviilll et al. (2003); [Begelfor & Werman (2006)), and it can be shown that affine and linear shape spaces for specific configurations can be identified by points on the Grassmann manifold. By applying this model to inverse molecular design for stable shape generation to discover new molecular structures, they can contribute to the development of drug discovery, computational anatomy, and materials science SannhezzLLngellngg & Aspuru-Guzik 2018); Bilodeau et al. (2022)).
Continuous normalizing flows (CNFs; Chen et al. (018): Grathwohl et al. (2019); Lou et al. (2020);
Kim et al. (020);; Mathieu & Nickel (2220) are generative models that have attracted attention in recent years along with the variational auto-encoders (VAE; Kingma & Welling (2014)), generative
adversarial networks (GAN; Goodfellow et al. (2014)), autoregressive models (Germain al. (2015);
Oord et al. (2016)), and diffusion models (Sohl-Dickstein et al. | (2015); Ho et al. (2020 Song et al. (2021); Huang et al. (2022); De Bortoli et al. (2022)) The CNF is method with theoretically superior properties that allow rigorous inference and evaluation of the log- likelihood. It can be trained by maximum likelihood using the change of variables formula, which allows to specify complex normalized distribution implicitly by warping a normalized base distribution through an integral of continuous- tommtumeuss-tueeeeo dynamics. In this paper, we present novel approach with the explicit goal of generating stable shapes. We employ the CNF to achieve stable shape generation on a Grassmann manifold; however, previous studies have lacked the theoretical foundation to handle the CNF on the Grassmann manifold. To construct this, we focused on the quotient structure of a Grassmann manifold and translated the problem of flow learning on a Grassmann manifold into that of preserving equivariance for orthogonal groups on a Stiefel manifold, which is its total space. The contributions of this study are as follows.
A theory and general framework for learning flows Grassmann manifold are proposed. In our setting, we can train flows on a Grassmann manifold of arbitrary dimension. To the best of our knowledge, this is the first study in which CNF was constructed on Grassmann manifold via a unified approach, focusing on the quotient structure.
The validity of the proposed approach was demonstrated by learning densities on a Grass- mann manifold using multiple artificial datasets with complex data distributions. In par- ticular, orthogonally transformed data were proven to be correctly learned without data augmentations by showing that untrained transformed (i.e., rotated or mirrored) data can be generated from the trained model. The model was evaluated on multiple patterns of training data and performed well with small amount of training data.
The effectiveness of our approach was confirmed by its state-of- taite-of-thearrd fit-hhe-rtt art performance in a
molecular positional generation task.
2 Related Works
Symmetry-basedd Learning The concept of equivariance has been studied in recent years to
leverage the symmetries inherent in data (Cohen & Welling (017)) Kondor & Trivedi 2018 1 Cohen
et al.) (2018 2019b a): Finzi et al. (000)) Haan et al.J (2021) )). Early work by CCohen & Welling
(2017) indicated that when data are equivariant, they can be processed with a lower computational
| cost and fewer parameters. In the context of CNF' (Rezende & Mohamed (2015); Chen et al. (2018):
Grathwohl et al. )2019). which are generative models, Rezende et al. (2019). Köhler et al. (2020)
and Garcia Satorras et al. (202T) proposed equivariant normalizing flows to learn symmetric densities on Euclidean spaces. Furthermore. the symmetries that appear in learning densities a manifold were introduced by Boyyda et al. (2021b) and Katsman et al. (202I) as a conjugate equivariant flow on SU (n), which is quotient manifold, for use in lattice gauge theory. However, normalizing flow on a Grassmann manifold capable of handling subspace data has not yet been established.
Shape Data Analysis and Other Applications with Subspace on the Grassmann Manifold dimensional shape data (Begelfor& & Werman (2006); Yoshinuma et al. (016);; Haitman al. (202

---

Doronina et al. (2022)) such as point clouds are essentially represented as subspace (shape space (Sepiashvili et al. (2003); SSrivastaa et al. (2005); Begelior & Werman (200)); Yataka & Fukui (2017)) data on a Grassmann manifold. Furthermore, subspace data can be obtained on many types of data and can provide advantages such as practicability and noise robustness. For example, multiview
image sets or video data (Fan et al. (2011); Turaga et al. (20TT); Lui (2012); Alashkar et al. (2016)) signal data (Srivastava Klassen 2004);; Gatto et al. (2017); Souza et al. 2019); Yataka et al. 2019)),
and text data (Shimomoto et al. (021)) are often provided as a set of feature vectors. Such raw data in matrix form is not very useful owing to its considerable size and noise. The analysis of its eigenspace, column subspace, is important because alternatively, the raw data matrix can be well approximated by low-dimensional subspace with basis vectors corresponding to the maximum eigenvalues of the matrix. Therefore, they inevitably give rise to the necessity of analyzing them Grassmann manifold.
3
Mathematical Preliminaries
In this section, the basic mathematical concepts covered in this paper are described. Further details on the fundamentals of a Grassmann manifold are summarized in the Appendix D
3.1 Grassmann Manifold Defined as Quotient Manifold
Definition of a Grassmann Manifold A Grassmann manifold Gr(k, D) is a set of k- dimensional subspaces span(Y) (Y is a matrix of basis vectors and the function span(.) is onto Gr(k, D)) in the D-dimensional Euclidean space DR,, and is defined as Gr(k, D) {span( Y) C TR. dim (span(Y) = } (Absil et al. (0088). The span(Y) is the same subspace regardless of a ke-dimnnsonaa rotation or k-dimensional reflection applied to Y on which is spanned. With respect to the compact Stiefel manifold St D) == (Y E R Dxk Y I&} defined as the set of D x k-orthonormal basis matrices Y, the equivalence class of Y E St(k, D) determined from the equivalence relation ~ iii defined by [Y] := (Y) = (YQ E St(k, D) Q e O(k)), where (Y) is a continuous surjection referred to as quotient map. The equivalence class corresponds one-to-one with the k-dimensional subspace:
Y
X = span(Y) (Y)
span (X),
where Y E St(k, D) is the representative of [Y]. span( Y) is termed invariant under ~. The quotient set composed of such [Y] as whole can introduce the structure of a manifold (Sato & Iwail 2014)).
Definition 1. Grassmann manifold as a quotiens manifold is defined as follows:
Gr(k, D) == St(k, D) (k(k) = ([Y]= x (Y) Y E (t(k,D)}:
(2)
where St(k, D) /O(k) is the quotient manifold by the k-dimensional orthogonal group O(k) with the total space St(k, D), and π (Y) is the quotient map π. St(kk D) + St(k, D) (O()).
Tangent Space and Vector Field Let Tix Gr(k, D) be the tangent space of [Y]
Gr(k, D). As the point [Y] is not matrix, Tir: Gr(k, D) cannot be represented by matrix. Therefore, treating these directly in numerical calcula- tions is challenging. To solve this problem, we can use the representative Y E St(k, D) for [Y] and the tangent vector εx E T St(k, D), which is re- ferred to as the horizontal lift of εir] E {() Gr(k, D). for Ei-]- These facilitate computation with matri- ces (Absil et al. (2008). T St(k, D) is a sub- space of the tangent space Ty Si D) at Y. which
Ty St. (k. D) (t.
T St (k. D)
(k,
di (X)
Tyn Gr (k,
(k.
Ein
[r]
is referred to as a horizontal space, and εx is re- ferred to as a horizontal vector. The tangent bundle T GrG(k, D) UlTEE)((K Gryj D) that sums up the tangent spaces T'y Gr(k, D) form vector fields
Figure 1: Conceptual diagram of spaces with horizontal lift.
WWhen there exists some Q E O(k) such that X = Y Q. then X and Y are defined to be equivalences X Y. Appendix D.2 D.2 provides further details.
3

---

X: Gr(k, D) + TGr(k, D). A conceptual diagram of the various spaces is shown in Figure Further details on these concepts can be found in Appendix D.4
3.2 Manifold Normalizing Flow
Let (M,h) be a Riemannian manifold. We consider the time evolution a base point a = y (0), y oo) M, whose velocity is expressed by vector field X (t, (t)). Intuitively, (t, (t)) represents the direction and speed at which æ moves on the curve 7 (t). Let T be the tangent space and TM UsSM T. be the tangent bundle. The time evolution of point according to a vector field X M x R - TM is expressed by the differential equation dr(!) = X y (1)) 7 (0) = æ. Let Px, .M- + M be defined as the map from x .M to the evaluated value time on the curve (t) starting at a:. This map Fx.r is known as the flow of X (Lee) (2003)). Recently, |athieu & Nickel (2020) introduced the Riemann CNF (RCNF), wherein the random variable = (t) E M is assumed be time-dependent and the change in its log- aeelikelihood follows the instantancous change formula for the variable. This is extension of the CNF (Chen et al. (2018); Grathwohl et al.] (2019) to a Riemannian manifold. Specifically, when py is the density parameterized by 8, the derivative of the log- lilllikiihoodd is expressed as dlog po(z(t))
div (Xo (t, z (t))), where Xø is the vector field parameterized by and div (+) is the divergence. By integrating this over time, the sum of the changes in the log- og-liikelhood with flow Fx,t can be computed:
de
log pe (#1)) log (Fxine (= (tt))) - -1." div (Xo (t, (t))
(3)
4 Invariant Densities from Grassmann Manifold Flow
This section provides tractable and efficient method for learning densities on Grassmann manifold Gr(k, D). The method for preserving the flow on Gr(k, D) is non-trivial. Therefore, we derive the following implications.
1. Vector field on Gr(k, D) < Flow on Gr(k, D) (Proposition)
2. Flow on Gr(k, D) Probability density Gr(k, D) (Proposition 2
3. Construction of a prior probability density for an efficient sampling (Proposition
B
(Homogeneity Property (4) with regard to The essence of the proof's is to show that Corollary 2 QQ E O(k), which is cquivariance in the horizontal space) is satisfied.
o E
where Ty horizontal lift representative Y relative to εir E Tin Gr(k, D). We defer the proofs of all propositions to Appendix B These series of propositions show that by using prior distribution on Gr(k, D) that can be easily sampled, flow that generates a complex probability density distribution on Gr(k, D) can be obtained.
4.1 Construction of Flow from Vector Field
To construct flows Gr(k, D), particular, there a natural correspondence between the vector fields on Gr(k, D) and the flows on Gr(k, D). This is formalized in the following proposition.
tools in the theory of manifold differential equations. In
Proposition 1. Let Gr(k, D) be Grassmann manifold, X be any time-dependent vector field Gr(k, D), and Fx.r be flow X. Let X be any time-dependent horizontal lift and F Fx be flow R X is vector field on St(k: D) if and only if F X,r is a flow on St(k, D) and satisfies invariance condition X & for all F Fx: T Therefore, X is vector field on Gr(k, D) if and only if Fx. = Fx is a flow on Gr(k, D), and vice versa.

---

Algorithm 1 Random Sampling from ()) (Xx); M U,
Require: A mean matrix M E St(k; D), a rows covariance matrix E PDD and columms covariance
matrix V B
1: Sample Vec (Z) DK from Gaussian Distribution NDx (0, V @ and reshape to E Rxx
2: Compute projected horizontal vector EMr
M (M 2) E The St(k, D).
3: Compute a representative Y = RM [mM] M + EM of the equivalence class [Y] Gr(k; D).
{ + + (Ix + Ens
-T E EM
4.2 Construction of Probability Densities with Flow
We show that the flow on Gr(kk D) induces density on Gr(k, D).
Proposition 2. Let Gr(k, D) be a Grassmann manifold. Let p be the probability density on Gr(k, D) and F be the flow Gr(k, D). Suppose p density on St(k, D) and F is flow on St.kk D)- Then, the distribution Pr after transformations by F is also a density on (t, D). Further, the invariance condition By ~ Tr is satisfied for all F ~ F.. Therefore, pF == Prl is a distribution on D),
In the context of the RCNF, Proposition 2 iimplis that the application of a flow on Gr(k, D) to prior distribution on Gr(k, D) results in probability density on Gr(k, D). Thus, the problem of constructing a probability density on Gr(k, D) is reduced to that of constructing vector field.
4.3 Prior Probability Density Function
To construct flow Gr(k; D), prior distribution that is easy sample basis for the transformation is required, although the method for constructing such a distribution is non-trivial. In this study, a distribution based the matrix-variate Gaussian distribution is introduced as prior on Gr(k, D) that is easy to sample, and a flow on Gr(k, D) is constructed.
Proposition The distribution D) on Grassmann manifold Gr(k, D) based on the matrix- variate Gaussian distribution MN can be expressed as follows.
P) ([X); [M], U, VGra.D)MNN' (5"r:
/et (VeL,RM))
RMM
(5)
where M an orthonormal basis matrix denoting the mean of the distribution, U a positive definite matrix denoting the row directional variance, V is positive definite matrix denoting the column directional variance, and EM is a random sample from MN in an (D - k) x k-dimensional horizontal space The St(k, D). VGe(k.D) denotes total volume of Gr(k, D) defined by (113), RM
b
denotes the horizontal relraction at M, and det (VE,, RM) denotes the Jacobian.
retraction is map for communicating data between manifold and its tangent bundles and is first-order approximation of an exponential map (Zhu & Sato (202T). Various methods of retraction have been proposed at (Absil al. (2008); Fiori et al. (2015 5); Zhu & Sato (20TT); );; in particular, the one based on the Cayley transform is differentiable and does not require matrix decomposition. We use (40) as Cayley transform based horizontal retraction. Further details are provided in Appendix D.6
5
Learning Probability Densities with Flow
5.1 Training Paradigms
Using the results of Section a flow model on a Grassmann manifold Gr(k, D) is constructed. Herein, the flow model on Gr D) expressed as GrCNE, In the proposed GrCNE, the probability density function in Section 4.3 is first constructed as prior distribution. Subsequently, the vector
field X, : Gr(k, D) + Tr D) that generates the flow model Fx,.7 constructed using a neural network, wherein stepwise integration on the manifold occurs in accordance with B. The
5

---

Vec
()
MILs
HorP
Grad
Fout
Py (antograd)
da (cr
St D)
Gz(k D)
Tin Gr(k, D)-
[X]
Eri
Horizontal Vector Vector Field
Figure 2: Conceptual diagram of the vector field calculation procedure. HorP denotes the horizontal projection layer, MILs denotes the multiple intermediate layers, and Grad denotes the horizontal lift
layer. The sequence of procedure represents that, given an input of Y E (Y) a horizontal vector 5y which is equivalent to determining Eiy] since Ey corresponds one-to-one with E/y is obtained result.
RCNF (Mahhiu & Nickel (2020)) framework is used for loss function, integration and divergence calculations. the integration steps, the ordinary differential equation (ODE) is solved using the ODE solver with orthogonal integration. The learnable parameters updated using backpropagation to maximize the sum of the computed log-likelihood. In order to efficiently compute gradients, we introduce adjoint method calculate analytically the derivative of manifold ODE, based on the results of Lou et al. 2020).
5.2 Sampling Algorithm from Prior
We propose a sampling algorithm on Gr(k, D) derived from the Cayley transform, according to the results of Section 4.3 The algorithm of sampling from a distribution on PGr(k,D) described in Algorithm [ Vec denotes the map of vertically concatenating matrices and converting them into vector and In a D D identity matrix.
5.3 Construction of Vector Field
We describe the method for constructing the vector field Xa. The learnable parameters are feedfor- ward neural networks, which accept the representative Y St(k, D) of [Y] Gr(k, D) inputs and
output horizontal lift Ex E The St(k, D) of {Ιy] E Tiyj Gr(k, D) that satisfies (4). The structure of X₂ important because directly determines the ability of the distribution be represented. To address these geometric properties, the following specific input and intermediate layers are used to obtain a (kk) invariant function value Yout- Figure shows a conceptual diagram.
Input Layer The input layer maps the input representative Y to point ςt T St(k, D). A horizontal projection HorP St(k, D) T Su(k, D) = (D=x))x is constructed an input layer:
HorP (Y) W -Y W.
(6)
where W is the output of NN obtained with YYT input. HorP is well-defined because it is uniquely determined regardless of how the representative Y E [Y] chosen.
To model the dynamics over the horizontal space, intermediate layers based neural networks constructed. The layers particularly important for the ability represent the distribution, and various configurations are possible such as concatsquash (CS) layer (Grathwohl al. (2019)). The layer can be stacked multiply, and the overall power of expression increases with the number of layers. The input æ of the first intermediate layer is æ Vec HorP. The last intermediate layer (one layer before the output layer) must be constructed such that it exhibits potential function value. This is to obtain the gradient for the potential function in the output layer.
Intermediate Layer
Output Layer We construct gradient Grad (IR, St(k, D)) + T St(k, D) output layer. In Grad, t E The St.k,, D) is obtained by taking the gradient of the potential function Vout:
Grad (Vout, Y) Pt (autograd (Vout, Y)) s.t. Pt (Z) Z Y sym (YTZ). Z)
(7)

---

where autograd (Vout. Y) denotes automatic differentiation Passke et al. (0117) denotes the projection of Z onto Ty St(k, D) and sym (B)
deat
Pt (Z)
BY
B B
2 is symmetric part of
B. Grad is well-defined on Gr(k, D) by the horizontal lift (Absil et al. 2008
5.4 ODE Solver with Orthogonal Integration
We present an ordinary differential equation (ODE) solver on Gr(k, D). Celledoni Owren 2002) proposed an intrinsic ODE solver that was applicable to St(k, D) and did not assume an outer Euclidean space. This solver works on St(k, D). thus must be reformulated into a solver suitable for Gr D). We introduce solver Gr(k; D) via ODE operating T St(k, D), based on results from Section[] [ This is an intrinsic approach regardless of whether the manifold has been embedded in bigger space with corresponding extension of the vector field. The ODE defined on Ty (k; D) is obtained as:
dNecc (e (1) = V. Vec (Xa (4)7(4)) s.t. 7 - - Ry (e
(4))
(8)
dt
where : [0, oo) + T St(k, D) the curve on T St(k, D), Ry denotes the horizontal retraction, V, can be calculated using (55) and Vec denotes the map of vertically concatenating matrices and converting them into a single vector. Further details on our ODE can be found in Appendix C.I
5.5 Loss Function
The total change in log- ligelike using GrCNF can be calculated using the following equation.
Po (Y (t1)) log D) (Fx (Y (tz))) - L.² div (Xo (t, Y (t)))
(9)
where Px is flow. In this study, we defined the loss function Loss for maximizing the log- likelihood: Loss = NLL - Po (Y (12)) where NLL denotes negative log-likelihood.
Experimental Results
In this section, discuss the results of several experiments for validating GrCNF. Further details regarding each experimental condition, such as the architectures and hyperparameters used in training, can be found in Appendix C.3
6.1 Generation and Density Estimation Artificial Textures
We first trained GrCNF on five different Gr(1,3) (I-dimensional subspace in RR, that is, a line through the origin) data visualize the model and the trained dynamics. The five datasets"] were spirals, a swissroll, circles, 2 sines, and Target. We represented Gr(1,3) with a sphere of radius centered at the origin by mapping 1- 1-dimensional subspace to two points on the sphere (a point the sphere and its antipodal point) for visualization.
The five types of probability densities transformed by GrCNF and the results of data generation are shown in Figure 3 The top, middle, and bottom rows in the figure present the correct data, the generated data with GrCNF when the left-most column is the prior distribution, and the probability density on a Gr(1, 3) obtained by training when the leftmost column was the prior distribution, respectively. Regarding the probability density, brighter region corresponds to higher probability. As shown in the figure. GrCNF can generate high-quality samples that are sufficiently accurate to the distribution of the correct data. To investigate whether GrCNF learns the distribution on Gr(L (1, 3), we generated antipodal points in Figure 3 with the trained GrCNF used in Figure 3 As shown in Figure 4] GrCNF generated exactly the same high quality samples as the original for the untrained antipodal points. This experimental result implies that the proposed flow accurately learns the distribution on Gr(k, D) and that all the orthogonal transformed samples can be obtained with equal quality by only training with arbitrary representatives.
Result
"The code for the data distributions is presented in Appendix C.3.1
7

---

Prior
2 spirals
Swissroll
2 circles
2 sines
Target
Data
Generated
Density
Figure 3: Generated samples and probability densities using the GrCNF trained on each of the five distributions. (top) Ground truth data, (middle) Generated data with the GrCNF when the leftmost column represents the prior distribution, and (bottom) densities obtained via training.
Figure 4: Results of the GrCNF transformation for the antipodal point of the prior in the middle panel of the Figure The antipodal points those that pass through the origin of the sphere and appear in the southern hemisphere (leftmost). It is evident that the antipodal points are generated with exactly the quality in the Figure 3 for the un-trained antipodal points.
Comparison with Conventional Methods for DW4 and IJJ3
Comparative experiments were performed using DW4 and LJ13-the the two systems presented by Köhler al. (2020). These datasets generated synthetically sampling from their respective energy functions using Markov chain Monte Carlo (MCMC) methods. Both energy functions (DW4 and LJ13) ideal for analyzing the advantages of various methods when they exist data that are equivariant for rotation and reflection, respectively. We used the datasets generated by MCMC that were employed in Köhler al. (2020). The orthonormalized data Y were used for this study by applying Gram-Schmide orthogonalizzation to each column of P such that the D-dimensional data was a matrix with & orthonormal basis vectors (DW4: D 4, 2; LJ13: D 13, 3).
To match the experiment conducted in Garcia Satorras et al. 2021). .000 validation and testing samples each
used for both datasets. For DW4 and LJ13, different numbers of training samples, ie., { [102, 103 104, 105) and 110, 102, 103, 10* }. respectively, were selected, and their performance for each amount of data examined. The proposed approach compared with the state-of-s the- art E(n) equivariant flow (E-NF) presented by Garcia Satorras et al.) (202T) and simple dynamics presented Köhler al. (2020). In addition, comparisons with graph normalizing flow (GNF), GNF with attention (GNF-att), and GNF with attention and data augmentation (GNF-att-aure (data augmentation by rotation), which non-equivariant variants of E-NF, were performed. All the reported values are averages of cross-validations (three runs), Otherwise, the network structures of all these conventional methods were the same that used in Garcia Satorras et (2021)
Result Table presents the results of the cross-validated experiments (negative log-likelihood: NLL) for the test samples. The proposed GrCNF outperformed both the conventional non-equivariant models (GNE, GNF-att, and GNF-att-aug) and the conventional equivariant models (Simple dynamics, E-NF) in all data domains.

---

Table Negative log- likelihood comparison on the test partition of different methods for DW4 and LJ13 datasets for different amount of training samples averaged over runs.
DW4
LJ13
# Samples
10a
10
10a
10a
10
10a
10a
10
GNF
-2.30
-7.04
-7.19
-7.93
6.77
-0.76
-4.26
-12.43
GNF-att
-2.02
-4.13
-5.25
-6.74
6.91
1.40
-6.81
-12.05
GNF-att-aug
-3.11
-4.04
-6.51
-9,42
2.95
-6.11
-13.94
-15.74
Simple dynamics
-1.22
-1.28
-1.36
-1.39
-1.10
-3.87
-3.72
-3.59
E-NF
-0.54
-9.89
-12.15
-15.29
-12.86
-15.75
-31.51
-32.83
GrCNF
-12.53
-13.74
-14.09
-16.07
-23.64
-44.24
-58.02
-58.71
Table 2: Results for the QM9 Positional dataset. (Left) Negative log-likelihood, ELBOO and JSD for the QM9 Positional dataset on the test data, and (Right) normalized histogram of relative distances between atoms for QM9 Positional and generated samples with GrCNE
NLL
ELBO
JSD
Simple dynamics
73.0
0,086
Kernel dynamics
38.6
0.029
GNF
-00,9
0.011
GNF--att
-26.6
0.007
GNF-atttaug
-33.5
0.006
60
E-NF
-70.2
0.006
GrCNF
NLL S -85.3
-85.3
0.005
1
6.3 Comparison with Conventional Methods for QM9 Positional
comparative experiment performed using the QM9 Positional-aa subset of the QM9 molecular dataset that considers only positional information. The purpose of this experiment was to evaluate the feasibility of generating practical point cloud. The QM9 Positional comprises only molecules with 19 atoms/nodes, and each node has a 3-dimensional position vector associated with it. However, the likelihood of the molecules must be invariant translation and rotation in 3-dimensional space; thus, the proposed model is suitable for this type of data. The dataset consisted of 13,831, 2501, and 1813 training, validation, and testing samples, respectively. metrics based on the NLL and Jensen-Shee Shannon divergence (JSD: Lin (199I)), in accordance with the experiments conducted Garcia Satorras al. (202T). The JSD calculates the distance between the normalized histograms that are generated from the model and obtained from the training set, by creating histogram of the relative distances among all the node pairs in each molecule.
this experiment,
used evaluation
GrCNF handles subspace data. Therefore, orthonormalized data Y was used only in the proposed GrCNF, in PP ~ Y AAY"
A is diagonal (Huang et 2015 ). such that the k-dimensional point cloud data P of N points N x k matrix is matrix with A orthonormal basis vectors (D 19, k- 3).), However, complete point cloud generating task, such as QM9 Positional, must also store scale parameter / VA such that estimate / VA. In addition, the proposed GrCNF encounters difficulties in computing the NLL in distribution (PP of P: thus, new loss function is required to address this. In this study, the evidence lower bound (ELBO), i.e., the lower bound of the likeliiod log Py: (P). was maximized using a variational inference framework. This is equal to the minimization of ELBO (P). Further details regarding ELBO can be found Appendix
3).
Y VA. Therefore, GrCNF incorporates an additional architecture
We compared our GrCNF with the GNE, GNF-att, GNF-att-aug, Simple dynamics, Kernel dynamics, and E-NF methods. The network structure and experimental conditions for all these conventional methods were identical to those used in Garcia Satorras et al. (2021). In all the experiments, the training was performed for 160 epochs. The JSD values were the averages of the last 10 epochs for all the models.
Result Table presents the NLLL and JSD cross-validated against the test data. Although the proposed GrCNF could not directly compute the NLL in this problem setting, it outperformed all

---

GrCNF
E-NF
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample 3
Sample 4
Sample
Nearest QM9
3.8
Generated QM9
x.y
ZAX
Figure 5: Visualization results of generated samples in Section 6.3 We visualize Generated QM9 by GrCNF and E-NF, well as the aligned data points from the QM9 dataset that closest the Generated QM9 samples. These visualizations illustrate that GrCNF is capable of generating realistic data points that align well with the existing dataset.
the other algorithms because the relation NLL < ELBO is true in general. With regard to the JSD, GrCNF achieved the best performance. Figure 2 presents the normalized histograms of the QM9 Positional molecular data and the data generated by GrCNF, in relation the JSD of GrCNF. As shown, the histograms the molecules generated by GrCNF are close to the histograms of the data set; thus, GrCNF generate realistic molecules stably.
7 Conclusion
We proposed the concept of CNF on a Grassmann manifold (GrCNF) for stable shape generation. The proposed model is generative model capable of handling subspace data; i.c., is a CNF that considers the equivariance on the Stiefel manifold. Through suitable experiments, the ability of the proposed GrCNF to generate qualitatively I-dimensional subspace datasets 1 was confirmed. Further, GrCNF significantly outperformed existing normalizing flows methods in terms of the ligelikel ELBO for DW4, LJ13, and QM9 Positional. It was shown that GrCNF can be used to generate realistic data more stably.
Societal Impact and Limitations Our work is concerned with accurately modeling the data topology; thus, we do not expect there to be any negative consequence in our application field. The proposed theory and implementation are valid in a remarkably general setting, although there are limitations that can be addressed in future works: 1. The proposed method requires calculation of the Kronecker product each ODE step (Section 5.4). which is computationally expensive. 2. Our current paper does not include experiments should be conducted using higher dimensional data, such as point clouds that represent general object geometry. However, the results presented in this paper still provide numerous valuable insights in the field stable shape generation.
high-dimensional data. Therefore, future experiments
10

---

References
P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ, 2008. ISBN 978-0-691-13298--3 URL https
//press, wwwnnctton.edd/abaa.
Taleb Alashkar, Boulbaba Ben Amor, Mohamed Daoudi, and Stefano Berretti. A grassmann frame- work for 4d facial shape analysis. Pattern Recognition. 57:21-30, 2016. ISSN 0031-3203. URL
https://doi org/ 10. patco................. 2016 03.003..000 013
E. Begelfor and M. Werman. Affine invariance revisited. In 2006 IEEE Computer Society Confer- ence on Computer Vision and Pattern Recognitiin (CVPR 066), volume 2, pp. 2087-2094, IEEE Computer Society, 2006. htl.httss://dde htt.///oi.ooooo org/ 10. 10.1109/CVPP, 2006. 50
Camille Bilodeau, Wengong Jin, Tommi Jaakkola, Regina Barzilay, and Klavs F. Jensen. Generative models for molecular discovery: Recent advances and challenges. WIREs Compulational Molecular Science, pp. e1608, 2022. URL hattps://bbi.org//
1608
Denis Boyda, Gurtej Kanwar, Sébastien Racanière, Danilo Jimenez Rezende, Michael S. Albergo, Kyle Cranmer, Daniel C. Hackett, and Phiala E. Shanahan. Sampling using su(n) (n) gauge equivariant flows. Phys. Rev. D, 103(7):074504, 2021a. URL htttps://doi. org 1103 1et.l1113/PPPhP PhysRevD 103 074504
Denis Boyda, Gurtej Kanwar, Sébastien Racanière, Danilo Jimenez Rezende, Michael S. Albergo, Kyle Cranmer, Daniel C. Hackett, and Phiala E. Shanahan. Sampling using su(n) gauge equiv- ariant flows. Physical Review D, 103(7), 2021b. URL https://iik: aps. ort./oi/o1/oo/..//////// 1103/
PhysRevD. 103. 074504
Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Velickovic. Geometric deep learning: Grids, groups, graphs geodesics, and gauges. CoRR, 13442804,.447, 2021. URL https
//arxiv org/ abs/ 2104. 13478
Elena Celledoni and Brynjulf Owren. A class of intrinsic schemes for orthogonal integration.
SIAM Journal on Numerical Analysis, 40(6):2069-2008 2002. URL URL htttp://ww..jstttt org/
stable/4100984
Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems, volume 31, pp. 6571-6583. Curran Associates, Inc., 2018. URL httpt::::/pproceediiie
www.ipe..p.... cct.pperrccpppp/r/////pp pdf
Yasuko Chikuse. Statistics on Special
Manifolds. Speciial Lecture Notes in Statistics, volume 174. Springer,
01 2003.
Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling. Gauge equivariant convolutional networks and the icosahedral CNN, In Proceedings of the 36th International Conference on Machine Learning, volume 97, pp. 1321-1330. PMLR, 2019a. URL (tRps:t:::// htttps::/proceedinggg
mlr. ptr.spres/v9////
http:/99//cooee/1// cohen html
Taco S. Cohen and Max Welling. Group equivariant convolutional networks. In Proceedings of The 33rd International Conference on Machine Learning, volume 48, pp. 2990-2999, PMLR, 2016.
URL https://proceedingg.. mlr wwwpppresssviee httess///////henc166 cww.nc16.....cc htmll
Taco S. Cohen and Max Welling, Steerable cmns. In International Conference on Learning Represen- tations, 2017. URL httpp::tt:///eeerreee http:////gppppaeeeee.aae/////trp wwt.:eeeeeeeeooo/m/ddd///ESSSSS
Taco S. Cohen, Mario Geiger, Jonas Köhler, and Max Welling. Spherical cnns. In International Conference on Learning Representations, 2018. URL htttps://opeereeee httprriwwww.weeeeeeee met/forum? id= Hkbd5x2Rbl
Taco S. Cohen, Mario Geiger, and Maurice Weiler. general theory of equivariant cnns on ho-
mogeneous spaces. In Advances in Neural Information Processing Systems, volume 32, Cur- ran Associates, Inc., 9h URI http://ppppppppoecennn..eeoeeeee
Dbere880002771300000007777777733 .pff

---

Valentin De Bortoli, Emile Mathieu, Michael Hutchinson, James Thornoon, Yee Whye Teh, and Amaud Doucet. Riemannian score-based generative modelling. In Advances in Neural Information Processing Systems, volume 35, pp. 2406-2422. Curran Associates, Inc., 2022. URL httpps://proceeddngss www.:ig.......p...... cc cappe.rr..pp......e.. _aper__PpP___ wwweslllppppeeeeeee pttle///ppprr/22//Eiieee Confer--oooeeeeee pdf
R. J. Dormand and P. Prince. A family of embedded Lange-Kuute, formulae. Journal of Com- putational and Applied Mathematics, pp. 19-26, 1980. URL https://doii wwt.oioorgr..... 1016/ 0771-050X(80) 9
Olga Doronina, 7achary Grey, and Andrew Glaws. Grassmannian shape representations for aerody- namic applications. In AAAI 2022 Workshop on Al for Design and Manufacturing (ADAM), 2022.
URL https::/ wttp:/ooruriddddd/uoooodd
Alan Edelman, Tomás A. Arias, and Steven T. Smith. The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2):003-33,, 1998. URL
https://doi. org/10 10. 011173//88857779559999559
Lawrence Craig Evans and Gariepy Ronald. Measure Theory and Fine Properties of Functions, Revised Edition (Ist ed.). CRC Press, Revised Edition, 2015. URL https://doi. http:///ooooorg/110/ oto//o.....1.. 1200//
b18833
Yong Fan, Yong Liu, Hong Wu, Yihui Hao, Haihong Liu, Zhening Liu, and Tianzi Jiang. Discriminant analysis of functional connectivity patterns on grassmann manifold. Neurolmage, 56(4):2058-206)
2011. URL http:: http::::///oi.rr/00...1 1117/12. 844495
Mare Finzi, Samuel Stanton, Pavel Izmailov, and Andrew Gordon Wilson, Generalizing convolutional neural networks for equivariance to lie groups on arbitrary continuous data. In Proceedings of the 37th International Conference on Machine Learning, volume 119, pp. 3165-3176, 2020. URL
https://dl.
acm. www.aaa........................a /ttp///o0/10..555/3333333 55555/3245353333 3524938 35225255
Simone Fiori, Tetsuya Kaneko, and Toshihisa Tanaka. Tangent-bundle maps on the grassmann manifold: Application to empirical arithmetic averaging. IEEE Transactions on Signal Processing, 63(1):155-166, 2015. URL https:://oi..ooo org/ 10. 1109/TSP...... 2014. 2365764
Victor Garcia Satorras, Emiel Hoogeboom, Fabian Fuchs, Ingmar Posner, and Max Welling, E(n) equivariant normalizing flows. In Advances in Neural Information Processing Systems, volume 34, pp. 4181-4192 Curran Associates, Inc., 2021. URL https::/ //proce//prooeeiinggs wewiiingnnnneiiss...ee..mu. cc/
Paper ppf
Bernardo B. Gatto, Juan G. Colonna, Eulanda M. dos Santos, and Eduardo F. Nakamura. Mutual singular spectrum analysis for bioacoustics classification. In 2017 IEEE 27th International Workshop on Machine Learning for Signal Processing (MLSP), pp. 1-6, 2017. URL https
//àoi.org/10 11191/19MMSSSS
8168113
Evan S. Gawlik and M. Leok. High-order retractions on matrix manifolds using projected poly- nomials. SIAM J. Matrix Anal. Appl., 39:801-828, 201 8a. httpttt///o/iiorrg 1137/ 17M1130459
Evan S. Gawlik and Melvin Leok. High-order retractions matrix manifolds using projected polynomials. SIAM Journal on Matrix Analysis and Applications, 39(2):801-82) 0
2018b. URL
https://doi org/ 10. 1137/17HI 17M11104554
Mathieu Germain, Karol Gregor, lain Murray, and Hugo Larochelle. Made: Masked autoencoder for distribution estimation. In Proceedingy of the 32nd International Conference Machine Learning, volume 37, pp. 881-889, PMLR, 2015. URL http: http::/pproceednngs mlr. mir.press/ press/ v37 /57/germain15. html
Ian Goodfellow, Jean Pouget-Abadie Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Informa- tion Processing Systems, volume 27. Curran Associates, Inc., 2014. http:/tttttss/prceeeeinn Pap.se3--ppppppe pdf
12

---

Simon Graham, David B. A. Epstein, and Nasir M. Rajpoot. Dense steerable filter cnns for exploiting rotational symmetry histology images. IEEE Trans. Medical Imaging. 39112):12244136 2020.
URL /tttps://doi..o org/10. 1109/TMI, 3013246
Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, llya Sutskever, and David Duvenaud. FFJORD: free-form continuous dynamics for scalable reversible generative models. In 7th International Conference on Learning Representations, ICLR 2019. New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL httpttttt///oererrtii
aet//forummidd<<xxkkccca
Pim De Haan, Maurice Weiler, Taco Cohen, and Max Welling. Gauge equivariant mesh cnns: Anisotropic convolutions on geometric graphs. In International Conference on Learning Represen- tations, 2021. URL http::://ppeereeieeet
http:////ppppeeeeeeeeeeee/r/rraa mett//oronnidddnsaaaaaaa IIE
Erst Hairer. Geometric Numerical Integration: Structure- Peeeture-rreeevving Algorithms for Ordinary Differential Equations. Springer, 2nd ed. edition, 2006. URL https: /tops:://oo.... org. 10. 1007 3-
Yuval Haitman, Joseph M. Francos, and Louis L. Scharf. Grassmannian dimensionality reduction for optimized universal manifold embedding representation of 3d point clouds. In 2021 IEEE/CVF International Conference Computer Vision Workshops (ICCVW), pp. 4196-4204, 2021. URL https://doi.
org/10. 10. 11191 ICCVW54120. 2021 00468
Jihun Hamm and Daniel D. Lee. Grassmann discriminant analysis: A unifying view on subspace- based learning. In Proceedings of the 25th International Conference on Machine Learning, ICML '08, pp. 376-383, Association for Computing Machinery, 2008. URL htt.h:tt::/.. : /ttt:::://doooor/1000/ 1145/1390156. 1390204
Mchrtash T. Harandi, Conrad Sanderson, Sarch Shirazi, and Brian C. Lovell. Graph embedding discriminant analysis on grassmannian manifolds for improved image set matching. In CVPR 2011, 2011. URL https://: htt.d/io.oogg////// 11.1111.1.1110.11 1109/CVPR 2011. 5995564
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Advances in Neural Information Processing Systems, volume 33, pp. 6840-6851. Curran Associates, Inc., 2020. URL
neurips httpppp..c/cpppp///////i/e/
pdf
Chin-Wei Huang. Milad Aghajohari, Joey Bose, Prakash Panangaden, and Aaron C Courville. Rie- mannian diffusion models. In Advances in Neural Information Processing Systems, volume 35, pp. 2750-2761. Curran Associates, Inc., 2022. URL htt.lttttt///poceediiiiii neurips cc/paper_ pdf
Zhiwu Huang, Ruiping Wang. Shiguang Shan, and Xilin Chen. Projection metric learning on grassmann manifold with application to video based face recognition. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 149, 2015. https://::///o
org/10.r 1109/CVPR. 2015. 7298609
Zhiwu Huang, Jiqing Wu, and Luc Van Gool. Building deep networks grassmann manifolds. In Proceedings of the Thirty-Secome AAAI Conference on Artificial Intelligence (AAAI-18), pp. 3279-3286 2018 URL (tLLhttss://dl acm. wocmmorg//ee www.oo////////oo....oooo 50tp///100005555504055555 355444440000033344336333 3504436
Sergey Toffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 448-456. PMLR, 2015. URL
mlr www.spreess..vee https///v///oofffle html
Arich Iserles, Hans Z. Munthe- Kaas, Syvert P. Nørsett, and Antonella Zanna. Lic-group methods. http://////aaa.ororoo0o.10/////
Acta Numerica, 9:215-365, 2000.
Isay Katsman, Aaron Lou, Derek Lim, Qingxuan Jiang, Ser Nam Lim, and Christopher M De Sa. Equivariant manifold flows. In Advances in Neural Information Processing Systems, volume 34, pp. 10600-10612, Curran Associates, Inc., 2021. URL https:ttps:/. http::::/ccooosooooeoeo www.eediiiinnn...ii...eee ec/
13

---

Hyeongju Kim, Hyeonseung Lee, Woo Hyun Kang, Joun Yeop Lee, and Nam Soo Kim. Softflow: Probabilistic framework for normalizing flow on manifolds. In Ad- vances in Neural Information Processing Systems, volume 33, pp- 16388-16397, Curran Associates, Inc., 2020. URI.
negggss.euiipss. wwtuurpps.cpppppeereeee.o /ecpppeer/200//filee/ pdf
bdbca288fee7f92F22 01112//2222001112277700000--0000
Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In 2nd International Conference on Learning Representations, ICLR 2014, 2014. URL http:://axxii org/ abs/
1312.6115 6114
Jonas Köhler, Leon Klein, and Frank Noé. Equivariant flows: Exact likelihood generative learn- ing for symmetric densities. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pp. 5361-5370. PMLR, 2020. URL httt::::/ppocceiinggss mwr.prreee. v119/kohler20a. html
Risi Kondor and Shubhendu Trivedi. On the generalization of equivariance and convolution in neural networks the action of compact groups. In Proceedings of the 35th International
Conference on Machine Learning, volume 80, pp. 2747-2755. PMLR, 2018. URL https
//proceedings.u mlr wwtppp/eess/odoreoooeepre/eo html
J.M. Lee. Introduction Smooth Manifolds. Graduate Texts in Mathematics. Springer, 2003. ISBN 9780387954486.
J. Lin. Divergence measures based on the shannon entropy. IEEE Transactions on Information Theory, 37(1)1145-15, 1991. httLittss///oi..oo org/ 10. 1109/18. 61111
Xiuwen Liu, A. Srivastava, and K. Gallivan. Optimal linear representations of images for object recognition. In 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recog- nition, 2003. Proceedings., volume 1, pp. H 2003. URL httpp:///dii.rgg/// 10. 1109/VVP 2003. 1211358
Aaron Lou, Derek Lim, Isay Katsman, Leo Huang, Qingxuan Jiang, Ser Nam Lim, and Christopher M De Sa. Neural manifold ordinary differential equations. In Ad- vances in Neural Information Processing Systems, volume 33, pp. 17548-17558. Curran Associates, Inc., 2020. URL httpps:/prroceddingg/ neuri weeurips.cccppppr//0000 pc/pp//////ppp///////iile//// Q05570044655221355344443334 Paper pdf
Yui Man Lui. Advances in matrix manifolds for computer vision. Image Vision Comput., 30(6-7):
380-388, 2012. https::////doi http://a.i...//0..11111. 1116//0...1................... iww.is................m......... 2011.00..1......1 002
J.R. Magnus and H. Neudecker. Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley Series in Probability and Statistics. Wiley, 2019. httpp://::///o org/ 10. 00.10022/98111154122
Arak M. Mathai, Serge B. Provost, and Hans J. Haubold. The In Multivariate Statistical Analysis in the Real and Complex Domains, pp. 217-288. Springer International Publishing, 2022. URL https://dti...oo org 10. 1001 10000000097777 00000//98--3-00 95864-0_4
gaussian distribution.
Emile Mathieu and Maximilian Nickel. Riemannian continuous normalizing flows. In Proceedings of the 34th International Conference on Neural Information Processing Systems, volume 33. http:/ipss..cppppp////////////// 125210066c6772447770005551112333 Paper pdf
Curran Associates, Inc., 2020. wtwptttss::tt../mm httpLattpps/pprooeenng..oomm
Cleve Moler and Charles Van Loan. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45(1):3 49, 2003. URL https:///oi. org/ /01 1137/
500361445024180
Hans Munthe-Kaas. High order runge-kutta methods on manifolds. Applied Numerical MattieeriicaMMMaataaataaaaa Mathematics, 29(1):115-127, 1999.
Aaron Van Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In Proceedings of The 33rd International Conference on Machine Learning, volume 48, pp- 1747-1756. PMLR, 2016. http: SRRL/ttpp///rrceedingg mlr. wttp://ws...n.mnnneeeeeeeee wwwpss/ese/oordaeaeoo html
14

---

Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary Vito,o Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. In NIPS 2017 Workshop on Autodiff, 2017. Btll:tttp:::/pereeviiie htl.lpperreviieeeee/ftrrrr
id=BJJsrmfCZ
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang. Zachary Vitoto, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, nigh-performance deep learning library. In Advances in Neural Information Processing Systems, volume 32. Cur- ran Associates, Inc., 2019. URL https: http://ppooeeednn...ee www.iinnnnnnnuussse.eee httprr/ppp..cppppp////////1//.1/ pappr//099/ttileee babca2288fee7f92f2DD http:///22222...0000002727770020 pa
Vic Patrangenaru and Kanti V. Mardia. Affine shape analysis and image analysis. In 22nd Leeds Annual Statistics Research Workshop, 2003, URL httpp:/wwww maths leeds. ac. uk/
Omri Puny, Matan Atzmon, Edward J. Smith, Ishan Misra, Aditya Grover, Heli Ben-Hamu, and Yaron Lipman. Frame averaging for invariant and equivariant network design. In International Conference on Learning Representations, 2022. URL Wtllhttps:tc httttt:s///peeeeee /owpr//eieeeereieeeeeoeome.
yyxtm61nU__6
Prajit Ramachandran, Barret Zoph, and Quoe V. Le. Searching for activation functions. CoRR, abs/1710.05941, 2017. URL
05941
Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 1530-1538. PMLR, 2015. URL http::tttt////oeeediiie mlr. www.mmrrreeeee pressa v37 v37/rezendeil html
Danilo Jimenez Rezende, Sébastien Racanière, Irina Higgins, and Peter Toth. Equivariant hamiltonian flows, 2019. /tt/sttss/////i
48550/ arXiv. 1909. 13739
Benjamin Sanchez-Lengeling and Alán Aspuru-Guzik. Inverse molecular design using machine learning: Generative models for matter engineering. Science, 361 36166)))):30--66, 2018. URL
httpp::/www..ciencee. science. org//oi//aaal 10. 1112///ciieee aat26666633 aat2663
Hiroyuki Sato and Toshihiro Iwa. Optimization algorithms on the grassmann manifold with appli- cation to matrix eigenvalue problems. Japan Journal of Industrial and Applied Mathematics, 31: 355-400, 2014. URL
Víctor Garcia Satorras, Emiel Hoogeboom, and Max Welling. e(n) equivariant graph neural networks, In Proceedings of the 38th International Conference on Machine Learning, vol- ume 139 of Proceedings of Machine Learning Research, pp. 9323 9332. PMLR, 2021. URL
https: ctttps:/proocedding.. mlr. www.prresssssee ceeess/vvv//ese satorras2a..aaaaaaa html
D. Sepiashvili, J.M.F. Moura, and V.H.S. Ha. Affine- permutaermttationn symmetry: invariance and shape space. In IEEE Workshop on Statistical Signal Processing, 2003, pp. 307-310, 2003, URL
https://doi. orriiir.rrgg111/ 10. 1109/SSP 1289406
Erica K. Shimomoto, François Portet, and Kazuhiro Fukui. Text classification based on the word subspace representation. Pattern Analysis and Applications, 24(3):1075-1093, 2021. URL
https://dol.orrg/1 1007/s10044-0220 000-0000000000000-----6--00
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsu- pervised learning using nonequilibrium thermodynamics. In Proceedings of the 32nd Inter- national Conference on Machine Learning, volume 37 of Proceedings of Machine Learn- ing Research, pp. 2256-2265. PMLR, 2015. URL httpps:/prroceedangg. www.ss.rrreee/37//mmm sohl-dicksteim15 html
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In In- lernational Conference on Learning Representations, 2021. URI. https:ttpp/.t.... /tttpp://operreview
Forum?id-Stigaaree
15

---

Lincon S. Souza, Bernardo B. Gatto, and Kazuhiro Fukui. Classification of bioacoustic signals with tangent singular spectrum analysis. ICASSP 2019 2209 IEEE International Conference on Acoustics, Speech and Signal P Processing (ICASSP), pp. 351-355, 2019.
10. 10..109/ICASSP 2019. 8682493
Lincon S. Souza, Bernardo B. Gatto, Jing- Hag Xue, and Kazuhiro Fukui. Enhanced grassmann discriminant analysis with randomized time warping for motion recognition. Pattern Recognition, 97:107028, 2020. httLt:/:://doii org 10. 1016/j patcoo 2019.107028
Lincon S. Souza, Naoya Sogi, Bernardo B. Gatto, Takumi Kobayashi, and Kazuhiro Fukui. Grass- mannian learning mutual subspace method for image set recognition. Neurocomputing, 2022.
ISSN 0925-2312. URL httpbhtp:://doo org/ 10. 1016/ neucom. 2022. 10.040
A. Srivastava, S.H. Joshi, W. Mio, and Xiuwen Liu. Statistical shape analysis: clustering. learning, and testing. IEEE Transactions Pattern Analysis and Machine Intelligence, 27(4):590-60 602, 2005. URL https: //doii org/10......... 1111/1TAAAAAT AMI... 2005 86
Anuj Srivastava and Eric Klassen. Bayesian and geometric subspace tracking. Advances in Applied Probability, 36( 3611),33-56, 2004. URL http:://ww. jstor org/sttable/148835
Pavan Turaga, Ashok Veeraraghavan, and Rama Chellappa. Statistical analysis on stiefel and
grassmann manifolds with applications in computer vision. In 2008 IEEE Conference Computer Vision and Puttern Recognition, pp. 1-8, 2008. URL. httph::tt//eet org/ wtt../////....///PPP...... 1009/CVPR.2000000 2008
4587733
Pavan Turaga, Ashok Veeraraghavan, Anuj Srivastava, and Rama Chellappa. Statistical computations on grassmann and stiefel manifolds for image and video-based recognition. IEEE Transactions on Puttern Analysis and Machine Intelligence, 33(1 (3(11))2273-2866 2011. URL https://d//d org/
10. 1001119/TTTT TPAMI. 2011.52
Maurice Weiler and Gabriele Cesa. General e(2)-equivariant steerable cnns. In Advances Neural Information ciates, Inc., 2019. URL Processing Systems, volume 32. Curran Asso-
www.ggs..errip...ooo www.eeww.....aappeppeeeeeeeeeeee http://...//pppr///////11ee pdff
Zaiwen Wen and Wotao Yin. A feasible method for optimization with orthogonality con- straints. Mathematical Programming. 142:397-434, 2013. URL https://::///o org/10.11077
610107-012-05841
Zhu Xiaojing. A riemannian conjugate gradient method for optimization on the stiefel manifold. Computationall Optimization and Applications, 1107 2017. /ttp:attss///oi..oo
1007/s10589 016- 9883-4
Ryoma Yataka and Kazuhiro Fukui. Three-dimensional object recognition via subspace rep- resentation on grassmann manifold. In Proceedings of the 6th International Conference on Pattern Recognition Applications and Methods, pp. 208-216. SciTePress, 2017. URL
https://doi.co org/ 00..//2220///2222 00062007020802225
Ryoma Yataka, Kazuki Hirashima, Takafumi Matsuda, Tai Tanaka, Masato Gocho, and Masashi Shiraishi. Projection metric learning of updated-subspaces for radar target classification. In 2019 16th European Radar Conference (EuRAD). pp. 5-8, 2019. URL /ttptttts//eeeeeppoor ieee
Takao Yoshinuma, Hideitsu Hino, and Kazuhiro Fukui. Personal authentication based on 3d configuration of micro-feature points on facial surface. In Image and Video Technology, pp. 433 446. Springer International Publishing, 2016. URL https://doi. org /0110000.2 1007/ 978-3- 319 29451-3_35
Xiaojing Zhu and Chun-Yar Duan. On matrix exponentials and their approximations related to optimization on the stiefel manifold. Optimization Letters, 13, 07 2019. httttt/:: org/ 10. 10.1107//11190- 1341-z
Xiaojing Zhu and Hiroyuki Sato. Cayley-transforr--aese gradient and conjugate gradient algorithms on grassmann manifolds. Adv. Comput. Math., 47(4), 2021. URI, httpp:://d: h/tp://////ooo.//1100//000 100044021100000
16

---

A Appendix Overview
First, we present the details of the various proof' of Section iin Appendix B Next, in Appendix C we describe the network layers for the building GrCNF, and the detailed model architectures, hyper- parameters, and implementation on the experiments. Finally, in Appendix D we provide a summary of the fundamentals of a Grassmann manifold, which is the core concept of this study.
B Proofs
B.1 Proposition 1
First, we invoked the following two corollaries.
Corollary (Diffeomorphism Invariance of Flows). Let F M - N be diffemmorphism. If is smooth vector field over M and 0 the flow of X, then the flow of F, x 17: Fo 0g F-! with domain N, = FMM,) (.M,)for for each E R.
Proof. See Lee 2003 Corollary 9.14).
Corollary 2 (Homogeneity Property). The horizontal tifi Ey at representative Y St(k, D) relative to ε E T(x Gr(k; D) satisfies the following homogeneity (equivariance) property with regard to "Q O()).
dh zrq= = -rQ
Proof. x (Y)= = (YQ) is true for "Y E Su(k, D) Q O(k). Therefore, π (Y) (n g) (Y)
true when defined as q (Y) = YQ. When the derivative dm (+) [] of both sides applied to the horizontal lift Ey of f the following is obtained:
dm Sx d (T 4) (Y)
dr (g (Y)) (dy (Y)
< [x]]] du (YQ) [
(10)
Moreover, from (98) which definition of horizontal lift, the following equation is true.
εY dm (YQ) [
ςΙو dm
(11)
Subsequently, we obtain the following equation.
h [ dm (YQ) 5 Q
εςΛ - dr (YQ)
(12)
Finally, the uniqueness of the horizontal lift yields Erq Q.Q.
Proposition I Let Gr(k, D) be a Grassmann manifold, be any dependent vector field on Gr(k, D), and Fx.r be a flow
Let X be any time-dependent horizontal lift and F x,1 be a flow X. & is vector field on Su(k, D) if and only if F x is a flow on St(k, D) and satisfes invariance condition X X for all Fxr Fx,T. Therefore, X is vector field Gr(k, D) if and only if Fx. = F Fx is a flow on Gr(k, D), and vice versa.
Proof. Flow Fx,T on Gr(k, D) = Vector Field X on Gr(k, D). Let Si(k: D) x O(k) Su(k, D), (Y, Q) + YQ be map representing the right action of the orthogonal group. In addition, let Fx, be a flow on Gr(k, D) and F T be flow on St(k, D). These satisfy F Fxq.T
F. denotes the pushforward, that is, another notation for the differential of F.

---

Fxq. E Fx.T, Fxr Fx,r.
x (1,, ( xQ. (8Q) = - (x
(13)
di (x
(14)
di (0 Fxt (8)
(15)
d(0)y dt Fxx (8)
(16)
d(0)r (x(1,xx,()) = -x(1 Q
(17)
(18)
Thus, X ~ XQ is true. Therefore, X is the horizontal lift of the vector field X on Gr(k, D) and is unique for X.
Flow Fx,r on Gr(k, D) = Vector Field X on Gr(k, D). Let 0 St(k, D) x O(k) + St(k, (Y.())Q))) + YQ be map representing the right action of the orthogonal group. In addition, let X be vector field over horizontal bundle T St(k, D) on St(k, D) and Fx,r be its flow. From the Corollary D
x, o 0-1 F O. of 0 प,T
F F,, = =
(19)
(20)
F X,I o0 0 F X,T
(21)
F FxQ.1 (YQ) = F X,r (
(22)
Note that d(0) XQ is derived from the Corollary 2 and (4) in Zhu & Sato 2021 This indicates that F T ~ Fx is true for any x,X E T St(k, D) that satisfies X ~ X. Thus, new flow
can be defined as Fx,r = Fx This is flow on Gr(k, D). Because X a vector field in horizontal bundle T St(k, D) on St(k, D), is a horizontal lift of the vector field on Gr(k, D) and is therefore unique for X.
Thus, the proof is complete.
B.2 Propasition 2
Proposition D Let Gr(k, D) be Grassmann manifold. Let p be the probability density on Gr(k, D) and F be the flow on Gr(k, D). Suppose is density on Si(k, D) and F is flow on St(k, D). Then, the distribution DF after transformations by F is also density on St.kk D). Further, the invariance condition PF ~ Pr is satisfied for all F F'. Therefore, pr = [PF) a distribution on Gr(k, D).

---

Proof. Let 0 St(k; D) x (kk) + St(k, D) (Y - YQ be map representing the right
action the orthogonal group.
Pr (0 Y) Pr (0 Y) ddet. {Jo (Y)H Po-toF ooF (Y)
(23)
det {Jo (X)
dee {Jo (Y)
dee {Jr 100 (Y)
((FF 00) (8))
(24)
Idet (Jo (Y))
|det (JooF-1 (Y)
(po ox( (
(25)
|det {Jo (X)
(det. {Jo (F-1 (Y)) JF-1 (8)
0 p F (-1)
(26)
deet (Jo (Y)H
det {o (F (Y)) }1 |det {Jr-1 (Y))
00 (70 F (8)
(27)
|det {Jo (Y)
det {Jo (8)((()))
0 ( (8) /det {Jp-1 (8)
(28)
det (Je (Y)
det {Jo (F-1 (Y))H Idet (Jo (Y)}
op (Y)
(29)
0 PF (Y),
(30)
where Idet {Jo (X)H 1 true because 0 is the action of the orthogonal group. Therefore, as Dr (0 o Y) = o Dy (Y) is true, Dy Q E pp is true. Based on this, it can be concluded that the subject is satisfied.
B.3 Proposition
Proposition 3. The distribution D) Grassmann manifold Gr(k, D) based on the matrix- variate Gaussian distribution. MN can be expressed as follows.
D) [ U, V) VG.(..))MN (Exri det (VEN, Rm)
(31)
where M an orthonormal basis matrix denoting the mean of the distribution, U a positive definite matrix denoting the row directional variance, V is positive definite matrix denoting the column directional variance, and EM is random sample from MN in an (D k)) x k-dimensional horizomaal space Th St(k, D). VGr(k,D) denotes the total volume of Gr(k, D) defined by (113), RM denotes the horizontal retraction at M, and det (Vg) RM) denotes the Jacobian.
EM
Proof. Let per ([X)) be probability density function Gr(k; D). From [01p. let (dX) ) be the invariant measure on Gr(k, D) and dE'rs be the Lebesgue measure on The Su(k, D). Subsequently, a change of variables was performed according to the following:
P) (IX)) (dX) PGr(k.D) ( [TAr (4"x)] dE'r
(32)
dE'r
PGr(k.D) ([X)) PGr(k.D)
TRi h (EAr) det
(33)
dR
PGr(k,D) ([X)) PGr(k.D)
RRv (5M)) det
(Ve)
RN
)
(34)
Ens
3

---

Suppose PGr ([x]) is integrable with the probability measure [dX]] on Gr(k, D) defined by (14).
Then, we obtain the following relation.
I Dock. D) ([X]) [dx]
(35)
Ven(..D) I Sonkk.)) D) ([X)) (dx)
(36)
V) Jry. St(k,D) P)
RAM (EMI)) det (V End RM))
dE
(37)
In addition, we obtain the following equation based on [d.]].
SCHA.D PGr(K.D) ((x)) [dx]
/
MN (EMs) dEM = 1,
(38)
T SU(k.D)
where MN (tmx) denotes the matrix-variate Gaussian distribution (Mathai et al. (2022)). Thus, the probability density function on Gr(k, D) can be expressed as follows:
PGr(k,D) [RM (tm))) VGr( (ε"r) /det (V RM)
(39)
The Jacobian can be represented as det (V
Eid RM) = R |act (aRM)) (ORM) from Evans
& Ronald 2015), where QRM BRM Further, 0RM can be computed follows. First, we define a the horizontal retraction Ry : Th St(k: D) + St((k, D) based on the Cayley transform from Zhu Sato (021).
X = RM
- (EM)) = M + EM
1 1 +
4 {m
(
(
S EM
h T EM
EM-
(40)
This is fixed time = 1) version of 122 Next, for improved visibility in subsequent calculations,
let E Eur F= } M + E, G
H E E. In addition, let D be defined as the operator for the derivative of a matrix by a matrix. Then, the derivative as follows:
I 1 H
RM DX by E is EM
RM = DX = DM + DE D (FGH).
(41)
EM
Finally, each derivative can be calculated as follows:
DM =0,
(42)
DE DK:
(43)
D(FGH) =D (F (GH))
(44)
{(GH) @ &D DF + (I, F)D(G))
(45)
{(GH)*
8I DF+ () ((H ( DG (4 ( GH)
(46)
{GH))
@ID DF
+
+
F) H
@ I
DG
(47)
(G
H
In)
DF
H
FF (4 8 DG)
(48)
DF -
M
1
+D
1
E
1
DE
(49)
2
4
DG (G
8 GG
(50)
DH (Ls + Kx,t) (I& @ ET)DEE
(51)
where denotes the Kronecker product. KD. is Dk Dk matrix K EM E (Li, @ LTJ) referred to as the commutation matrix, which denotes the transposi-
tion operation of D x Further, Li,j D matrix whose (i.j) component is whereas all other components are 0.

---

deg-
dog 30
day 60
dey-90 90"
dry 120
dag 160"
Figure 6: PGr(1.3) ([X]) with M - 0.00, 0.0) -U- 1 - 0.5. Each sphere in the figure indicates Gr(1,3), with brighter spheres representing higher densities.
For details the formulae for matrix derivatives used in this proof, please refer to Magnus & Neudecker (2019).
P) ([X]) P) ([X]: [M] U, V) is probability distribution following mean [M] and matrix variance U, V. Using Gr(1, 3)
example, qualitatively confirmed through visualization that PGr(113) ([X]) is a density on Gr(1,3). Gr(1, 3) is a 1-dimensional subspace in 3-dimensional space; that is, space whose elements are lines passing through the origin in 3-dimensional space. For the visualization, we expressed Gr(1, 3) by mapping a 1-dimensional subspace
points sphere (one point the sphere and its antipodal point) of radius centered at the origin.
Figure 6 shows the density of PGr(1.3) (()) with M 01.0,0.0.0.0)
-
0.5. Each sphere in the figure indicates Gr(1,3), 3), with brighter spheres representing higher densities. The leftmost figure shows M as viewed from the front diagonally above, and the other figures present views when the viewpoint rotated clockwise around the z-axis by 30 to 150% with movement to the right. In the leftmost figure, the density is highly spread around M. In the other figures (particularly the rightmost one), the antipodal point (-M out. This implies that when only one M is specified as the representative of the equivalence class [M], the density around the other elements in the equivalence class [M] is as high as that around the representative. Thus, we can confirm that P) ([X]) has density of Gr(1.3).
1.0, 0.0, 0.0) is densely spread

---

C Experimental Details for Learning GrCNF
C.1 Details on ODE Solver with Orthogonal Integration
We explain in detail an ordinary differential equation (ODE) solver on Gr(k, D). Several studies on ODE solvers employed on a manifold M have been reported (Munthe- Kaas (999 : Iserles et al.) (2000); |airer (2006)). |Hirer (2006) proposed a simple projection method that projected onto the manifold at each step and the symmetric projection method suitable for long- integration. However, this method requires M to be a submanifold in Euclidean space, and thus cannot be applied on Gr(k, D). In contrast, Celledoni & Owren (2002) proposed an intrinsic ODE solver that was applicable to a Stiefel manifold and did not assume an outer Euclidean space. This solver works on a Stiefel manifold, thus it must be reformulated into a solver suitable for Gr(k, D), which is our problem setting. We introduce below solver on Gr(k, D) via ODE operating on the horizontal space T St(kk, D), based on results from Celledoni & Owren (2002) and Section [ This is an intrinsic approach regardless of whether MM has been embedded a larger space with a corresponding extension of the vector field.
First, consider an ODE expressed using vector field Xo curve y (t) (0, St(kk D) such that for each time t.
dry (t) = X (t, (t),,
(52)
dt
where Xp is constructed by a neural network with parameter 0, as described in Section 5.3 retraction Ry which is defined (122) and is described in Appendix D.6, defines local coordinates of Gr(k, D) in a neighborhood of the point [Y]. Thus, the solution of ODE can be expressed as:
Horizontal
Y (t) = Ry (e (t))
(53)
where E: : o0) + T St(k, D) is the curve on T St(k, D). By differentiating 53 with t, the following equation is obtained:
dr (1) d Ry (c))) Xo (l)
(54)
dt
Therefore, the ODE defined on 72 St(k:, D) is obtained (8
deecc (e (t)) Ry) -1 Vec (Xo (t, Ry (e (t))))
dt
Ry Vec (Xo (4, Ry (e (()));
where Vec denotes the map of vertically concatenating matrices and converting them into a single vector. V,l can be calculated using the derivative of (23)
V,RY 2NN & ID) VM +2 2 (I @ V
(55)
where X x (t), M Y X N I YY, VM- IDk - (I XXXX and
VN-1 = (N-Ta & N-1) (I& 8 T)) Because (8) is an ODE on To St(k, D) = xx
from Absil et al. 2008), it can be solved using ODE solver such Runge-Kutta methods that operate on Euclidean space. This study used Algorithm 5.1 presented in Celleedoni & Owren (2002).
In each step, first, (8) was solved using the Runge-Kutte method of order as in Dormand & Prince
(1800). Subsequently, the solution of ODE 522) was obtained by applying the solution e to (53).
C.2 Loss Function based on Variational Inference
In the setting in Section 6.3 we used orthonormalized data Y as in PP -AYT where A is diagonal (Huang et al. (2015), ). such that the k-dimensional point cloud data of N points N x matrix is matrix with k orthonormal basis vectors. Thus, generating complete point cloud requires the estimation of the scale parameters VA to be P- Y Y A and loss function that incorporates this. In this study, we approximated by maximizing the evidence lower bound (ELBO), which is the lower bound of the overall log- likelihood log p (P) of p P), using variational inference framework. The loss function is the variational energy- ELBO (P) with negative ELBO.
NLL log pa (P) < ELBO (P) Loss
(56)

---

Rg((5x (167
(N
Y
(EY.)
Br. (16v.)
Sv. (16x))
Ty, St(R D)
da (tr..
Golk. D)
Rori (15mr)
St(kk D)
[C]]]
1
Epra
St(R,
Tira Gr(k,
(Xa)
(a)
(b)
Figure 7: The proposed ODE solver the Grassmann manifold and the ODE solver the Stiefel manifold (Celledoni & Owren (2022). (a) The proposed ODE solver working on a Grassmann manifold. The Ey obtained by the proposed neural architecture NN the horizontal lift (98). that is, dm proposed ODE solver maps and updates the tangent vector čiy the point [Y] onto the Grassmann manifold at each step using horizontal retraction [2I]). On other hand, (b) the solver Celledooni Stiefel manifolds, updates in each step by mapping the tangent vector EY at the point Y onto the Stiefel manifold. In other words, the difference between the proposed ODE solver and the ODE solver on Stiefel manifolds that the ODE solver on Stiefel manifolds works only on Stiefel manifolds, while the proposed ODE solver always updates in each step with the Stiefel manifold and Grassmann manifolds linked together.
zh
Eir]. of the tangent vector E(y] at the point YY] the Grassmann manifold. The
& Owren 2002). working
ELBO (P) can be decomposed as follows.
ELBO (P) = log (P) DEL (Y (Y (P)P))
(57) (58)
E [log& - DRL (1o (p)
where go Y ( is the inference model with parameter d, Pa (P is the decoder model with parameter v. Pv. (Y) PP) is the posterior distribution with parameter v and pa (Y) is the prior distribution with parameter Further, DKL (Y (P)()p((Y) can be formulated using differential entropy as follows.
DEL (1s PY)p) (Y)) -E 1E) (po H las [P).
(59)
Thus, the final loss function is as follows.
Loss ELBO (P)
(60)
- Byg (P) [logp YP - EydY [Yo) H (Ys P)).
(61)
Each term of the loss function can be calculated as follows.
PP Expectation of log-likelihood Eg [log P (P) Y)) is the reconstruction log- log-likelihood of P.
The expectation is estimated by Monte Carlo sampling.
Differential entropy In the decomposition of point cloud P, there exists arbitrariness in the choice of Y and A, in PPT Y TA In this study, we assumed that the diagonal components of A are in descending order, and we restricted the decomposition arbitrariness to be an action Q E (k).
we suppose that the action Q follows a uniform distribution when Y -XQ holds,
then Y also follows uniform distribution in the k-dimensioonl subspace span(Y) Although
this is a uniform distribution on St, k), we can consider uniform distribution on ((k) because
St(k, k) O(k). The probability density function of Q is represented by Uou) (Q) in [11]
7

---

Therefore, the differential entropy of the decoder model can be calculated as follows.
H 190 Y P)]
E - log q0 (Y P)]
(62)
I low
Uo( (Q) log Uo()) (Q)
(63)
Sors) Vo(
1 log Vo(k)
[dQ]
(64)
log Vo(k)
Vo(M) So(e) [dQ]
(65)
log Vo(k)
(66)
Vo(k)
Expectation of prior distribution We used (9) for the prior distribution po()). Further, re- parameterization was used to enable differentiable Monte Carlo estimation of expectations.
BY P) (Y) =
po (XQi) s.t. X~ Y, Q1 (K)),
(67)
where it was assumed that Q is sampled from uniform distribution, which follows Haar measure
(k). L is set
C.3
Implementation Details and Experimental setting
The following sections present more details about the network architectures, training hyperparameters, and experimental conditions for each of the experiments in Section 6
C.3.1 Artificial Textures
The vector field was constructed with the specific input, intermediate, and output layers described in Section 5.3 The GrCNF architecture is shown top in Table Layers are denoted as Layer in the table, and were processed from top to bottom. Norm. and Act. denote the normalization and activation functions to be applied immediately after the Layer, and the Norm. and Act. were applied that order. Further, Out Size denotes the output size after Act. Vec denotes the map of vertically concatenating matrices and converting them into single vector. Moreover, only row Input denotes the size of the input data, not the input layer (HorP). was used for the loss function.
Network Architecture
The mean M and covariances U and V in the prior distribution the Grassmann manifold were set M (1.0, 0.0, 0.0),, U o² and V IT 0.3. respectively. Other hyperparameters used during the training of GrCNF are shown in Table 4
Hyper-parameters
Implementation We used PyTorch (Paszke et al. (019)) to implement the model and run the experiments. The CNF is based on the implementation in Chen al. (2018) and the framework of the RCNF (Mathicu & Nickel (2020)) Thus. the ODE was solved using the explicit and adaptive Runge-King Kutta method (Dormand & Prince (980)) of order 5, and worked by projecting each step onto manifold (Hairer (2006)). The autograd in ( was calculated with borch.autograd.o .grrddd.gdddddd (Paszke)
al. (2017) ) in Py Torch. The experimental hardware was built with an Intel Core i7-9700 CPU and single NVIDIA GTX 1060 GPU with 6 GB of RAM.
The code used in the experiment to generate the data distributions on Gr(1,3) is shown in Listing This implementation of the data distributions is based on the codes in Kim et al. (0200) and Grathwon!
al. 2019
'We used the authors" implementation: http://wwww...ae...e..o//.////aa
"We used the authors' implementations:
http::///ggttaaa.ccoo///aaa///// and https:
//github.. wttp////ooochcee/ffforrr..dd

---

Listing 1: Code for the data distributions.
Import numpy a8 up
def get_ dat_data_att batch (batch_ dist):
rng up. randon. RandonState
"Zepirals";
np. (np
// 2, 1)) 540 (2 mp. )i) / 360
d1x -ap,, -ap.cos(a) (a)
up rendon. rand (batchhh.. // 2, 1) 0.1
// x up.vstack ((np hatack (ddxx, dly)), up. hatack bastack((Eat)) ((-dix, -dly)))) aample_2d
dly ap.sin
ap randon. raad (aatch_____eiee
0.1
np.randon. randn (4x. shapa) 0.1
"sviseroll":
data - sklearn. datasets. matee make make_aviss_ron (on (n..aapllerrbbatc....ia noisem.3) [0]
data data, astype (" flost32") ("tooot))) [t, [o, 2]]
cample_2d = data
elif "2circles":
dats - aakeear dataaete caate_circlee (u.eanplles-atccce.eee
factor= moise=0.05 [o]
data - data. astyaa
sample_2d data
alif dist == "2aines":
(rug. rand (batchbeiie))
up.pi
(rag. binonial (1, 0.5, batch_size) 0.5)
y up. nx)
+= up. randaaannaaaaae (*x. shape) 0.1
up- randon randn (xy, shype) 0.1
sample_2d ap.stack((x y),
elif dist "target";
shapes randon. randint (7, randi nask
size=batch,eizo)
for in (7):
nask append ((ehapee 1) 1.)
theta np. lipplacaacee 2 pi.pi, batch, size, endpoint-Falee)
(maee [0] mask mask
(rng rand (xncc (-mark [3] mask [4] + 0.0 m5))
* oneo (batch__size)) +
nask[6] ap.cos(theta))
(masx [3] nask mask [5]) (rng rand (batch_alzo 0.5)
(-maek [0] mask [1] 0.0 maek [2]) onee
ones (batch_.iee)
nask
(thhett)
x +- up randon randn (*x shape) 0.1 y ap. randon...... randn shape)
sampla_2d = stack ((x, y).
sample_20 ua. (up. linalg norm (sample_2d, sxis-1)) sample_3d uр concatenate ((np. ones onee ((batch_eize 1)), norm), axie-1) sample up. linalg norn (saaplee_ axis-1) [:, up. nevaxis]
C.3.2 DW4 and LJ13
Network Architecture As in Appendix C the vector field was constructed with the specific input, intermediate, and output layers described in Section 5.3 The GrCNF architecture shown on the bottom left and right in Table The bottom left and right were used for experiments and LJ13 datasets, respectively. The views presented in the table is the same as in Appendix C (9) was used for the loss function. In addition, for architectures in methods other than GrCNF, please refer Garcia Satorras et al. (2021).
the DW4
Hyperparameters The mean M and covariances and in the prior distribution on the Grass- mann manifold were set to M I 1xx,, U の² 1, and V = = M 11333, o² and V = o² , o = 0.3 for LJ13, respectively. Other hyperparameters used during the training of GrCNF methods other than GrCNF, please refer to Garcia Satorras ef al. (20IT)
0.3 for DW4 and
shown in Table 4 In addition, for the hyperparameters in
The experimental hardware was built using single NVIDIA Quadro RTX 8000 GPU with 48 GB of GDDR6 RAM. The other environments were the same in Appendix C.3.I
Implementation

---

Table Network architectures for each experiment; (left) for the Simple Texture dataset, (middle) for the DW4 dataset, (right) for the LJ13 dataset.
GrCNF for Textures
Layer
Out Size
Norm/Act.
Input
3x1
HorP
3x1
-/Tanh
Vec
-/-
CS
64
-/Tanh
CS
64
-/Tanh
CS
-/Tanh
Grad
3x1
GrCNF for DW4
GrCNF for LJJ3
Layer
Out Size
Norm/Act.
Layer
Out Size
Norm./Act.
Input
4x2
Input
13x3
HorP
4x2
-/Tanh
HorP
13x3
-/Tanh
Vec
8
Vec
39
+/-
CS
64
-/Tanh
CS
32
-/aanh
CS
64
-/Tanh
CS
32
-/Tanh
CS
64
-/Tanh
CS
32
-Tann
CS
-/Tanh
CS
-/Tanh
Grad
4x2
Grad
13x3
Full results In Tables 5 and 6. the same DW4 and LJ13 averaged results from Section 6.2 were reported; however, they included the standard deviations over the three runs.
C.3.3 QM9 Positional
Network Architecture On the QM9 Positional, we addressed the task of generating the molecular
P by estimating the scale parameter VĀ = diag (lag(((())) { in addition to the generation of the orthonormal basis matrix Y with GrCNF, Because the molecular generation task requires specialized loss function based on variational inference, we used (56), as explained in Appendix C.2 We designed two networks achieve this. The first the same GrCNF architecture as in previous experiments, and the second is scale estimator. sshows the architectures. The left side of the table shows the GrCNF architecture and the right side shows the scale estimator. The scale estimator estimated one scale parameter from each of the three orthonormal basis vectors Y = {y: E 1219 Ji=1'
for total of three parameters { VA, R } With the orthonormal orthogonal basis matrix Y and the estimated scale parameter / A, generated a point cloud P V In this study, the overall architecture that generates P is also named GrCNE.
Hyperparameters mann manifold were set to M = T19x3, U = I and V = I²I 0.3, respectively. In addition, for the hyperparameters in methods other than GrCNE, please refer to Garcia Satorras et (2021).
The mean M and covariances and V in the prior distribution on the Grass-
Implementation
The experimental hardware was built using single NVIDIA A100 GPU with 80GB PCLe of GDDR6 RAM.
10

---

Table 4: List of hyperparameters used in various experiments. A nL" indicates that the hyperparameter unused.
Textures
DW4
LJ13
QM9
Train
109/10//10/100
10/102/10/100
13, 831
of Data
Validation
500
1000
1000
2,501
Test
500
1000
1000
1,813
Name
Adam
Adam
Adam
Adam
betal
0.9
0.9
0.9
0.9
Optimizer
beta2
0.999
0.999
0.999
0.999
Weight Decay
1.0e-12
1.0e-12
1.0e- 12
Learning Rate
1.0e-3
1.0e-4
1.0e-4
5.0e-4
Epoch
72000
1000/300/50/6
500/1000/300/50
160
Schedule
LR Step with 0.1
20000
Batch Size
500
100
10/100/100/100
128
Integration Time
Training
Training
Training
0.1
atol
1.0e-5
1.0e-5
1.0e-5
1.0e-5
NeuralODE
rtol
1.0e-5
1.0e-5
1.0e-5
1.0e-5
Adjoint
X
Table 5: Negative log-likelihood comparison the test partition of DW4 dataset for different amounts of training samples; averaged over runs and including standard deviations.
DW4
of Samples
102
10s
10a
10
GNF
-2.30 +: 1.59
-7.04
0.64
-7.19 + 0.99
-7.93 1.10
GNF-att
-2.02 + 1.34
-4.13 1.20
-5.25 + 0.89
-6.74 0.89
GNF-att-aug
-3.11 2.15
4.04
+ 0.49
-9.42 = 1.15
Simple dynamics
-1.22 + 0.05
-1.28
-1.36 0.02
-1.39 + 0.04
E-NF
-0.54 + 0.45
-..99
-12.15 = 1.16
-15,29 0.53
GrCNF
-12.53 + 0.92
-13.74 0.30
-14.09 = 0.44
-16.07 = 0.46
Table 6: Negative log- likeliiid comparison on the test partition LJ13 dataset for different amounts of training samples; averaged over runs and including standard deviations.
LJ13
Samples
10
10a
103
104
GNF
6.77 + 0.39
-0.76 + 1.12
-4.66 +: 2.76
-12.43 + 1.21
GNF-att
6.91 + 0.17
1.40 0.79
-6.81 + 2.09
-12.05 2.28
GNF-att-aug
2.95 + 0.55
1.1+1.11
-13.44
-15.74 0.58
Simple dynamics
-1.10
3.87
-3.72 - 0.08
.3.59 0.52
E-NF
-1.86 1.86+3.66
.15.75 5.02
31.511 1.19
-3.833 1.98
GrCNF
-23.64 + 2.23
-4..44 4.26
-58.02 4 5.43
-58.71 4.71
11

---

Table 7: Network architectures for QM9 Positional. (Left) GrCNF architecture, (Right) scale es- timation architecture. The scale estimator estimates one scale parameter from each of the three orthonormal basis vectors Y y E 1
J for vA = diag ([v)));=) with three param-
eters {vAs E ]] Using the orthonormal orthogonal basis matrix Y and the estimated scale parameter VA, we generate point cloud P = A.A. SiLU is an activation function proposed in Ramachandran et al. (217)) and BatchNorm. is batch normalization layer in (loffe & Szegedy
(0155).
GrCNF for QM9 Positional
Scale Estimator for QM9 Positional
Layer
Out Size
Norm/Act.
Layer
Out Size
Norm./Act.
Input HorP
19x3
Input
19x3
19x3
-/Tanh
FC
128
BatchNorm//SLU
Vec
57
-/-
FC
256
BatchNorm
CS
32
-/Tanh
FC
256
BatchNorm./Sil.
CS
-/Tanh
FC
128
BatchNorm//SLU
CS
32
-/Tanh
FC
-/ReLU
CS
-/Tanh
Grad
19x3
12

---

D Fundamentals of Concepts Associated with Grassmann Manifold
D.1 Definition of Stiefel Manifold
Definition 2. An (orthogonal or compact) Stiefel manifold St(k, D) is defined as the set of orthonor- mal bases of k-dimensional subspaces in the Euclidean space _. in 68).
Su(k, ()k = {Y E R Y Ix]
(68)
For Y E St.k: D), the space span (Y) spanned by its column vectors is the element of Gr(k, D).
I St(k D) + Gr(k; D) Y + span
(69)
St(k, D) is Dk k(k+1) dimensional compact manifold (Absil et al. 2008
D.2 Equivalence Relation
To define the equivalence relation on a Stiefel manifold St(k, D), we introduce the following two lemmas.
Lemma 1.
The necessary and sufficient conditions for span Y;) = span Y2) to hold for Y1, X2 e D) are as follows.
3Q OKk)
Y2 Y.Q.
(70)
Proof. span(Yi) = span (Y2)
+ Y2 = Y.Q. From the definition, Y Y, = Y Y22 = Is. the
following is obtained:
I = = Y Y
(71)
Thus, there exists a k-dimensional orthogonal matrix Q E O(k). As the subspace span (Y) is invariant to coordinate transformations by orthogonal matrices, sY = span( (2)) is true.
span( Yi) sP) = Y2 = Y.Q. From the assumption, we immediately concluded that Y22 =YQ for Q OKk).
Lemma 2. We define the equivalence relation ~ on St(k: D) be Y, Y2 whenever 77 is satisfied with respect to Y,, Y2 E St(k, D). The fact that a binary relation is an equivalence relation ~ implies that the following three statements hold for v Y1. Y2, Ys E St(k, D).
Reflexivity Y1 Yr.
Symmetry Y, Y2 Y2 Y,
Transitivity Y1 ~ Y2 A Y. = Y
Proof. With Lemma D we can confirm that it is valid as follows:
Reflexivity From Y Y1I, I O, we obtain Yi ~ Y1.
Symmetry As Y2 Y is obtained from Y, Y2, and YºQ Y,, Q E O true, then Y2 ~ Y1 is obtained.
Transitivity Y3 r2QQQ with Y. YiQ and Y3 Y2Q2. Q1Q2 is (Q1Q2) (Q1Q2) Q 2 QiQq = Q Q2 = I. Moreover, as (Q1 (Q1Q2) = Q.Q:Q.Q = holds, QiQ2 E O is true. Therefore, concluded Y1 Ys.
The equivalence class Y E St(k: D) denoted by YY. In other words, Y is the set of all elements of St(k,, D) that equivalent to Y. and the equivalence relation St(kk D) divides St(k: D) into equivalence classes with no intersection. Thus, Y is then referred to as the representative of the equivalence class [Y). The set of equivalence classes is denoted Si(k, D)/~ and is referred to the
Reflexivee symmetric and transitive binary relations. one equivalence relation divides (classifies) the set into equivalence classes. Note that H is a binary relation in the set X if for any y E X. only either is related to by the relation R. or x is not related to based on the relation that A occurs. We write is related y by relation Re as afy.
consequence of these properties,
given set,
13

---

quotient of St(k, D) by the equivalence relation r. In addition, ग :- St(k, D) + St(k, D)/~ is the natural projection that maps Y St(k, D) onto its equivalence class [Y). This projection surjection.
D.3 Quotient Manifold
Definition 3. Let M be manifold with equivalence relation ~.. The quolient space M/~ with ~ of M is the set of all equivalence classes. Thus, M M/>:=fo { (x) I E M), where л M ++ M/ is the natural projection and π (τ) == ( M M y~ æ}. Then, M is referred to as the total space or the total manifold. Moreover, M/ is referred to quotient manifold of M if M/~ admits differentiable structure.
Let M M// be a quotient manifold. Further, suppose that M endowed with Riemann metric g. and let x = (ar). The horizontal space T the orthogonal complement of the vertical space M = Tx (x) in the tangent space T&M and defined as the follows:
T MTM
TT
TAM
Fx (5xz The - 0, E M.
(72)
h E T of the tangent vector EN E M at point T E π (a) is a tangent
The horizontal lift & vector that is uniquely determined as do
5mm
Exx (Absil et al. (008)).
D.4
Grassmann Manifold Exploiting the Quotient Structure
D.4.1 Tangent Space on Stiefel Manifold
We describe the relationship between tangent space Gr(jGr(k D) Gr(k, D) and tangent space St-St(k: D) on Su(k, D) to relate the tangent vectors of a Grassmann manifold Gr(k, D) to the tangent vectors of a Stiefel manifold (k: D) in a matrix representation. We take the derivative on both sides of Y (1) Y (1) = Ip in 66 by / and solve for t O.
(Y()) Y (+) d dt
d
Ip
(73)
d =Y Y (1) T Y (1) + Y (1)
Y (t) -0 dl
(74)
di
d
d
Y (0) Y (0) + Y (0)
y (0) 0
(75)
dt
dt
s y + Y &x 0,
(76)
where yy di Y (0) is the tangent vector at Y
Definition 4. Define the tangent space Ty St,k D) at Y on the Stiefel manifold follows:
St D) Er E Dxk EY +YY+YTEN
Y =0p
(77)
Let matrix Y, E DD--k) be matrix satisfying the following:
Y Ip YY YY +YYY = Ip.
(78)
As Y Y1 is an orthogonal matrix the column vectors of Y and Yı form orthonormal basis in P Thus, any D k matrix be written in terms of the C R and B (D- coefficient matrices as follows:
xx*k
YC + YLB,
(79)
'The tangent space is defined independently for each point of the manifold; hence, the subscript Y, as in Ey is clearly stated to emphasize that it tangent vector Y.
o Y Y. [ Y Y ] Y Y
Y Y Y== Y ][ Yi ] Ip.
14

---

where Ey = YC Y_B Bi is inserted. The following equation is obtained.
ErY +Y y (YC+ Y YY (
(80)
-BYY D +C
(81)
Y +C
(82)
=C
(83)
=
(84)
Thus, the following equation is derived.
+
(85)
Thus, C is a ho x i skew-symmetre matrix Skew (k:). Therefore, obtain the following another
representation of the tangent space on St(k, D).
St. D) = +C C'e (k), B E (D-)))))
(86)
D.4.2 Riemannian Metric on a Stiefel Manifold
In the tangent space M defined at each point a E .M on the manifold M, the inner product is endowed bilinear map. This h is referred to Riemannian metric manifold, and the manifold M on which the Riemannian metric h is endowed is referred to as a Riemannian manifold (M, We define the Riemannian metric y on St,k; D) follows.
TY (Ex: :r) = (EVTTT)
s.t. Er, Fy Ty D), Y E St(kk D).
(87)
This is the standard inner product of IR Dxk induced by Ty St(k, D), with Ty St(k, D) C Px
D.4.3 Tangent Space Grassmann Manifold
We describe the relation between tangent spaces Gr(k, D) and St. D) to relate tangent vectors in tangent spaces Gry), D) on Gr(k, D) to tangent vectors E E Ty St(k,D)
First, define the vertical space Tÿ St(k, D) subspace of Ty St.k, D) follows.
TÝ St(k, D) == Ty ((Y),
(88)
where St(k:, D) - Gr(k, D) is the natural projection defined by æ (Y) converges all Y St(k, D) such that Y ~ Y' to point [Y] on Gr(k, D). Therefore, using ( 88 can be transformed as follows.
Y
Thus.
Ty St(k, D) Ty (YQ E O(k))
(89)
However, Er E ETY St(k, D) can be written as Ex = YS with S E TT.
Tr. O(k) = Tr, St(k,)
(90)
= =I + (1a1 B 0z) C Skew (k), B E R (R (k====0)xkk
(91)
= =kkew (k))
(92)
Whee inner product A C-C C ATC of a vector is typically referred to as the standard inner product. Further, matrices similarly defined with standard inner product, defined as A C (A C). A space R P such that the & matrix space can constructed by a matrix wherein only one element matrix space is a linear space because it satisfies the linearity that is similar to that in case of a linear vector space.
element is referred
matrix space. The standard basis of the matrix matrix is and remaining are 0. The
"When a submanifold of Riemannian manifold (M,g) g). we define the Riemannian metric g of N be:
I= (E. m) = 9x (E.n), x E A' C M, E T&N C T&M. is an induced metric and (N, 7) is Riemannian submanifold of (M, As St(kk, D) is submanifold of Dxk we can define the standard inner product A C = (A" Cf the induced metric 7. Thus, St(k, D) a Riemannian submanifold Px
"Suppose set is given suitable equivalence relation. A natural projection is map that sends each element of set to the equivalence class which it belongs.
15

---

Thus, we obtain the following formula.
Ty = C E Skew (k))
(93)
Next, we define the horizontal space T& St.kk D as the orthogonal complement of Ty St(k: D) in Ty St,k D) endowed with the inner product (87
r& St(k, D) (TV St(k,D))
(94)
hT = - E St D) tr 5r Th = 0, mir TÝ St DI}.
(95)
Based on the fact that T. St(k, D) is a subspace Ty D) and T& St(k: D) is defined as its orthogonal complement, the direct sum decomposition is as follows.
Ty St,, Tł St(k: D) e T D),
(96)
where @ denotes direct sum. Moreover, the tangent space is linear space Absil et al. (86), element YC of TV St(k, D) corresponds to the first term of (33) thus, (9) tormulated as follows:
2008 From
T St(k; D) { Y B
(97)
h Note that the horizontal vector Ey not necessarily an orthogonal matrix.
Finally, define the element Ey E Th St(k, D) of the horizontal space at Y St(k, D) for the tangent vector Err] E Tix] Gr(k, D) at [Y] Gr(k; D) as satisfying the following formula.
dm (Y)
[x E(r,
(98)
where drr (Y): Tv St(k D) + Tl Gr(k, D) is the derivative dr
dY of . Su(k, D) - Gr(k, D)
at Y E D). The Ex T St(k, D) is referred to as the horizontal lift at Y E St(k, D) of [Y] Gr(k, D).
We describe the tangent space of Gr(k, D) with the concept of horizontal lift.
Definition 5. Let T4 St(k: D) be horizontal space on St(k, D). Then, define the tangent space
Tyy Gr(k, D) of the Gr(k, D) as follows.
Tm Gr(kk D) {ir/) du (Y)
Eh Ev Eir), Ey E T Stk
(99)
2h From the above. Eiyπ Tir] Gr(k, D) is obtained from the map dx (Y) Ey when &y is obtained. The f is defined by an equivalence class and cannot be treated numerically in matrix form; however, it is sufficient to obtain the Ey for actual numerical calculations. For ε|-] T(y] Gr(k, D), there
th exists εy T St(k, D) that uniquely satisfies (98). In other words, we can handle it in matrix form by using elements of the horizontal space of Stietel manifolds through the concept of horizontal lifting. Figure conceptual diagram of the tangent space representation of Grassmann manifold by horizontal lift.
D.4.4 Riemannian Metric on Grassmann Manifold
We define the Riemannian metric g of Gr(k, D) through the concept of horizontal lift.
Definition 6. Let Et and niy be the horizontal lifts that become dm (Y) [
εςy] and
du (Y) [nt]] = my), respectively. Then, we define the Riemannian metric on Gr(k, D) as fol- lows:
g|Vl (Ex) mxn = 9Y cr, / = tr -BT
(100)
where B and D are matrices that are Ey Y1 B and nt Y D. respectively.
16

---

D.5
Invariant Measures
Let the column vectors of matrix Y fyr.... yk E R be the orthonormal basis that span the subspace span( Y) Gr(k, D) in I and the column vectors of Yi (, yn DDDD be the orthogonal complementary space span(Y1) of syan(Y), respectively. Then, the following differential form can be defined.
D-k k
(dY)
A
A
dy:
(101)
3 jm1 1m1
19 dyn A... 1k+1dyss A.. (ybdy) A ybdy)):
(102)
where the wedge product and the relation satisfies wi Aw; = wy The above equation is in k (D oorrr differential form, which is an invariant measure on Gr(k, D) (Chikuse (2003)).
wy Awy and wi Awy -w Aw;
If we define the matrix X, to be differential form for invariant measure on St(k; D) is defined follows.
x
X1 ] for any point X ( ,..) E Stikk D), the
D-k k
(dx) = A
A x
k+y
dæ; A Atjddx; dx; = (dQ)
(103)
j=1
where (dQ) is the invariant measure of Olk). The integral of [103 that is, the volume of St D), can be evaluated as follows:
VSU(K.D) =
(dX).
(104)
Du(k,
(144) can be computed as follows. First, the surface Sn of the D-dimensional unit sphere can be defined as follows:
SD d
D
2T
VD = DVD [
(105)
dr
+1)
(
r=1
where VD is the volume of a D-dimensional sphere Then, the following equation obtained.
x +(
D and (2) is the gamma function.
SSLA.D) (dX) = SD
SS /
(SC(k-1,D-1) (dX1),
(106)
where (dX,) is the differential form of St(k 1, 11) Thus,
104 can be transformed as follows.
b II SD i=1
2 D
ViS(K,D)
(SS(E.D) (dx)
(107)
T
where T& (2) is the multidimenssonal gamma function. The invariant measure (dX) is
normalized measure. A measure normalized to be probability measure can be formulated as follows:
1
dx
(dX).
(108)
V)s(k,D)
This is a uniform distribution on St D). As St(k, k) = Ok:), the volume Vo(k) of O(k) can be represented using (dQ) as follows.
2 π 22
Vo(k) Vsu(kk)
(109)
Ts (
Furthermore, a measure normalized to probability measure can be represented by the following:
1
[dQ]
(dQ).
(110)
Voú.
17

---

From the above, the probability density function (o(K)) (Q) of the uniform distribution on (kk is as follows:
1
UO(K) (Q)
s.t. QEO(K).
(111)
Vo(
JOss UOa) (dQ)
I
(dQ) /
JO(k) Vo(k)
[dQ] =
(112)
((k)
As Gr(k, D) is defined quotient manifold St(k: D) (k) as in the volume (ar(k. of Gr(k, D) can be defined as follows.
V) / SCHOA.) (dY) = VIHCE.D) VS(K.D)
TA (4)
(113)
Vo(K)
Vis(k.k)
I' (2)
The measure normalized to be a probability measure is expressed as:
[dY]
(dY).
(114)
V)
D.6 Retraction
In general, the points except the origin (0) of the tangent space M the manifold M are not elements on .M (t) E & 0). Therefore, the result of the operation on the tangent space be used another point on the manifold M,, it is necessary map p (8) to the manifold M. The map from a tangent space to a manifold is referred to as an exponential map. However, because the exponential map is computationally expensive, retraction based on numerical linear algebra is often used as an alternative (Zhu & Sato (202I)). Retraction is a method for approximating an exponential map to first order while maintaining global convergence in optimization algorithms on Riemannian manifolds. The most commonly used retractions on Gr(k. D) are methods based on QR decomposition or singular-value decomposition (SVD) (Absil al. (2008); Zhu & Sato (021) )
In addition, retraction based on the Cayley transform is introduced in Zhu & Sato (2021) 1.. This retraction is closely related the Cayley transform on (k. D) (Wen Ymn (203); x (207))
Zhu & Duan (2019)) and the Projected polynomial retraction (Gawlik & Leok (2018))
D.6.1 Exponential Map and Retraction
Geodesics on Gr(k, D) can be expressed as the equivalence class |exp Gr (IEx ). where
expy (tEr) = Y1 cxp (+9B ) I D
(115)
- BB
Here, exp on the right-hand side the matrix exponential, and B
skew (D),
B
0D-k
where B satisfies Ey = Y, B. We can use a following exponential map that is mathematically
equivalent to 115 (Edelman et al. 1998)):
expy Gr (tEx) (YV cos (Et) + U sin (Et) VT,
(116)
where U, Σ, V SVD (Ey).
Further, we can use the Padé approximation to approximate geodesics on Grassmann manifolds as follows:
Y()= = Y1 I' (DB) expx (lEy),
(117)
where Fm (X) is the mathoorde diagonal Padé approximation to the matrix exponential exp (X). See the expression of I'm (X) in Moler & Loan (2003). The simplest member of this class is surely the first-order Padé approximation
Ry (IEx) = Y Y1 Iri (IT) I
(118)
= Y Y. In 2
-1
I
Tn +
{3 2
I
(119)
which is also known as the Cayley transform. From the error expression exp (Y) = (Y) O [|Y|22++] of the Padé approximation, we have
+
Ry (tEx) = expx which is also given by Theorem 3 in Gawlik & Leok (0018)).
Gr (tεy) + 2m++1 || 2m+11
(120)
18

---

D.6.2 Horizontal Retraction
From Definition 3 in Zhu & Sato (021)) 119 is horizontal retraction, and
R|r ((É()) = |i (55x)
(121)
retraction Gr(k, D) quotient manifold defined by (2) This is because R satisfies the invariance condition that Ry ((Ex) Ry ((5x)) for all Y E St(kk, D) Y' St(k, D) Ey-
Th St(k,, D) and Ey, E The St(k, D) such that Y Y and Ey and Ey, are horizontal lifts of f(y) E Tin Gr(k, D) at Y and Y', respectively.
In low-rank cases,
obtain economical version of 119 follows (Zhu & Sato (021
Ry (16)
Y + tEx
(
Y 2
h )
12 hT h In +
75x
Sy
fx
hT EY
EY
(122)
4
The inverse retraction
Rix) St(k, D) + T St D) of Ry
(5x)
is the following:
PTy (xx)) h -Ry' (X)
(123)
-2Y1 (Is + + X)
(124)
-2 -2(X-YY' x) (I++
(125)
19


Reviews:
[{"review_id": "cRxETGlpYj", "reviewer": "Reviewer_gPKP", "summary": "This paper is on developing techniques for sampling from probability distributions on Grassmann manifolds and utilizing it to generating shapes. The shapes are associated with the probability distributions on Grassmann and sampling from the distribution generates shapes. The sampling itself it based on flows under vector fields associated with the gradient of potentials (negative logs of pdfs). A large amount of paper is dedicated towards defining vectors fields and flows on a Grassmann manifold assuming one knows how to do it on a Stiefel manifold. Some of these constructions are implemented using deep neural networks. The paper provides some examples of shape and texture generation and compares it to some similar methods. ", "questions": "\nSee my question listed above. \n\nThe authors need to clarify in simple layman's terms the big picture of their approach. This can be done without using undefined terminology and stating ideas precisely. ", "limitations": "\nYes, they have listed some limitations of the computational steps of their framework. ", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "-- The paper is trying to make a connection between a number of distinct and important mathematical quantities: shapes, distributions on Grassmann manifolds, random sampling, gradient flows, etc. \n\n-- The geometry part of the paper is solid. The use of horizontal lift to remove the nuisance group is a standard technique in analysis on quotient spaces. \n\n", "weaknesses": "\n-- I find this paper somewhat confusing to read. The presentation relies on CNF and Riemannian CNF etc, while these quantities are standard flows under given gradient fields (associated with given pdfs). \n\n\n-- Perhaps the authors can clarify what is different in CNFs from a integral curves or flows under vector fields such as gradient vector fields. See for example, the paper by Liu et al., Optimal Linear Representations of Images for Object Recognition, IEEE TPAMI, May 2004. These are very standard constructions for either Stiefel or Grassmann manifolds. \n\n-- The connection with shapes is also unclear. What do the authors mean by a shape? They should explain precisely (preferably in mathematical terms) on how shapes are represented and how sampling from a distribution on a Grassmann becomes generation of a shape. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed comments and valuable feedback! \nWe have prepared detailed responses to your questions below. \nWe sincerely hope that these responses will address and alleviate any concerns.\n\n****\n> **Weakness:1. I find this paper somewhat confusing to read. The presentation relies on CNF and Riemannian CNF etc, while these quantities are standard flows under given gradient fields (associated with given pdfs).**\n\nand\n\n> **Weakness:2. Perhaps the authors can clarify what is different in CNFs from a integral curves or flows under vector fields such as gradient vector fields. See for example, the paper by Liu et al., Optimal Linear Representations of Images for Object Recognition, IEEE TPAMI, May 2004. These are very standard constructions for either Stiefel or Grassmann manifolds.**\n\nWe greatly appreciate your provision of the reference. \nThe foundational theory of our proposed method lies in the principles of differential geometry, encompassing integral curves and gradient vector fields known as flows. \nHowever, finding a convenient approach to learning densities on a manifold that adhere to the symmetry imposed by the orthogonal group is not straightforward and presents non-trivial challenges. \nMoreover, the construction of manifold continuous normalizing flows that maintain the symmetry has remained unclear.\nTo address these issues, we deduced implications akin to Propositions 1-3, which paved the way for a practical solution. \nThis solution, predicated on obtaining the gradient of the potential function, allows for the derivation of a CNF. \nSpecifically, leveraging these insights, we constructed the algorithm (GrCNF) by parametrically modeling continuous normalizing flows and realizing a modelization for computing gradients (referred to as $\\mathsf{Grad}$) with respect to the potential function $v_{\\operatorname{out}}=\\mathsf{MILs}\\left(\\mathsf{HorP}\\left(\\boldsymbol{Y}\\right)\\right)$ within the proposed neural architecture.\nWe assert that constructing potential functions invariant under orthogonal groups on the Grassmann manifold is notably more flexible and straightforward compared to directly parameterizing complex densities invariant under orthogonal groups or flows equivariant under orthogonal groups, as attempted in Liu et al.'s work.\n\n****\n> **Weakness 3: The connection with shapes is also unclear. What do the authors mean by a shape? They should explain precisely (preferably in mathematical terms) on how shapes are represented and how sampling from a distribution on a Grassmann becomes generation of a shape.**\n\nIn shape theory, \"shape\" refers to the equivalence class $\\left[\\boldsymbol{Y}\\right]$ of all specific elements $\\boldsymbol{YQ}$ obtained through transformations $\\boldsymbol{Q}$ (such as affine transformations) of a single base shape $\\boldsymbol{Y}$ belonging to a particular class. As shown in Eq.(1), the equivalence class is equivalent to the linear subspace $\\operatorname{span}\\left(\\boldsymbol{Y}\\right)$, making it an element of the Grassmann manifold. According to [57], shape is defined as the shape space $S\\left[x_1,\\cdots,x_D\\right] = \\\\{ \\xi=\\left(\\xi^1,\\cdots,\\xi^D\\right), \\sum_{j=1}^{j=D}\\xi^jx_j=0, \\sum_{j=1}^{j=D}\\xi^j=0 \\\\},$ where $\\left(x_1,\\cdots,x_D\\right)$ are systems of point $x_j \\in \\mathbb{R}^k$, which is precisely $\\operatorname{span}\\left(\\boldsymbol{Y}\\right)$.\nGrCNF treats this equivalence class $\\left[\\boldsymbol{Y}\\right]$, i.e., \"shape,\" as data for learning and generation. Theoretically, sampling is done for obtaining the equivalence class $\\left[\\boldsymbol{Y}\\right]$, while on the computer, the base shape (representative $\\boldsymbol{Y}$) is obtained.\nTherefore, a noteworthy characteristic of the proposed method is that the \"shape\" itself is learned during the learning and generation processes. During generation, one can obtain any base shape $\\boldsymbol{Y}$ using a trained model on the computer."}, {"review_id": "TkeLXbqhp1", "reviewer": "Reviewer_ioxc", "summary": "The paper presents a novel concept: Continuous Normalizing Flows on a Grassmann Manifold (GrCNF), a model that presents a more efficient and accurate way to capture and represent the geometric features of data. The unique proposition of GrCNF is its application of the Grassmann Manifold to grasp data equivariance, a key aspect that delineates the transformation response of data. GrCNF's efficacy is demonstrated via several experiments. The results affirm its capability to accurately generate data that depict complex geometric shapes, and outperform current Normalizing Flow models in metrics like log-likelihood and ELBO for diverse test datasets.", "questions": " In the artificial texture experiments, the antipodal points were generated using the trained GrCNF. While this approach demonstrates the learning capabilities of the model, it does not guarantee the accurate representation of all antipodal points in different contexts or datasets.", "limitations": "1. The proposed GrCNF has been tested on a limited variety of data types. Further exploration is required to understand its potential application and effectiveness on different data types, such as image, audio, and textual data.\n2. In the experiments, GrCNF is mainly compared with existing normalizing flows methods. The inclusion of other machine learning models, such as generative adversarial networks (GANs) or autoencoders, could have provided a broader perspective on GrCNF's performance.\n", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "+ GrCNF is a novel model to bring continuous normalizing flows (CNF) into the field of Grassmann manifolds, expanding its applications to data manifolds with geometric structures.\n+ GrCNF showed superior performance when compared to existing Normalizing Flow models in experiments with different datasets. It not only generated high-quality samples but also effectively learned the distribution on Gr(k, D).\n+ GrCNF is able to handle subspace data, making it a versatile tool for handling various types of complex data.\n", "weaknesses": "1. The use of the Kronecker product at each ODE step is a demanding process, which suggests potential improvements in computational efficiency could be beneficial. This exploration could lead to exciting developments in scalability and applicability.\n2. The study primarily focused on lower-dimensional data, and an examination of the GrCNF model's performance with higher-dimensional data could provide additional valuable insights.  \n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the detailed comments and positive feedback indicating that the presented framework is non-trivial and novel!\nWe have prepared detailed responses to your questions below. \nWe sincerely hope that these responses will address and alleviate any concerns.\n\n****\n> **Question:1. In the artificial texture experiments, the antipodal points were generated using the trained GrCNF. While this approach demonstrates the learning capabilities of the model, it does not guarantee the accurate representation of all antipodal points in different contexts or datasets.**\n\nOur model can guarantee this property by (9) in the Appendix. \nThe (9) represents the invariance of the model, ensuring that data transformed by the orthogonal group $\\mathcal{O}\\left(k\\right)$, i.e., any reflection of the data, is learned without the need for data augmentation.\n\n****\n> **Limitation:1. The proposed GrCNF has been tested on a limited variety of data types. Further exploration is required to understand its potential application and effectiveness on different data types, such as image, audio, and textual data.**\n\nand\n\n> **Limitation:2. In the experiments, GrCNF is mainly compared with existing normalizing flows methods. The inclusion of other machine learning models, such as generative adversarial networks (GANs) or autoencoders, could have provided a broader perspective on GrCNF's performance.**\n\nThank you for your valuable feedback.\nWe wholeheartedly agree with your assertion that evaluation is indeed necessary.\nAs a response to your feedback, we are currently planning to evaluate our method using the ShapeNet[82]. \nHowever, due to the short two-week timeframe, conducting a comprehensive evaluation using such datasets is challenging because of time constraints. \nThus, the results of these experiments will be included in the Camera-ready version of the paper.\nFinally, we would like to emphasize that this paper primarily focuses on establishing fundamental theories. \nWithin our algorithms, we have introduced theoretical guarantees, laying the foundation for the next step, which involves conducting large-scale experiments.\n\n[82] Angel X. et al., ShapeNet: An Information-Rich 3D Model Repository. arXiv, 2015."}, {"review_id": "JaY0UUpNTv", "reviewer": "Reviewer_SWNf", "summary": "This paper extends the previous Riemannian CNF to subspaces, i.e., Grassmann Manifold Gr(k,D), and tackle stable shape generation with nice transformation-invariant property. They derive the first and full mathematical formulation for learning invariant densities and flow on a Grassmann manifold. They successfully bring the complete model to life by using Stiefel Manifold to help construct the vector field effectively, with adapted ODE solver on Gr(k, D). They have also shown the superior performance of the proposed method on both toy and well-adopted public benchmarks compared to several existing seminar works. ", "questions": "1. I am also wondering about the computation overhead, besides the expensive ODE solver, how many steps does it take to generate the final sample from a Gaussian density?\n\n2. When comparing with the baselines, the proposed method will instead take the orthonormalized data, I am wondering whether this is a fair comparison setting. ", "limitations": "NA", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "[**originality**] the proposed work is the first one to explore continuous normalizing flow on subspace on the Grassmann Manifold. They not only creatively combine the success of quotient manifold for shape analysis, and the recent Riemannian CNF, but also contributed an elegant way of constructing the vector field through the connection to Stiefel manifold with horizontal lift. \n\n[**quality & clarity**] The writing quality is over the bar, with clean explanation on the motivation, intuition, and design. As a person who is not that familiar with manifold theory, I found that the mathematical derivation is straight-forward to understand. They have presented complete and detailed propositions and proofs throughout the main paper and supp. \n\n[**significance**]  The proposed framework is a pioneer work on exploring continuous normalizing flow on subspaces with nice equivariance-preserving property. Compared to existing baselines, this method is achieving SOTA performance on simple shape generation. It has great potential to benefit 3D molecules design, other high-dimensional geometry data generation including common objects. ", "weaknesses": "1. From Figure 2, all that has been showing is on the Stiefel Manifold, without showing how it is connected to the target Grassmann manifold, which is confusing and not informative. Without this connection, I am not clear how the nice O(k)-invariant output v_out is achieved. The author might want to update the figure, or provide a better clarification;\n\n2. As the author also mentions, it will make the proposed work more significant if showing experimental results on real-world object shapes.   More importantly, I am interested in knowing the major challenges to use the developed framework to handle such object shape data;\n\n3. On the public benchmarks, the quantitative numbers are better than the baselines, it will make the comparison more complete if we can further show the visualizations of the generated samples;", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the detailed comments and positive feedback indicating that the presented framework is non-trivial and novel!\nWe have prepared detailed responses to your questions below. \nWe sincerely hope that these responses will address and alleviate any concerns.\n\n****\n> **Weakness:1. From Figure 2, all that has been showing is on the Stiefel Manifold, without showing how it is connected to the target Grassmann manifold, which is confusing and not informative. Without this connection, I am not clear how the nice O(k)-invariant output v_out is achieved. The author might want to update the figure, or provide a better clarification.**\n\n**We have uploaded a PDF containing this figure in the Author Rebuttal (global response).\nPlease refer to Figure 1 in the Author Rebuttal's PDF.**\n\n****\n> **Weakness:2. As the author also mentions, it will make the proposed work more significant if showing experimental results on real-world object shapes. More importantly, I am interested in knowing the major challenges to use the developed framework to handle such object shape data.**\n\nThank you for your valuable feedback. We wholeheartedly agree with your assertion that evaluation is indeed necessary. \nAs a response to your feedback, we are currently planning to evaluate our method using the ShapeNet[82]. \nHowever, due to the short two-week timeframe, conducting a comprehensive evaluation using such datasets is challenging because of time constraints. \nThus, the results of these experiments will be included in the Camera-ready version of the paper. \nFinally, we would like to emphasize that this paper primarily focuses on establishing fundamental theories. \nWithin our algorithms, we have introduced theoretical guarantees, laying the foundation for the next step, which involves conducting large-scale experiments.\n\n[82] Angel X. et al., ShapeNet: An Information-Rich 3D Model Repository. arXiv, 2015.\n\n****\n> **Weakness:3. On the public benchmarks, the quantitative numbers are better than the baselines, it will make the comparison more complete if we can further show the visualizations of the generated samples.**\n\nWe have uploaded a PDF containing this figure in the Author Rebuttal (global response).\n**Please refer to Figure 3 in the Author Rebuttal's PDF.**\n\n****\n> **Question:1. I am also wondering about the computation overhead, besides the expensive ODE solver, how many steps does it take to generate the final sample from a Gaussian density?**\n\nIn our experiments, the number of steps for the trained models is as follows: 150 steps for Experiment 6.1, 160 steps for Experiment 6.2, and 220 steps for Experiment 6.3.\n\n****\n> **Question:2. When comparing with the baselines, the proposed method will instead take the orthonormalized data, I am wondering whether this is a fair comparison setting.**\n\nWe believe that we have constructed the experiments to be fair and equitable. \nIn Experiment 6.2, all the comparative methods are evaluated using data transformed into orthonormal basis matrices. \nMoreover, in Experiment 6.3, by incorporating scale information into the ELBO computation in GrCNF, we demonstrated that even the ELBO surpasses traditional methods. \nThese measures ensure that the comparison conditions are properly aligned for conducting the experiments."}, {"review_id": "QY55OjyE9D", "reviewer": "Reviewer_5FfM", "summary": "This paper presents the application of continuous normalizing flow (CNF) to the Grassmann manifold for shape generation. The authors establish the theoretical foundations for learning distributions on the Grassmann manifold using CNF, enabling the generation of stable shapes. The proposed approach offers a promising generative model capable of generating subspace data. ", "questions": "There should indeed be a comma rather than a period before finishing the equation, as seen in Eq. 8 on lines 232-233.\n\nRegarding the performances for different D and k values on DWR, LJ13, and QM9, it would be valuable for the paper to provide an analysis and evaluation of the proposed method's performance in these scenarios. This would enhance the understanding of the method's effectiveness across different datasets and settings.\n\nThe reason why the proposed GrCNF outperforms the counterpart methods in Table 1 should be clarified in the paper. It would be beneficial for the authors to provide an explanation or analysis that highlights the specific advantages or characteristics of the proposed method that contribute to its superior performance.\n\nRegarding the stability of shape generation in the proposed method compared to other methods, the paper should provide a clearer explanation to address this question. It would be helpful for the authors to discuss the specific mechanisms or techniques employed in the proposed method that lead to more stable shape generation results.\n\nThe feasibility and necessity of bringing invertibility to normalizing flows could be an interesting aspect to explore. It would be valuable for the authors to discuss the potential benefits and drawbacks of incorporating invertibility into the proposed method, considering the existing literature and the specific requirements and objectives of the task at hand.", "limitations": "The paper does not provide a thorough comparison with existing methods, such as [51], to demonstrate the superiority of the proposed approach. A comparative analysis would have helped assess the advancements of the proposed method over previous work.\n\nThe paper lacks empirical studies on real-world, high-dimensional data, such as point clouds. This limitation restricts the generalizability and applicability of the proposed approach to practical scenarios, as the effectiveness of the method has only been demonstrated on low-dimensional data. \n\nIt is not adequately explained how the proposed approach technically differs from the Stiefel-based ODE [7]. A clearer illustration of the transfer from Stiefel-based ODE to Grassmann-based ODE would have provided a better understanding of the method's technical contributions. \n\nWhile the paper claims that the proposed GrCNF outperforms counterpart methods, it does not provide a comprehensive explanation for this superiority. It would be beneficial to discuss and analyze the specific factors or characteristics of the proposed method that contribute to its improved performance. \n", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The study represents the first exploration of applying a continuous normalizing flow (CNF) to the Grassmann manifold for shape generation.  \n\nThe experiments conducted demonstrate the effectiveness of the proposed approach in generating promising shapes, particularly subspace data with low dimensions. ", "weaknesses": "I have concerns regarding the significant contribution of the paper to the domain, given the existence of [51]. To address these concerns, it would be beneficial for the authors to include a comparative analysis with [51] in the experiments to demonstrate any potential superiority of the proposed method. \n  \nFurthermore, it is not clear how the proposed approach technically differs from the Stiefel-based ODE [7]. To provide a clearer understanding, it would be helpful if the authors could provide an intuitive illustration of the transfer from the Stiefel-based ODE to the Grassmann-based ODE. \n  \nAdditionally, the paper's validation is limited to low-dimensional data, and there is a lack of empirical study on real-world, high-dimensional data such as point clouds. As a result, the current validation may not be sufficiently convincing to establish the effectiveness of the proposed approach in practical scenarios. Conducting experiments on high-dimensional data, such as point clouds, would enhance the credibility and generalizability of the proposed method. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed comments and valuable feedback! We have prepared detailed responses to your questions below. We sincerely hope that these responses will address and alleviate any concerns.\n\n****\n> **Weakness:1. I have concerns regarding the significant contribution of the paper to the domain, given the existence of [51]. ...**\n\nTypically, density estimation is performed in Euclidean spaces (e.g., GAN, VAE). \nGeometric deep learning has recently aimed to extend these traditional Euclidean methods to the realm of Riemannian manifolds, with a representative approach found in [51]. \nHowever, [51] only presents a highly general framework that includes Grassmann manifolds but does not address any practical methodologies for computations on the Grassmann manifold. \nIn particular, it does not maintain symmetry. In fact, constructing models on the Grassmann manifold requires extensive discussions on how to represent subspace data on computers and ensure model symmetry. \nTo tackle these issues, we have provided rigorous proofs in Proposition 1 to 3.\n\n****\n> **Weakness:2. Furthermore, it is not clear how the proposed approach technically differs from the Stiefel-based ODE [7]. To provide a clearer understanding, it would be helpful if the authors could provide an intuitive illustration of the transfer from the Stiefel-based ODE to the Grassmann-based ODE.**\n\n**We have uploaded a PDF containing this figure in the Author Rebuttal (global response). \nPlease refer to Figure 2 in the Author Rebuttal's PDF.**\nThe most significant difference lies in the constraint of the vector field to the horizontal space, causing the ODE solver to operate while transitioning between the horizontal space and the Stiefel manifold at each time step. \nProposition 1 to 3 ensures that the updates at each time step are simultaneously updated on the Grassmann manifold.\n\n****\n> **Weakness:3. Additionally, the paper's validation is limited to low-dimensional data, and there is a lack of empirical study on real-world, high-dimensional data such as point clouds. As a result, the current validation may not be sufficiently convincing to establish the effectiveness of the proposed approach in practical scenarios...**\n\nThank you for your valuable feedback. We wholeheartedly agree with your assertion that evaluation is indeed necessary. \nAs a response to your feedback, we are currently planning to evaluate our method using the ShapeNet[82].\nHowever, due to the short two-week timeframe, conducting a comprehensive evaluation using such datasets is challenging because of time constraints. \nThus, the results of these experiments will be included in the Camera-ready version of the paper. \nFinally, we would like to emphasize that this paper primarily focuses on establishing fundamental theories. \nWithin our algorithms, we have introduced theoretical guarantees, laying the foundation for the next step, which involves conducting large-scale experiments.\n\n[82] Angel X. et al., ShapeNet: An Information-Rich 3D Model Repository. arXiv, 2015.\n\n****\n> **Question:1. There should indeed be a comma rather than a period before finishing the equation, as seen in Eq. 8 on lines 232-233.**\n\nThank you for your feedback. We have made the necessary updates. You will be able to see it in the Camera-ready version.\n\n****\n> **Question:2. Regarding the performances for different D and k values on DWR, LJ13, and QM9, it would be valuable for the paper to provide an analysis and evaluation of the proposed method's performance in these scenarios. ...**\n\nThank you for your valuable input. \nWe believe that the experiments conducted with the DWR, LJ13, and QM9 datasets, each having different $k$ and $D$ values, have confirmed the fundamental characteristics of our method. \nThis is supported by the fact that our approach consistently outperforms the conventional methods across all these datasets with varying $k$ and $D$ values. \nParticularly noteworthy is the significantly improved performance compared to GNF, which requires data augmentation.\nWe acknowledge your concerns regarding the behavior of GrCNF with $k > 3$, and we fully agree that a detailed investigation is necessary. \nThe datasets used in our current experiments are limited to $k \\leq 3$, which means addressing the challenges you mentioned requires the selection of new datasets. \nAs one of our future directions, we plan to include such evaluations in our analysis. \nWe appreciate your feedback, and we are committed to enhancing the comprehensiveness of our research.\n\n****\n> **Question:3. The reason why the proposed GrCNF outperforms the counterpart methods in Table 1 should be clarified in the paper.**\n\nIn shape theory, a shape refers to the equivalence class of all configurations obtained by specific transformations (e.g., affine transformations) applied to a single base shape. \nGrCNF treats these equivalence classes as data for learning and generation. \nAffine transformations involve rotations, scalings, reflections, and shears, and learning in the space affected by these actions is very challenging from a combinatorial perspective. \nOn the other hand, GrCNF achieves learning that entirely ignores the actions of affine transformations, thanks to the invariance provided by orthogonal transformations to orthonormal basis matrices. \nThis capability allows GrCNF to overcome such difficulties, leading us to believe that in evaluations, GrCNF can exhibit higher performance compared to other traditional methods. \n\n****\n> **Question:4. Regarding the stability of shape generation in the proposed method compared to other methods, the paper should provide a clearer explanation to address this question. ...**\n\nSimilar to the previous question, the proposed method ensures stability by completely eliminating the possibility of unlearned orthogonal-transformed data in the trained model."}, {"review_id": "q2ynhaxBL5", "reviewer": "Reviewer_D7Hq", "summary": "This paper proposes a normalizing flow approach applicable to generate data on Grassmann manifold. The proposed method utilize the fact that Grassmann manifold is a quotient space of Stiefel manifold on which numerical calculation is feasible. Specifically, the method first constructs the flow from vector field and then transform a probabilistic density using the flow. The vector field is learned through a neural network. A prior probabilistic density is defined as a matrix-variate Gaussian distribution on Grassmann manifold. The experiments are conducted on three different types of datasets among which the first one is synthetic and the last one is molecular point clouds. The experimental results show that the proposed GrCNF can generate the distribution on Grassmann manifold. ", "questions": "In general:\n\n1. What's the meaning of \"stable shape\" in the title? Why does the generation process is stable?\n\nMethod parts:\n\n2. In Figure 1, what's the relation between T^v_YSt(k, D) and T^h_YSt(k, D) and why do we use T^h_Y, not use T^v_Y?\n\n3. In Line 106, it mentions that \"we use the representative Y ...\", so how to choose the representative Y?\n\n4. In Proposition 1, what's the relation between a vector field and a flow? I.e., how to compute a flow F_{X, T} given a vector field X?\n\n5. Why do we need retraction in Eq.(4) in Proposition 3?\n\n6. In Section 5.3 and Figure 2. does the output (through a neural network) is ensured in the tangent space of T^h_YSt(k, D) theoretically?\n\n7. In Section 5.4, what's the goal of the ODE solver? Could you use one or two sentences to summarize it.\n\nExperiment parts:\n\n8. In the experiment in Section 6.1, I'm not sure why the spiral (and swissroll and circle ...) are 1-dimensional data in R^3, as they contain x and y two coordinates.\n\n9. In the experiment in Section 6.2 and 6.3, is it possible to visualize the generated data, similar to Figure 3.\n\nImplementation parts:\n\n10. As the method is complicated in theory, will the code be released?\n\n\n\n", "limitations": "The authors mentions the two limitations of the proposed method in Line 324 - 328.", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "This paper aims to resolve the problem of shape generation when the high-dimensional shapes intrinsically embed in a low-dimensional manifold, e.g., Grassmann manifold, using normalizing flow (NF). Extending NF from common Euclidean space to non-linear manifold is novel and meaningful, as the manifold hypothesis assumes that most high-dimensional data usually embeds in a low-dimensional manifold. The theoretical part looks sound. Experiments under three different settings verify the effectiveness of the proposed GrCNF to learn distributions on Grassmann manifold.", "weaknesses": "The main weakness (also my concern) is, as also mentioned by the authors in Line 325 - 328, whether the proposed GrCNF is applicable to  high-dimensional data? E.g., point clouds containing thousands of points? If yes, it will make the paper very strong in both theoretical and application aspects. (I also want to know what real-world high-dimensional data embeds in Grassmann manifold?) If not applicable to high-dimensional data, it will weaken the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the detailed comments and positive feedback indicating that the presented framework is non-trivial and novel!\nWe have prepared detailed responses to your questions below. \nWe sincerely hope that these responses will address and alleviate any concerns.\n\n****\n> **Question:1**\n\nWhen we refer to \"Stable,\" we mean that our approach ensures the complete elimination of the possibility of unlearned orthogonal-transformed data in the trained model. \nAs a result, it guarantees the generation of data after any orthogonal transformations during the data generation process. \nIn this context, stability implies the robustness of the model to handle various orthogonal transformations and produce accurate data generation results.\n\n****\n> **Question:2**\n\nAs stated in (94) of the Appendix, the vertical and horizontal spaces are orthogonal to each other. \nAdditionally, the tangent space on the Stiefel manifold, as shown in (96), is a direct sum of the vertical and horizontal spaces. \nThe vertical space represents the components of $\\mathcal{O}\\left(k\\right)$, while the horizontal space, orthogonal to the vertical space, realizes the invariance with respect to $\\mathcal{O}\\left(k\\right)$ (as shown in (9) of the Appendix). \nUtilizing only the horizontal space is equivalent to manipulating points on the Grassmann manifold. \nFor detailed mathematical discussions on this matter, please refer to Appendix D.4.3. \n**We have uploaded a PDF containing this figure in the Author Rebuttal (global response).\nPlease refer to Figure 1&2 in the Author Rebuttal's PDF.**\n\n****\n> **Question:3**\n\nWe have the option to set the data within the dataset as representatives. \nIn the ODE, each step is updated from the initial value $\\boldsymbol{Y}$, and by applying the horizontal retraction at each step of the algorithm, we always obtain specific representative $\\boldsymbol{Y}$ on the computer alongside their corresponding theoretical equivalence class $\\left[\\boldsymbol{Y}\\right]$. \nTheoretically, any element of $\\left[\\boldsymbol{Y}\\right]$ can be chosen as a representative.\n\n****\n> **Question:4**\n\nA flow refers to the continuous movement of points over time. \nThe direction of motion at each point is determined by the vector field X assigned to that point. \nTherefore, a flow is essentially a mapping determined by the vector field. \nIn our proposed method, the flow is generated using ODE solver with numerous time steps. \nIn other words, the flow can be viewed as the ODE solver itself, which utilizes the vector field to solve the differential equation.\n\n****\n> **Question:5**\n\nThe reason for requiring retraction in (4) is that when sampling from this distribution on a computer, it involves traversing the horizontal space. \nThe desired sample point is the representative $\\boldsymbol{Y}$ on the Grassmann manifold represented by the point $\\left[\\boldsymbol{Y}\\right]$. \nTherefore, a mapping (retraction) from the horizontal space to the Stiefel manifold, where the representative exists, is necessary. \nThe primary reason for passing through the horizontal space lies in the theoretical concept of horizontal lift, which allows us to represent the Grassmann manifold. \nHowever, intuitively, using the horizontal space simplifies the process of converting the matrix-variate gaussian distribution on the easier-to-sample Euclidean space into the representative $\\boldsymbol{Y}$. (Please refer to Algorithm 1 for more details).\nFor a more theoretical derivation, please see the proof of Proposition 3 in Appendix B.3.\nAdditionally, We have uploaded a PDF containing this figure in the Author Rebuttal (global response).\n**Please refer to Figure 2 in the Author Rebuttal's PDF.**\n\n****\n> **Question:6**\n\nThe proposed ODE solver uses retraction to map the variables to the horizontal space after each ODE step. \nThis ensures that the output at each ODE step exists in the horizontal space. \nWe have uploaded a PDF containing figures related to your question in the global response. \n**Please refer to Figure 2 in the Author Rebuttal's PDF.**\n\n****\n> **Question:7**\n\nThe objective of our ODE solver is to obtain the value of $\\gamma$ as given in (7). \nTo achieve this, the ODE in (7) is solved using our ODE solver, and the resulting $\\epsilon$ is mapped to obtain $\\gamma$ through retraction.\n\n****\n> **Question:8**\n\nThe spiral shown in Figure 3 itself is not one-dimensional data. Instead, each individual white point within the spiral represents one-dimensional data. \nIn a three-dimensional space, the one-dimensional data is visualized as a thin cyan line connecting the blue point and the origin, as depicted by the leftmost sphere in Figure 4. \nSince one-dimensional data lies on a line, it includes its antipodal points (i.e., points obtained through orthogonal transformations). \nThe advantage of learning on the Grassmann manifold is that such antipodal points are learned without the need for data augmentation. \nFor visualization purposes, drawing numerous lines passing through the origin might clutter the image, so in Figure 3, we represent the one-dimensional data as white points.\n\n****\n> **Question:9**\n\nWe have uploaded a PDF containing this figure in the Author Rebuttal (global response). \n**Please refer to Figure 3 in the Author Rebuttal's PDF.**\n\n****\n> **Question:10**\n\nWe apologize for any inconvenience, but according to our organization's policies, we do not have plans to release the code to the public.\nInstead, we have provided specific components that will allow you to fully reproduce our results from our paper.\nThese components include a concrete sampling algorithm from the prior distribution (Algorithm 1), detailed network architectures (Table 3), ODE solver (7), specific loss functions ((56), (57), and original CNF implementations), comprehensive experimental conditions (Appendix C.3), and details of the data generation method (Listing 1). \nWith access to these components, we believe you will be able to completely replicate our paper's findings."}]

Author rebuttal:
We have included a PDF with the figures as part of our response. Kindly review the PDF to access the pertinent information and visual illustrations.
Please note that the figures in the uploaded PDF are presented in a smaller size due to limitations in available space.
Rest assured, in the Camera-ready version, we will furnish larger and more visually distinct figures for your convenience.
Should you have any additional inquiries, please don't hesitate to contact us. We value your engagement and eagerly anticipate addressing any inquiries you might have.

Meta-review: This paper generalizes Continuous Normalizing Flows (CNFs) to work on Grassmann Manifolds, by representing flows on these manifolds using tangent vector fields and constructing a prior density with an efficient sampling. The Grassmann data is used to represent and learn distributions on shapes, where a “shape” $Y$ with equivalent representations $YQ$ can be represented as an element $span(Y)$ in the Grassmann. The paper presents several contributions such as modeling of shapes with Grassmann manifold, adapting CNFs to this manifold, and presenting applications to 3D problems with superior performance. Some weaknesses of this paper include experiments that are mostly low dimensional and not "real world", computational demanding divergence computation (inherited/similar to other CNFs training). Please include discussion/relation to Stiefel based CNFs in camera ready as well as incorporating other requested/promised changes during rebuttal period.
Decision: Accept (poster)