Siegel Neural Networks
Authors: Xuan Son Nguyen
Aymeric Histace
Nistor Grozavu
Venue: NeurIPS 2025 poster ()
Keywords: Neural networks
Siegel spaces
Symmetric spaces
Riemannian manifolds
Primary area: deep_learning

Abstract: Riemannian symmetric spaces (RSS) such as hyperbolic spaces and symmetric positive definite (SPD) manifolds have become popular spaces for representation learning. In this paper, we propose a novel approach for building discriminative neural networks on Siegel spaces, a family of RSS that is largely unexplored in machine learning tasks. For classification applications, one focus of recent works is the construction of multiclass logistic regression (MLR) and fully-connected (FC) layers for hyperbolic and SPD neural networks. Here we show how to build such layers for Siegel neural networks. Our approach relies on the quotient structure of those spaces and the notation of vector-valued distance on
RSS. We demonstrate the relevance of our approach on two applications, i.e., radar signal classification and node classification. Our results successfully demonstrate state-of-the-art performance across all datasets.

Siegel Neural Networks
Nistor Grozavu ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS aegam-son.nguyeer intrenaammen aymeric.. hwst.c.....teeeeeeeeeeeeeeeeeeee histaacee@ensea...f. nistor. gruzavdcyuuuu
Xuan Son Nguyen Aymeric Histace
Abstract
Riemannian symmetric spaces (RSS) such hyperbolic spaces and symmetric positive definite (SPD) manifolds have become popular spaces for representation learning. In this paper, propose novel approach for building discriminative neural networks on Siegel spaces, family of RSS that is largely unexplored in machine learning tasks. For classification applications, one focus of recent works is the construction of multiclass logistic regression (MLR) and fully-connected (FC) layers for hyperbolic and SPD neural networks. Here we show how build such layers for Siegel neural networks. Our approach relies on the quotient structure of those spaces and the notation of vector-valued distance RSS. We demonstrate the relevance of our approach on two applications, i.e., radar clutter classification and node classification. Our results successfully demonstrate state-of- the-art performance across all datasets.
1
Introduction
Deep neural networks are generally built upon the assumption that the data or features at hand exhibit Euclidean latent structure. Unfortunately, this assumption does not hold in many applications [1, 16] where the data or features lie on a multidimensional curved surface which is locally Euclidean. For such applications, those models often produce unsatisfactory results because their building blocks based on Euclidean geometry break the geometric stability principle that plays a crucial role in geometric deep learning architectures [8]. To deal with this issue, many Riemannian neural networks have been developed for solving wide variety of machine learning problems [16, 22, 25, 29, 30, 32]. In this paper, we restrict our attention to discriminative neural networks with manifold-valued output.
Early works focus either on hyperbolic spaces [16, 38] or on matrix manifolds [20, 21, 22]. In an attempt develop unified framework for more general setting, the authors of [31, 33, 34] leverage the gyro-structure of certain Riemannian manifolds. However, in the general case, their methods cannot provide an explicit form for the point-to-hhe -onn-tto-yyeepllae distance which is at the heart of their proposed network building blocks [38] since the distance must be derived with respect to specific Riemannian metric. The work in [35] alleviates this issue by deriving closed form for the point-to-hyperplane distance associated with G-invariant Riemannian metrics on RSS. Although this work applicable to Siegel spaces, the construction of the MLR and FC layers [35] from the derived distance is heavily based on the maximal abelian subspaces of those spaces. This can affect the ability of the resulting networks learn rich representations and complex decision boundaries.
In this paper, we propose a novel approach for building neural networks on Siegel spaces. Those are among the most versatile RSS [27] and have many attractive theoretical properties. The two well- established models of Siegel spaces, i.e., the Siegel upper space and the Siegel disk [39] generalize the complex Poincaré upper plane and the complex Poincaré disk [18] spaces of symmetric complex matrices [36]. SPD manifolds associated with affine-invariant Riemannian metrics [37] also special cases of Siegel spaces associated with Siegel metrics [39]. Despite the potential of Siegel spaces in capturing rich geometrical structures, they are much less studied in the context
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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of deep learning compared to other RSS. Although few recent works [27, 41] use Siegel spaces as representation spaces, they only focus on learning and visualizing embeddings natural language processing and graph tasks. Therefore, an effective framework for building discriminative Siegel neural networks is still missing. summary, our contributions are the following:
We propose a novel formulation of MLR layers for Siegel neural networks based on a gyrovector approach which has proven effective in building hyperbolic and SPD neural networks [16, 31, 38). This formulation is then extended to the product space setting.
We show that the notion of vector-valued distance [24] which captures the complete G- congruence invariant of pairs of points on RSS enables another formulation of MLR layers for Siegel neural networks in a natural way. This formulation leads more compact MLR layers than those obtained by the first formulation.
We introduce two variants of FC layers for Siegel neural networks.
We build the first discriminative Siegel neural networks and evaluate them on the radar clutter classification and node classification tasks.
2 Mathematical Background
2.1 Siegel Spaces
The Siegel upper space Sinm defined as
SHm {x u u iv u E Symm, Sym
where Symm, and Sym denote the space of m real symmetric matrices and that of m m SPD matrices, respectively.
Another model for Siegel spaces is the Siegel disk defined by
SD, w Symm,c I' ww. H
where In the m x m identity matrix, Symm, and H denote the space of m x m complex symmetric matrices and that of m Hermitian positive definite (HPD) matrices, respectively, and H is the conjugate transpose w.
One can convert a point E SHm to SDm using the matrix Cayley transformation defined as (xx) (æ m + imm))
The inverse matrix Cayley transformation that converts point w E SID,, to Slilm is given by (w) i(Im + (mm w)-
In the following, we shall focus on the Siegel upper space model.
Quotient Structure Denote by
Spam {[o
a
b) d
ab"
= ba" NaaT,co cdd = de ad -bo" Im
the real symplectic group. This group acts transitively on SH,, by the action s[xr] = (ax + b) aa b)
E Spzm and E SHm. The stabilizer group of x = ilm SHim is the subgroup of symplectic orthogonal matrices SpO2m defined as:
d) where
c d
SpO = [[= a aa bT Im, Im,aTb E Symm Spzm 002m=
where Ozm is the group of orthogonal matrices. We thus have the identification SH,, Sp2m / Sp02mm The element in Spzn that transforms i!m to x = + in E Sli via the group action is given by the map d6-) in the following identification:
1 SlH.. - Spam / SpO 2m
un +
SpO = (((x) SpO
0
v

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Riemannian Metric The Riemannian metric (also referred to as Symplectic metric) for the Siegel upper space model given [39] by
- = 2 Tr
(1)
where x u + iu E m and Tr(-) is the matrix trace (see Appendix 3.2 for further discussions).
The associated Riemannian distance das (z. v) between two points 2, y E SHlm is given by
1
deca(( v) =
log²
where rj. I,..., , m are the eigenvalues of the cross-ratio R(x, y) defined
R(a, y) (xr y) y) f) y) y)
where x denotes the complex conjugate of a.
2.2 Riemannian Symmetric Spaces
This section briefly reviews key concepts from the theory of RSS for our work. We refer the interested reader to Appendix 3.1 for further discussions.
A symmetric space is a connected Riemannian manifold X with geodesic-reversing isometry at each point. In other words, for each point x e X there is an isometry ow of X such that 02 (x) and the differential of 0₂ at x is multiplication by -1 [7]. Siegel spaces belong to a family of RSS referred to as symmetric spaces of noncompact type or noncompact RSS. In the following, we refer to noncompact RSS as RSS or symmetric spaces. Let G be connected noncompact semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Then a symmetric space X consists of the left cosets
x= = G/K == (1 g g G).
The action of G on x G is defined as g|z] = g|hk] ghK for I = hK E X, g, E G. Let o be the origin K in X, then the map y gk + g/0l is diffeomorphism of G/K onto X.
Let d(.,-) be the distance induced by the Riemannian metric. A geodesic ray in X is a map 5 (c) X such that (((()),((())
1t 11,, VI, 2 0. A geodesic line in X is map 8: R - such that d((()),(())) = 11,, Vi, TR.
The geometry of X can be studied through the geometry its maximal flats [2, 7, 18, 24]. A subspace F C is called a flat of dimension (or k-flat) if it is isometric to IR* The subspace F is called a maximal flat if it is not contained in any flat of bigger dimension. Since all maximal flats in X are isometric [18], they can be simultaneously identified with a model (maximal) flat F Food-
Flats are decomposed into Weyl chambers. A Weyl chamber in maximal flat F with tip at x E F is a connected component of the set of points E FV \ {x² such that the geodesic line through x and x' is contained in a unique maximal flat [7]. Since G acts transitively on the set of Weyl chambers in X [18], they can be simultaneously identified with Weyl chamber A. The subgroup of isometries of F which are induced by elements of G is isomorphic to a semidirect product R x W. W is called the Weyl group of G and X.
Any symmetric space X is associated with a boundary at infinity 0X constructed as the set of equivalence classes of geodesic rays in X. Two rays are considered equivalent if their images are bounded distance apart [7]. The equivalence class of geodesic ray 5 is denoted by 8(2o).
3 Proposed Approach
Our proposed hhint-to-hyperplare distances based on the quotient structure of Siegel spaces and the vector-valued distance are presented in Sections 3.1 and 3.2, respectively. In Section 3.3, we present our MLR models and introduce two variants of FC layers for Siegel neural networks. In our work, we focus on Siegel spaces but many of our results can also be stated for other RSS. To simplify the notation, we use the letters X, G. and K (see Section 2.2) to denote the spaces associated with Siegel spaces (see Section 2. unless otherwise stated.
3

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3.1 Point-to-hyperplane Distances Based on the Quotient Structure of Siegel Spaces
3.1.1 Hyperplanes
We start with a formulation of Euclidean hyperplanes given as
Hggb {ar E II (a, a) - b= 0},
where RM \ (0), b R. and (., is the Euclidean inner product. Hyperplane Ha, can be reformulated [16] as
Ha, b = f Rm (-p+
where p R" and (p.a) =
To generalize Euclidean hyperplanes to our setting, we follow the approach in [33, 34] which relies on binary operation, an inverse operation, and an inner product defined on the target space. Let x gK,y h. E X where g. E G. the case of mH 9 ((x). h Φ(y) where v2 uv 17 u iv E SIH
s(u iv)
Definition 3.1 ([35)). The binary operation @ and inverse operation è defined
# ggh,, Gx K.
propose the following inner product. Definition 3.2. The inner product (:,)) on X is defined as
(x.y)s (log(gg"), log(hh")))
where log(-) denotes the matrix logarithm.
Our proposed inner product is motivated by Proposition 3.3. (see Appendix 4.1 for its proof). Proposition 3.3. The inner product (:, -)s agrees with the Riemannian distance, i.e.,
e x @ ylls xx dga(x, y),
where x. y E X, and the norm ||s induced by the inner product (s Furthermore, the inner product (:, 1s is invariant under the action of K, i.e., for any k E K,
(x, y)s (le kly)ss
Note that both properties in Proposition 3.3 are satisfied by the inner products in [33, 35] and the second property is also satisfied by the one in [19]. Note also that these properties hold for the more general case in which G is the general linear group or its subgroup. and K is the group of orthogonal matrices or its subgroup (see Appendix 4.1). We are now ready to define hyperplanes.
Definition 3.4. Let P E X. Then hyperplanes on X are defined as
Ha.p fa E (Op @ x. a)s = 0}-
Segments of the form 3p can regarded as Siegel analogs of Euclidean lines. Thus, Ha.p has a similar interpretation as a Euclidean hyperplane, i.c., the former contains fixed point X and any point x X such that the segment Gp is orthogonal fixed direction a. Therefore. hyperplanes as given in Definition 3.4 are natural extensions of Euclidean hyperplanes.
3.1.2 Point-to-hyperplane Distance
The distance d(,rr, Ha.p) between point X and hyperplane Ha.p given in Definition 3.4 can be formulated [33] as
d(z, Ho.p) sin( (an(<xpq)d(a p)),
where Lapg is the gyroangle [33, 43] (see Appendix 3.4) between Op @ x and ep @ g, and q computed as
(Ep (Ep 24, s/s
q = arg max oE Ha.
le ( glsl|
e Ppp arls
By convention, sin(/rpg) for any 2, E Ha,p- Theorem 3.5 gives closed form for the point-to-hyperplant distance Siegel spaces (see Appendix 4.2 for its proof).

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Theorem 3,5. Let L, a, X and let Ha.p be hyperplane as given in Definition 3.4. Then
d(z, Ha.p) [ ((())(()))) (ogod(()))))
[og(
where ||
denotes the Euclidean norm, and the map Φ(-) is given in Section 2.1.
3.1.3 Product Spaces
We now extend the above method to the product space setting. Let X be defined as the Cartesian product X X1 X ... XL, where X; Gz/ j.j L..... are RSS, Gj is a connected noncompact semisimple Lie group with finite center, K, is a maximal compact subgroup of Gj- Here we focus on the Cartesian product of SPD and Siegel spaces. Each point E X can be described through its coordinates = II.). 2j E Xj.J 1,.................... L. In this setting, one has simple decompositions of the tangent space, the exponential map, and the squared Riemannian distance [15, 17, 42]. When X, is an SPD space, G₃ is the general linear group and K2 is the group of orthogonal matrices (see Appendix 3.3). Thus one can define the binary operation, inverse operation, and inner product on X, in Definitions 3.1 and 3.2, and the results in Proposition 3.3 still hold. By abuse of notation, we shall use the same notations for those operations as Section 3.1.1.
Definition 3.6. Let x (#1...., IL), y (y1...., WL) &, E Xz.J I,...,L L. The binary operation @ and inverse operation @ on X are defined as
x + = (x) 2 .1........ 2g + UL), ex = (e
Definition 3.7. The inner product (, on X is defined as
L
(x, vis = Σ (1g, J=1
The following theorem (see Appendix 4.3 for its proof) extends Theorem 3.5 to the considered setting.
Theorem 3.8. Let Ha.p be a hyperplane as given in Definition 3.4, where a = (a1..... aL), = (p1..... (P),,...PP),.4 aj WyK,. Pj K Xj, Wy hy E Gj.J 1,.... L, and let (cI....,ID. E X where Ij = Kj E Xx:99 E GG. Then
d(z, Ha.p) = E)=n (log(h; 9,9 h "), log(w,
E log(w, )1?
3.2 Point-to-hyperplane Distances Based on Vector-Valued Distances
As shown in [38], the formulation of Euclidean hyperplanes in Eq. (2) has an over parameterization issue, i.e., increases the number of parameters from m + to 2m each class. Our formulation of hyperplanes in Section 3.1.1 (see Definition 3.4) follows that formulation and thus suffers from similar issue. In this section, we propose another method for constructing the point-to-hyperplane distance which results in more compact MLR layers for Siegel neural networks.
3.2.1 Hyperplanes
We start with a similar formulation of Euclidean hyperplanes in Eq. (2) but use a different parameteri- zation. Given p Rm and & E ÖR"" the Euclidean hyperplane H. parameterized by p and & can be defined [35] by
H. E (p 2, 0} = (xx Im (vec((, p), = =0}
where & is the equivalence class of the geodesic ray 5(t) vec(2, p) = - x denotes the translation carrying x to p.
I (0). and the function ||||
In symmetric space, a natural analog of the function vec(..)) is the vector-valued distance func- tion [23, 24]. Given two points x, y E X, one computes G-invariant distance by first transforming (via the G-action) z and y to x' and y on the model flat Fmod. respectively, and then identifying the
5

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H
9
P
Figure 1: The distance between point x E IR and a Euclidean hyperplane H
translation modulo the action of the Weyl group carrying x' to v/. Note that in RT the projections of and on maximal flat are precisely x and p, respectively. The domain of the resulting distance function, which is fundamental domain for the action of the Weyl group on the translations, can be canonically identified with the Weyl chamber A. The above observation motivates the following definition.
Definition 3.9. Let da(:;)) Xxx + A be the vector-valued distance function on X. Let p X, E 0X, and lel ag A be such that &, is the equivalent class of the geodesic ray 5(1) = kexp(lae) K k E K. Then hyperplane Hep is defined as
Hep = far E: =
-].
A hyperplane given in Definition 3.9 has clear interpretation, i.e., it contains fixed point p X and any point x E X such that the vector-value, distance between I and is orthogonal to a fixed direction ag- We note that the notion of vector-valued distance has been employed in [27, 28] for learning and visualizing embeddings in natural language processing and graph tasks. However, none of those works reveals the analogies discussed above for defining Siegel hyperplanes.
3.2.2 Point-to-hyperplane Distance
Let E Rd and & E ORR" The distance d(r, H between a point æ E R" and hyperplane H can be computed (see Fig. 1) as
d(z, H g.p) d(x, p) cos
where 3 is the angle between the segments [x, and [x. 4] and is the projection of x on HE By convention, d(. H = O, Vo E HE The above equation can be rewritten as
d(z, H&,)) = d(z, p) cos (p. E),
where (a(p, {) denotes the angle at x between the geodesic segment [x. p] and the geodesic ray which issues from and is in the class E. We generalize the above equation our setting.
Definition 3.10. Let X, 0X, and let He, be hyperplane in X. Then the (signed) distance d(z, Hp between a point a E X and hyperplane Hg.pp is defined as
d(z, He,p) d(x, p) cos (p.
Deriving closed form of the point-to-hypeee to-hyperplaae distance for applications from Definition 3.10 is trivial. However, one obtain upper bound of this distance which is given in Proposition 3.11 (see Appendix 4.4 for its proof).
Proposition 3.11. Let x. p E X, & E Ox, and let ag E be such that & is the equivalent class of the geodesic ray 5(t) & cxp(ta K, where lo K and exp(.) is the matrix exponential. Then
d(x, Hg.w) S (da(x, ag)
Note that the point-to-hyperppane distance in Section 3.1 well as those in [16, 33, 34] are obtained by solving an optimization problem in Fuclidean spaces. This different from our method this section which estimates upper bound of the point-to-hyperplace distance on the target spaces.

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3.3 Neural Networks on Siegel Spaces
In this section, we show how to construct MLR layers for Siegel neural networks using the tools introduced in Sections 3.1 and 3.2. We also propose two types of FC layers which are crucial building blocks in the context of deep neural networks.
3.3.1 MLR Layers
We follow the approach in [16, 26] for building Riemannian MLR. Given M classes, (Euclidean) MLR computes the probability of each of the output classes
exp(af x 10--by bT))
ply - j| = Σn= explax
x x b,),
by) -b)
where is input sample, by E R. a, E RT
1....,,, M, and exp(.) is the ordinary exponential function (by abuse of notation), As shown [26], Eq. (3) can be rewritten as pvy -|z> x exp(sign(aş x - b |(tz, -0;i=/,,;l;aa, H,,,
where d(r, H.,, bby ) the distance from point to hyperplane He,, (see Section 3.1.1). our case, a hyperplane can be parameterized by two elements in (see Definition 3.4), or by an element X and an element in A (see Definition 3.9). We replace the expression in the argument of the function exp(.) by the distances Theorems 3.5 and 3.8 as well as the upper bound of the boin-to-hyperplane distance in Proposition 3.11. The final formulations of our MLR layers given in Appendix 1.
3.3.2 FC Layers
The FC layer with group action (AFC) Let + E SH.. Then the element in Spam mapping im to + ib via the group action (see Section 2.1) is given by
b2 ab ab- b-1
Φ(a ib)
Given an input E SHm, the output of the AFC layer is obtained by taking the group action á(a + [z]. This leads us to the following construction.
Definition 3.12. Let u iv Sllmm be the input of the AFC layer. Then the output of the AFC layer is given by:
t= (b² ubà + aa + ib² vb².
where a E Symm and E Sym are the parameters of the layer.
We have that bå ubà always outputs points on SHm. The transformation performed by the AFC layer can be interpreted as translation the input by a + ib (see Section 3.1.1).
Symm and ubà E Sym by construction. Hence, the AFC layer
The FC layer for dimensionality reduction (DFC) Based on the definition of the AFC layer, another type of FC layers for Siegel neural networks can also be built using a method similar to [20]. Definition 3.13. Let Stm,mz be the space of m x m2 real matrices (m > m2) with mutually orthogonal columns of unit length (the compact Stiefel manifold), and let + iv E SHHm be the input of the DFC layer. Then the output of the DFC layer is given by:
= (6" ub + a) + ib" vb,
where a E Symm, and E Stm,mz are the parameters of the layer.
Our FC layers generalize some FC layers in previous works. Specifically, when 0 and a = 0, the imaginary part b² v of the output of the AFC layer corresponds to the transformation performed by the affine-invarians translation layer [33], and the imaginary part bTeb eb of the output of the DFC layer corresponds to the transformation performed by the well-known Bimap layer [20]. In [40], the authors also proposed FC layers for neural networks on RSS. However, these layers are different from our FC layers in some aspects. First, the former include activation functions which are not used in the latter. Second, the former do not output points on the considered spaces, as opposed to the latter which always output points these spaces.
7

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Dataset
Dataset
Dataset
Dataset
Method
(3, 600, 3600)
(4. 100, 2000)
(5, 80, 1600)
(6, 50, 500)
KNN [11]
76.22=0.0
93.00=00.0
7..75=0.0
73.20+0.00
SPDNer [20]
63.44:10.11
41.50:+0.12
45.88
66.80-10.04
SPDNerBN [9]
62.67++0.10
40.10-00008
45.75-+0.15
68.40++0.004
MI.R-AT [33]
65.61+0.115
47,40++0.1 1.12
46.12-0.17
67.60-+0.04
GyroSpd++ [34]
62.244+0.16
46,20+-0. 14
48.25+0.19 19
67.80+-0.08
Siege/Net-DRC-QML.. Symd x (Ours)
40.78++0.23
82.70+0.18
74.88=0.21
71.20==0.08
Siege/Net-AFC-QMLR Symt, x (Ours)
80.94+0.14
96.50=#0112
91.00=0.18
85.60=0.06
Table 1: Results (mean accuracy standard deviation) computed over runs for radar clutter classification. The tuple (m, M, below each dataset indicates the signal dimension m, the number of classes M, and the size of the dataset s.
4 Related Work
Existing MLR models on Riemannian manifolds are generally built on either SPD manifolds [13,33]
and their low-rank counterparts [34] or hyperbolic spaces [5, 16, 26, 38]. Many of them [16, 33, 34, 38] leverage the gyro-structures of the Poincaré ball and SPD manifolds. The work in [35] proposes MLR and FC layers for neural network on RSS which rely the construction of Busemann functions. The work in [40] analyzes some existing hyperbolic and SPD neural networks from the perspective of harmonic analysis on RSS. It mainly concerns with a constructive proof of the universal approximation property of finite neural networks on RSS. Our method in Section 3.1 is inspired by the works [16, 33, 34] and focuses on Siegel spaces. Our method in Sections 3.2 explores the connection between the point-to-hypens been investigated in previous works.
distance and the vector-valued distance which has not
Experiments
This section reports results of our experiments on the radar clutter classification and node classification tasks. For further details, please refer Appendix 1 which we present more experimental results on human action recognition and Riemannian generative modeling.
5.1 Radar Clutter Classification
Radar clutter classification aims at recognizing different types of radar clutter which is the information recorded by a radar related to seas, forests, fields, cities and other environmental elements surrounding the radar [10]. Due the scarcity of publicly available radar datasets for the task, experiments are performed using simulated radar signals' which are commonly assumed to be stationary centered
autoregressive (AR) Gaussian time series [3,4, 4, 6, 10]. The AR model is given by
4 Uy Σ Unnn- Up:
where 9 (q > 1) is the order of the AR model, un C is the vector of signals time n, Cj Cmxm" .j 1,..., are the prediction coefficients (AR parameters), and en E C. is the prediction error at time which is assumed be a multidimensional Gaussian random variable (detailed descriptions of the construction of our datasets are provided in Appendix 1.1). To compute an input data for networks from time series,
parameterize the time series as (Po, W1,-... wy-1)
SD,,, (see Appendix 1.1). We note that methods dealing with data that lie on these product spaces have already been studied in previous works [4, 10, 11]. These representation spaces are endowed with natural metric inspired by information geometry [4, 10]. We discard the imaginary part of the component Po and map it to an SPD matrix ýo (see Appendix 1.1). Fach component w; is converted to z; E SIH, using the inverse matrix Cayley transformation (see Section 2.1). The input data is thus represented by point (Po, 22....,Zgg11. E Sym, x
H. SI where Po E H. and Vi......ppe.
'nttps: :/ thub wwwp////gyyyooooooooo////////eeo data
8

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Method
Glass
Iris
Zoo
KNN[1I]
29.65110.0
31.6610.0
33.33.10.0
LogEig classifier [27]
41.34.14.22
51.04.3.53 34.33.3.46
SiegeJNcx-BFC-BMLR [35]
41.12=3.86
37.26+2.53
48. 12=00888
SiegelNet-AFC--M. VMLR (Our))
42.06
36.94:1..68
3.286
SiegelNet-AFC-QMLL Rgn (Ours)
45.79 4.66
38.20+33.03
38,20
53.37 4.23
Table 2: Results (mean accuracy + standard deviation) computed over 10 runs for node classification.
Method
Glass
Iris
Zoo
SiegelNet-BMLL [35]
40.55++3.50
36.94+2.09
46,43 3.64
SiegelNee--VMr VMLR (Ours)
41.78+4.111
36.89+13.73
50.38-13.47
SiegelNNe-QMM Rgg... (Ours)
42.61+3.226
37.52+2.54
52,00--440
Table 3: Comparison (mean accuracy + standard deviation) of MLR models on Siegel spaces.
Each of our networks consists an FC (AFC or DFC) layer and a MLR layer built on the distance in Theorem 3.8. The sizes of the parameter b in the DFC layer are set x 2, x 3, x 3, and
6 x 4 for the experiments on datasets 1, 2. 3, and 4, respectively. We compare our approach to the following methods: (1) k-Nearest Neighbors (kNN) based on the Kähler distance [11] which is among the very few works for supervised classification in the product space H x SIDY-1 and (2) state-of-the-art SPD neural networks [9, 20, 33, 34] which use the real parts of the covariance matrices estimated from the time series input data (the real parts mapped to SPD matrices as above). We use default settings for SPD models as in the original papers (see Appendix 1.1). Results in Tab. show that SiegelNet-AFC-QMLLR
Symt, sym xsa2. yields the best performance in terms of mean accuracy across all the datasets. It is able to improve upon kNN, the second best method, by a margin of 4.71 %, 3.5%, 14.25% %, and 12.39% on datasets 1, 2, 3, and 4, respectively. There are large gaps in the performance of our models, yet in most cases, our worst model still outperforms SPD models by large margins. The results of our networks and KNN demonstrate the representation power of Siegel spaces the considered application. This is also confirmed by our experiments (see Appendix 1. 1.1) in which the performance of SiegelNet-ACC-QMML Symtt xSEy, drops drastically
when the coordinates associated with the product space SIH m (i.e., 21,..., Zq-1) are removed from the input data.
5.2 Node Classification
We perform node classification experiments on Glass, Iris, and Zoo datasets from the UCI Machine Learning Repository [14]². Like [27], our main aim is to demonstrate the applicability of our approach on Siegel spaces, and we do not necessarily seek state-of-the-ard results for the target task.
To create input data which are graph node embeddings on Siegel spaces, we optimize distance-based loss function [17, 27]. Given the distances (2)((ii1))))11 in between all pairs of connected nodes ji and j2, the loss function is given by:
M
des (M ( Kin)
[(x) =
Σ
31.32
where zj, and Ttz are the node representations on the embedding space of nodes ji and j2, respec- tively, and ds is the distance function given in Section 2. This loss function captures the average distortion. We use the cosine distance to compute complete input distance graph from the original features of the data points [12, 27]. After the node embeddings' are learned, they are used as input features for all methods. In our experiments, the embedding dimension is set to 6.
Each of our networks consists of AFC layer and MLR (QMLRgE,, VMLR) layer. The QML and VMLR layers are built using the distances Theorem 3.5 and the upper bound of d(z Hg,p) in Proposition 3.11, respectively. We compare our networks to the following methods:
hnttpa://archive ice. ics.uci........... uci.edu/datasete hnttps://gh thub wttp///ayaaxunnnnno//aaaaattaeee data

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(1) KNN based on the distance function dea(-: 3); (2) LogEig classifier [27]: and (3) SiegelNet- BFC-BMLR which consists of an FC (BFC) layer and a MLR (BMLR) layer based on Busemann functions [35]. Results in Tab. 2 show that MiegelNee-AFC--MI...
gives the best mean accuracies across all the datasets. In terms of mean accuracy, Siege/Net-FCC-VML. surpasses SiegelNet-BFC- BMLR on Glass and Zoo datasets. MiegelNet-AFC-VMI Glass and Iris datasets. Tab. reports the results of SiegelNet-Bres BFCC--BL and our networks without FC layers. It can be observed that our MLR models achieve higher (mean) accuracies than the BMLR model. Specifically, DeegelNet-OMLReeE improves upon SiegeINet-BMLR by a margin 2.06%, 0.58%, and 5.57% Glass, Iris, and Zoo datasets, respectively. able to improve by 3.17%, 0.67%, and 1.36% w.r.t SiegelNee- SML on Glass, Iris, and Zoo datasets, respectively. demonstrating the effectiveness of the AFC layer. Although SiegelNet- VMLR outperformed by SiegelNet-QMe -agee- about two times smaller than that of the latter (see Appendix 1.2).
also surpasses the LogEig classifier on
DegelNNe-AFC-OMLLRS
is
Rsu,, it is important to note that the model size of the former is
6 Limitation of Our Approach
limitation of our method in Section 3.1 that our formulation of Siegel hyperplanes suffers from an parrammrrtteritiioonn issue. We alleviate this problem by reparameterizing Siegel hyperplanes as proposed in Section 3.2. However, this new parameterization does not yield competitive performance compared to the original one.
Our methods rely operations Siegel spaces which generally expensive. Our method in Section 5.1 suffers from high computational cost in the setting of high-dimensional radar signals. Similarly, the loss function in Section 5.2 is based on the average distortion, for which the distances over all pairs of points must be computed during training. Since the computation of the Riemannian distance between two points on Siegel space (see Section 2. based on eigenvalue decomposition, our method in Section 5.2 is computationally expensive when it comes to learning on large graphs.
Like hyperbolic and SPD spaces, Siegel spaces are spaces of non-po non-positive curvature. Therefore, method Section 5.2 does not allow isometric embeddings of graphs with a different curvature property, e.g., non-negative curvature. Although it can still be applied in this case, the learned node embedding may not preserve the curvature property of the embedded graph, leading to poor performance. Furthermore, like other graph embedding approaches, low-dimensional embeddings on Siegel spaces are not able to capture complex relationships within data which can affect the performance of our method.
7 Conclusion
We have proposed Riemannian MLR and FC layers which enable the construction of effective Siegel neural networks. Our MLR layers are built upon the quotient structure of Siegel spaces and the concept of vector-valued distance on RSS. Our FC layers are based on the action of the real symplectic group Siegel spaces. We have provided experimental evaluations demonstrating state-of ofte-oo-ther -he--artt performance of our approach in the radar clutter classification and node classification tasks.
There are several potential improvements and extensions to Siegel neural networks that could be addressed future work. Based on experimental results, it can be observed that the DFC layer gives inferior performance compared to the AFC layer. It is therefore desirable to develop alternative layers for the DFC layer which are able achieve better performance. Also, important building blocks such as convolutional layers, batch normalization layers, pooling layers, and attention layers are not studied in our work. Those are crucial the development of effective deep Siegel neural networks.
Acknowledgments
We are grateful for the constructive comments and feedback from the anonymous reviewers.
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Reviews:
[{"review_id": "dWj07z0r4Z", "reviewer": "Reviewer_8RJ3", "summary": "The authors build neural networks for Siegel spaces (i.e., space of complex-valued matrices A+ iB such that A is real symmetric and B is real positive-definite symmetric). These spaces generalize the Poincare upper half-plane model of hyperbolic geometry. They propose multiclass logistic regression  (odds) and fully-connected layers for these spaces via gyrovector approach, and the paper is concluded with experiments.", "questions": "This is an interesting paper. The authors define distances on Siegel spaces which lead to multi-class odds in Equation 2 for classification in these spaces, as well as linear layers in Section 3.2.2. \n\n\n\n### Major Comments/Question:\n\n- The paper is less accessible to the community, as it involves a lot of deep mathematical definitions and hard to understand. Moreover, the theory while being plausible, is not well motivated. My main concern is about the applicability of the results in practice, and the relevance. \n\n \n### Other Comments:\n\n- Line 72: Can you explain a bit what sort of Riemannian metric induces this distance? \n\n- Line 84: I'm a bit confused; can we obtain any RSS via some G and some K?\n\n- Definition 3.2: How do you define $g^T$ for elements of the group? Are those coming from matrix representations of a Lie group? How do you define log in the same definition?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Pros:\n\n- interesting problem\n- well written\n\n\nCons:\n\n- this setting, while being theoretically interesting, is not well motivated in terms of applications", "was_revised": true, "final_justification": "I had a number of concerns/questions that have been addressed in part by the authors during the discussion period. I decided to update my score from 2 to 4 conditioned on applying the list of promised changes to the paper.", "rebuttal": "We thank the reviewer for their comments.\n\n*Line 72: Can you explain a bit what sort of Riemannian metric induces this distance?*\n\nThe distance in line 72 is induced by the following line element [34, 37]:\n\\begin{equation*}\nds^2 = 2 \\operatorname{Tr}(v^{-1}dxv^{-1}d\\bar{x}), \\hspace{3mm} x = u + iv \\in \\mathbb{SH}_m,\n\\end{equation*}\nwhere $\\operatorname{Tr}(\\cdot)$ is the matrix trace.\n\n*Line 84: Can we obtain any RSS via some G and some K?*\n\nPlease refer to Appendix 3.1 (lines 172 - 176) for details. \n\n*Definition 3.2: How do you define for elements of the group? Are those coming from matrix representations of a Lie group? \nHow do you define log in the same definition?*\n\nThe space $X$ can be expressed as a quotient of a Lie group. The elements of $X$ are specified from its quotient structure. For Siegel spaces, it is described in Section 2.1. The matrix representation of the Lie group and the map $\\phi(\\cdot)$ in Section 2.1 are used to give the matrix representation of the elements of $X$. \n\nThe function $\\log(\\cdot)$ denotes the ordinary matrix logarithm.\n\n*The paper is less accessible to the community, as it involves a lot of deep mathematical definitions and hard to understand. Moreover, the theory while being plausible, is not well motivated. My main concern is about the applicability of the results in practice, and the relevance.*\n\nThe concepts used in our paper are not new in the context of our work. Section 3.1 uses a gyrovector space approach which has been developed since the pioneering work in [17]. Section 3.2 is based on vector-valued distances which have been studied in [28, 29]. Our work is motivated by the lack of an effective framework for building discriminative Siegel networks. Please refer to Section 1 for our motivation. \n\nOur method in Section 5.1 could result in new tools for analyzing complex stationary centered autoregressive Gaussian time series and Block-Toeplitz matrices [4, 10].\n\nBelow we present experiments to further demonstrate the applicability of our method. \n\n## **Riemannian Generative Modeling**\nTo our knowledge, we are not aware of any generative methods that target Siegel spaces. Since we do not aim to propose new methods in this context, we consider SPD spaces and apply the method in [A1] based on flow matching to solve the task. Note that our methods are directly applicable to SPD spaces.\n\n### **Datasets and Experimental Settings**\n**BCIC-IV-2a [A2]**\nIt consists of electroencephalography (EEG) data captured from $9$ subjects. The cue-based BCI paradigm consists of $4$ different motor imagery tasks, namely the imagination of movement of the left hand (class $1$), right hand (class $2$), both feet (class $3$), and tongue (class $4$). Two sessions on different days are recorded for each subject. Each session is comprised of $6$ runs separated by short breaks. One run consists of $48$ trials ($12$ trials for each of the $4$ possible classes), yielding a total of $288$ trials per session.\n\n**MAMEM-SSVEP-II [A3]**\nIt consists of EEG data with $256$ channels captured from $11$ subjects executing a SSVEP-based experimental protocol. Five different frequencies ($6.66$, $7.50$, $8.57$, $10.00$ and $12.00$Hz) are used for the visual stimulation, and the EGI $300$ Geodesic EEG System (GES $300$), using a $256$-channel HydroCel Geodesic Sensor Net (HCGSN) and a sampling rate of $250$ Hz is used to capture the signals.\n\nWe create two datasets of SPD matrices by computing a covariance matrix from each signal of the above datasets. For BCIC-IV-2a, the session 1 data of subject 1 is used as the training set whose 1/8 is used as the validation set. The session 2 data of subject 1 is used as the test set [A2]. For MAMEM-SSVEP-II, the first 4 sessions of subject 1 are used as the training set whose 1/4 is used as the validation set. The fifth session of subject 1 is used as the test set [A3].\n\n### **Optimization and Hyperparameters**\nWe use the official code of [A1] for our experiments. Vector fields are parameterized as neural networks in the ambient space and are projected onto the tangent space at every point. They are normalized by the inverse of square root of the metric tensor to cancel out the effect of the metric tensor on the Riemannian norm [A1]. Inspired by [A4] which uses a Poincaré MLR layer to construct the vector field, we replace the first convolutional layer of the original vector field (a multilayer perceptron) with the VMLR and QMLR layers, as well as the MLR-LE and MRL-AI layers [31]. The vector-valued distance is computed as in [29]. We use $1000$ Euler steps with a projection after every step for evaluation. We use the checkpoint that gives the best negative log-likelihood (NLL) on the validation set to compute the NLL on the test set [A1]. The numbers of hidden units and layers of the vector field are set to $32$ and $4$, respectively. All networks are trained using AdamW optimizer with model exponential moving average. The learning rate and weight decay are set to $2e-4$ and $0.999$, respectively. The number of epochs and the batch size are set to $100$ and $128$, respectively. Experiments are performed using machines with an Intel Core i7-9700 CPU 3.00 GHz.\n\n| Method | BCIC-IV-2a | MAMEM-SSVEP-II | \n|-----------------------|-------------|-------------|\n|     RFM [A2]       |    -225.23$\\pm$3.42    |   -169.28$\\pm$1.24      | \n|   RFM-MLR-LE [31]  |   -231.64$\\pm$6.19    |    -170.37$\\pm$3.18    |\n|   RFM-MLR-AI [31]    |    -238.59$\\pm$6.38    |    -170.92$\\pm$4.05    |\n|   RFM-QMLR (Ours)    |    -252.94$\\pm$8.53    |    **-173.06**$\\pm$7.34   |  \n|   RFM-VMLR (Ours)    |    **-260.81**$\\pm$6.55    |    -172.53$\\pm$3.62    |  \n\nTable A: Test NLL (mean $\\pm$ standard deviation) computed over 5 runs (lower is better). The number of function evaluations is set to 1000.  The sizes of SPD matrices in BCIC-IV-2a and MAMEM-SSVEP-II are $22 \\times 22$ and $8 \\times 8$, respectively.\n\n### **Results**\n\nTable A reports estimates of NLL on the two datasets. Our proposed MLR layers significantly improve RFM on BCIC-IV-2a. We also see that these layers achieve the best performance on both the datasets.\n\n## **Action Recognition**\n### **Datasets and Experimental Settings**\nWe conduct experiments similar to those in Appendix 1.1.4.\n\n**HDM05 [A5]**\nIt has 2337 sequences of 3D skeleton data with $130$ classes. Each frame contains the 3D coordinates of $31$ body joints.  We use the experimental protocol in [A7] in which 2 subjects are used for training and the remaining 3 subjects are used for testing.\n\n**NTU-60 [A6]**\nIt has 56880 sequences of 3D skeleton data with $60$ classes. Each frame contains the 3D coordinates of $25$ or $50$ body joints. We use the interaction actions for classification (11 classes) and the cross-view protocol [A6] in which training data come from the camera views 2 and 3, and testing data come from the camera view 1.\n\nWe consider a challenging setting in which the input data contain only one of the three coordinates (channels) of human joints. The method in Section 5.1 is used to compute the input data which belong to products of an SPD space and a Siegel space. We focus on comparing our network and SPD neural networks.\n\n### **Optimization and Hyperparameters**\nWe use the same method in Appendix 1.1.2 for optimizing parameters. All networks are trained using cross-entropy loss and Adadelta optimizer for 2000 epochs. The learning rate is set to $1e-2$. We use a batch size of 32 for HDM05, and a batch size of 256 for NTU-60. Experiments are performed using machines with an Intel Core i7-9700 CPU 3.00 GHz.\n\n| Dataset | HDM05 | HDM05 | HDM05 | XView60 | XView60 | XView60 | \n|------------|----------|----------|--------------|-------------|---------------|-------------|\n|              | x-channel | y-channel | z-channel | x-channel | y-channel | z-channel |\n| SPDNet | 38.21$\\pm$0.34 | 52.66$\\pm$0.36 | 39.25$\\pm$0.26 | 64.91$\\pm$0.48 | 59.26$\\pm$0.40 | 47.36$\\pm$0.41 |\n| SPDNetBN | 40.74$\\pm$0.30 | 56.82$\\pm$0.33 | 42.48$\\pm$0.25 | 68.74$\\pm$0.46 | 61.97$\\pm$0.42 | 50.61$\\pm$0.38 |\n| MLR-AI  | 41.32$\\pm$0.32 | 59.54$\\pm$0.35 | **45.17**$\\pm$0.24 | 69.62$\\pm$0.50 | 63.87$\\pm$0.45 | 50.58$\\pm$0.43 |\n| GyroSpd++ | 41.09$\\pm$0.37 | 57.32$\\pm$0.39 | 43.65$\\pm$0.30 | 68.83$\\pm$0.51 | 62.12$\\pm$0.48 | 49.94$\\pm$0.46 |\n| SiegelNet-AFC-ProductQMLR | **42.17**$\\pm$0.85 | **66.91**$\\pm$0.69 | 44.92$\\pm$0.44 | **72.57**$\\pm$0.73 | **66.17**$\\pm$0.58 | **51.78**$\\pm$0.48 |\n\nTable B: Results (mean accuracy $\\pm$ standard deviation) computed over 5 runs. XView60 corresponds to the cross-view setting of NTU-60.\n\n### **Results**\nResults in Table B show that our method surpasses its competitors in most cases. The mean accuracy of  our network using only the y-coordinates (66.91\\%) is better than that of SPDNetBN using all the joint coordinates on HDM05 (62.54\\% [30]).\n\n**References**\n\n[A1] Ricky T. Q. Chen, Yaron Lipman: Flow Matching on General Geometries. ICLR 2024.\n\n[A2] Clemens Brunner, Robert Leeb, Gernot Müller-Putz, Alois Schlögl, and Gert Pfurtscheller. Bci competition 2008–graz data set a. Institute for Knowledge Discovery (Laboratory of Brain-Computer Interfaces), Graz University of Technology, 16:1–6, 2008.\n\n[A3] Spiros Nikolopoulos. MAMEM EEG SSVEP Dataset II (256 channels, 11 subjects, 5 frequencies presented simultaneously). 2021.\n\n[A4] Mathieu, E. and Nickel, M.. Riemannian continuous normalizing flows. Advances in Neural Information Processing Systems, 33:2503–2515, 2020.\n\n[A5] Müller, M.; Röder, T.; Clausen, M.; Eberhardt, B.; Krüger, B.; and Weber, A. 2007. Documentation Mocap Database HDM05. Technical Report CG-2007-2, Universität Bonn.\n\n[A6] Shahroudy, A.; Liu, J.; Ng, T.-T.; and Wang, G. 2016. NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysis. In CVPR, 1010–1019.\n\n[A7] Harandi, M.; Salzmann, M.; and Hartley, R. 2018. Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods. TPAMI, 40: 48–62."}, {"review_id": "xt2lXMjRQC", "reviewer": "Reviewer_7CnN", "summary": "The paper introduces SiegelNet, the first discriminative neural-network architecture whose layers operate natively on Siegel spaces, a broad family of Riemannian symmetric spaces that strictly generalizes both hyperbolic space and the SPD manifold.\n- Two multiclass-logistic-regression (MLR) layers are derived: QMLR, obtained from a new closed-form point-to-hyperplane distance based on the quotient structure of Siegel spaces, and VMLR, a more compact alternative built from vector-valued distance.\n- Two fully-connected layers (AFC, DFC) are proposed; DFC also performs dimensionality reduction.\n- Combining these ingredients yields SiegelNet, which attains state-of-the-art accuracy on (i) four simulated radar-clutter datasets and (ii) three UCI node-classification benchmarks, consistently outperforming k-NN, SPD and hyperbolic baselines.", "questions": "Please refer to the Weaknesses section", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n- This study extends the ideas behind gyrovector-based hyperbolic neural networks and neural networks on the SPD manifold to the Siegel domain. Its originality lies in the development of techniques that tackle the new geometric challenges that arise when broadening the class of manifolds up to Siegel domain.\n\n# Weaknesses\n- On the other hand, Sonoda et al. (2022) have already applied Helgason’s harmonic analysis to design geometric neural networks on general non-compact symmetric spaces and to show their universality. A discussion clarifying the differences between the present work and that study is therefore requested.\n  [SII2022] Sonoda-Ishikawa-Ikeda, Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis, ICML2022", "was_revised": true, "final_justification": "As the relation to prior research has been clarified and it is now clearer that this work is novel, I will raise my score.", "rebuttal": "We thank the reviewer for their insightful comments. \n\n*On the other hand, Sonoda et al. (2022) have already applied Helgason’s harmonic analysis to design geometric neural networks on general non-compact symmetric spaces and to show their universality. A discussion clarifying the differences between the present work and that study is therefore requested.*\n\nThe work in [A8] analyzes some existing hyperbolic and SPD neural networks from the perspective of harmonic analysis on RSS. This work mainly focuses on a constructive proof of the universal approximation property of finite neural networks on RSS, and it does not concern with practical applications on RSS. \n\nIn terms of network architecture, their considered FC layers are different from our FC layers in some aspects. First, the former include activation functions which are not used in the latter. Second, the former do not output points on the considered spaces, as opposed to the latter which always output points on these spaces. \n\nWe propose to add the first paragraph of the above discussion to Section 4 (Related work) and the second one to Section 3.3.2 (FC layers).\n\n**References**\n\n[A8] Sho Sonoda, Isao Ishikawa, and Masahiro Ikeda. Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis. In ICML, pp. 20405-20422, 2022."}, {"review_id": "D1XFfYUqaa", "reviewer": "Reviewer_ahAK", "summary": "This paper introduces Siegal Neural Networks, meant to function on the dataset of underlying siegal space structure, which is roughly a tensorial generalization of Hyperbolic space.  The paper establishes the analogue of FC layer(AFC, DFC) and classification readout layer(MLR) by closely aligning (1) the linear transformation in usual FC to the group transformation of Sp2m, (2) “distance to plane” component of the softmax to the “distance to the hyperplane” in SHm.  \nThe paper establishes closed form as well as the surrogate for these “distance” functions.\nThe paper provides empirical results showcasing the efficacy of the Neural network consisting of XFC and MLR.", "questions": "What is the true advantage of Weyl chamber based definition (3.12?). The mention is being made about the reduction of the parameter size, but it does neither changes the “order-level” cardinality of the parameter set  nor play out too well in the final performance.    Is it also a legitimate “distance”?  \n\nIn regards to the point made in the weakness section, is there not a dataset/task  that “only” accepts computations closed in SHm?  Are there any intuitive cases where the use of Siegel Networks would be natural?", "limitations": "Limitation is being mentioned in Appendix Section, with regard to the overparameterization.    I wonder if there is also possible limitation with respect to the representation power, caused by the structural inductive bias to the network.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strength: \n\nThe paper introduces mathematically principled extension of the usual MLP to Siegel space by theoretically assuring each intermediate output to live in SHm and by aligning the point-to-plane distance component of the softmax to the Siegel Geometry. \n\n\nThe paper provides closed forms and analogues and upper bounds that seems useful not only for the construction of their propose network but also to other applications on Siegel Space \n\n\nWeaknesses \n\nSome parts of the manuscript were simply hard to follow. For example, I presume that, in 3.2.2, the definition of dbar changes over the paragraph— starting from Def 3.10, it leverages the prop 3.11 to “redefine” dbar in 3.12.   This was very confusing.  The motivation of 3.2 was also not strong enough for me— it took a while until I realized that this section was dedicated for the proposal “approximate surrogate” as opposed to a strictly better version of 3.5.  Also,  Eq(1) referred wrongly in the intro of 3.2?  I believe the authors wanted to refer to definition 3.4?   It also confused me that QMLR appeared in 5.1 (it is first described in 5.2)\n\n\n\nWhile I am hesitant in evaluating the paper of theoretical significance as this research based on the quality of experiments, I am a little worried about the implications of the results— although the construction of Siegel Networks on their own is intellectually exciting and promising,  I am afraid “Why Siegel? Why not Hyperbolic?” is not fully justified in the end.  With such interesting results regarding Siegel Space, I believe the paper is obliged to demonstrate to the community the clear “practical” symmetric difference of Siegel space based approach from Hyperbolic space approach. \nFor example, it is being claimed in section 5.1 that  \"to the best of our knowledge , there is no baseline in our context to compare against\";  however, because the task on its own is a classification problem, I find it hard to believe that the dataset cannot be embedded to other spaces, such as Poincare Ball, Euclidean Space,  etc.  In such a case, Hyperbolic Networks and other conventional methods  would be the true adversary.  I wonder if the same applies to Node classification. 　For this ground I feel that the statements in abstract regarding the experimental performance is overstated.\nWhile I completely agree that the focus of this paper should not be the invention of a SOTA method for a popular scientific task,  I feel that it is important to bring about at least one case in which only the Siegel network can solve.", "was_revised": true, "final_justification": "I am suggesting the acceptance with score 4.  I could not raise the score to 5 because I was unable to obtain the clear answer  to the question of “Why Siegel? Why not Hyperbolic?” has not been fully answered.  The question was answered partially through its reference to [28], but the experiments were not conducted on the tasks discussed in [28], by which the research may be able to further strengthen its evidence of utility. At the same time, I still maintain my praise for the research in regards to its theoretical and fundamental contributions, as well as its provision of a new venue of framework.", "rebuttal": "We thank the reviewer for their insightful comments and questions. \n\n*Some parts of the manuscript were simply hard to follow. For example, I presume that, in 3.2.2, the definition of dbar changes over the paragraph— starting from Def 3.10, it leverages the prop 3.11 to “redefine” dbar in 3.12. This was very confusing.*\n\nTo avoid confusion, we propose to denote the upper bound of $\\bar{d}(x, \\mathcal{H}\\_{\\xi,p})$ by $\\tilde{d}(x, \\mathcal{H}\\_{\\xi,p})$ which is defined as $\\tilde{d}(x, \\mathcal{H}\\_{\\xi,p})=\\langle d\\_{\\Delta}(x,p), a\\_{\\xi}\\rangle$. We can then remove Definition 3.12 and in the experiments, we will state that the QMLR layer is built upon the upper bound of $\\bar{d}(x, \\mathcal{H}\\_{\\xi,p})$.\n\n*The motivation of 3.2 was also not strong enough for me— it took a while until I realized that this section was dedicated for the proposal “approximate surrogate” as opposed to a strictly better version of 3.5.*\n\nPlease see our answer to the question related to this comment below.\n\n*Also, Eq(1) referred wrongly in the intro of 3.2? I believe the authors wanted to refer to definition 3.4? It also confused me that QMLR appeared in 5.1 (it is first described in 5.2)*\n\nWe would like to state that \"the formulation in Eq. (1) has an over-parameterization issue, our formulation in Definition 3.4 follows that formulation and thus suffers from a similar issue.\" We think Eq. (1) is correctly referred, but we agree that there is a missing connection between the first two sentences of Section 3.2. We propose to rephrase them as above to avoid confusion.\n\nThe QMLR layer in Section 5.2 (constructed from the distance in Theorem 3.5) is only a special instance of the ProductQMLR layer in Section 5.1 (constructed from the distance in Theorem 3.8). This is because when the Cartesian product contains only one Siegel space, the distance in Theorem 3.8 is precisely the one in Theorem 3.5. To avoid confusion, we propose to call both layers as QMLR and use subscripts to indicate the components of the considered space. For instance, the ProductQMLR layer will be called as QMLR$\\_{\\operatorname{Sym}\\_m^+ \\times \\mathbb{SH}\\_m^L}$, and the QMLR layer will be called as QMLR$\\_{\\mathbb{SH}_m}$, where $L$ is the number of Siegel spaces in the Cartesian product.\n\n*I believe the paper is obliged to demonstrate to the community the clear “practical” symmetric difference of Siegel space based approach from Hyperbolic space approach.*\n\nThe practical advantage of Siegel spaces compared to Euclidean spaces, hyperbolic spaces, products of Euclidean and hyperbolic spaces, and SPD spaces has been demonstrated in [28] in the context of graph embedding for a variety of applications (e.g., graph reconstruction, recommender systems, analysis of the embedding space). It is observed from their experimental results that the use of Siegel spaces in combination with appropriate metrics generally results in the best performance. Please refer to [28] for more details. \n\nIn our work, the practical advantage of Siegel spaces compared to hyperbolic spaces can be seen in radar clutter classification in which the use of Siegel networks is natural and results in the best performance, whereas the use of hyperbolic networks is not trivial. We have also shown that the task of action recognition can be tackled by Siegel networks as opposed to hyperbolic networks which are not commonly adopted for solving such a problem. Please refer to our response to Reviewer 8RJ3 for more experimental results.\n\n*For example, it is being claimed in section 5.1 that \"to the best of our knowledge , there is no baseline in our context to compare against\"; however, because the task on its own is a classification problem, I find it hard to believe that the dataset cannot be embedded to other spaces, such as Poincare Ball, Euclidean Space. I wonder if the same applies to Node classification.*\n\nOne can use conventional models for classification of time series, e.g., RNN and LSTM in hyperbolic and Euclidean spaces to tackle the problem in Section 5.1. However, these models were originally designed to work with real-valued data, and it is not clear how effective they are for complex-valued data. In our experiments, we tested an LSTM for complex-valued data using the ComplexNN library implemented in Pytorch (we cannot provide its link due to the policy of NeurIPS 2025). However, we obtained very poor results on our datasets.  \n\nFor the task of node classification, the performance of our method depends not only on the proposed FC and MLR layers but also on the method used for learning node embeddings. Since the focus of our work is to build neural networks on RSS, we think it would be more important to validate these layers by comparing them with existing ones on Siegel spaces. \n\nSince there is possibility to use methods in hyperbolic and Euclidean spaces (or even other Riemannian manifolds) for comparison in Section 5.1 (although it is not guaranteed that they work well for our problem), we propose to remove the statement \"to the best of our knowledge, ...\" in this section.\n\n*What is the true advantage of Weyl chamber based definition (3.12?)... Is it also a legitimate “distance”?*\n\nFrom a theoretical aspect, the distance in Definition 3.12 has a nice property, i.e., it is $K$-invariant. This can be seen by first noting that $d\\_{\\Delta}(x,p) = d\\_{\\Delta}(k\\_1[x],k\\_1[p])$ for any $k\\_1 \\in K$ due to the $G$-invariant property of the vector-valued distance $d\\_{\\Delta}(\\cdot,\\cdot)$, and $a\\_{\\xi} = a\\_{k\\_1[\\xi]}$ since $k\\_1[\\xi]$ is the equivalent class of the geodesic ray $k\\_1[\\delta(t)] = k\\_1k\\exp(ta\\_{\\xi})K = k\\_2\\exp(ta\\_{\\xi})K$ where $k\\_2 \\in K$ (see the notations in Proposition 3.11). Therefore, $\\bar{d}(x,\\mathcal{H}\\_{\\xi,p}) = \\left\\langle d\\_{\\Delta}(x,p),a\\_{\\xi} \\right\\rangle = \\left\\langle d\\_{\\Delta}(k\\_1[x],k\\_1[p]),a\\_{k\\_1[\\xi]} \\right\\rangle = \\bar{d}(k\\_1[x],\\mathcal{H}\\_{k\\_1[\\xi],k\\_1[p]})$. It is easy to see that this property does not hold for the distance in Section 3.1. \n\nThe distance in Definition 3.12 when applied to Siegel spaces results in the VMLR layer which has the same memory and time complexities as the BMLR layer (please refer to our response to the last comment of Reviewer qfip). The former improves upon the latter on node classification in most cases (see Table 3). Furthermore, our experiments on Riemannian generative modeling demonstrate the effectiveness of this distance in such an application. Please refer to our response to Reviewer 8RJ3 for details on our experiments.\n\nSince the distance in Definition 3.12 is not the shortest distance between a given point and any point on a given Siegel hyperplane, \nit is not a legitimate point-to-hyperplane distance in this sense.\n\n*It is important to bring about at least one case in which only the Siegel network can solve. In regards to the point made in the weakness section, is there not a dataset/task that “only” accepts computations closed in SHm? Are there any intuitive cases where the use of Siegel Networks would be natural?*\n\nTo our knowledge, we are not aware of any dataset/task in which only input data/features on Siegel spaces are available. In our work, the use of Siegel networks for radar clutter classification is well-motivated. Algorithm 1 (see Appendix 1.1.2) is a robust method for estimating autocorrelation matrices which have proven effective in signal processing applications. This motivates its usage for computing the input data of our networks. Due to the assumption that radar signals are stationary centered autoregressive (AR) Gaussian time series, it has been shown in [10] (see Section 2.2.9, Theorem 3) that the output of Algorithm 1 belong to product space $\\mathbb{H}\\_m^+ \\times \\mathbb{SD}\\_m^{q-1}$. Therefore, the use of Siegel networks comes naturally from the effectiveness of autocorrelation matrices in capturing patterns within data and from the above assumption on radar signals. It has been shown in previous works [21, 22, 23] that different representation spaces (i.e., SPD manifolds, Lie groups, Grassmann manifolds) of the same input data can lead to enhanced performance using Riemannian neural networks.\n\nWe note that methods dealing with data that lie on product spaces $\\mathbb{H}\\_m^+ \\times \\mathbb{SD}\\_m^{q-1}$ have already been studied in previous works [4, 10, 11]. These representation spaces are endowed with a natural metric inspired by information geometry [4, 10]. \n\n*I wonder if there is also possible limitation with respect to the representation power, caused by the structural inductive bias to the network.*\n\nLike hyperbolic and SPD spaces, Siegel spaces are spaces of non-positive curvature. Therefore, our method in Section 5.2 does not allow isometric embeddings of graphs with a different curvature property, e.g., non-negative curvature. Although it can still be applied in this case, the learned node embedding may not preserve the curvature property of the embedded graph, leading to poor performance for node classification. \n\nFurthermore, like other graph embedding approaches, low-dimensional embeddings on Siegel spaces are not able to capture complex relationships within data which can affect the performance of node classification. \n\nIn addition to the above limitations with respect to the representation power, our methods rely on operations on Siegel spaces which are generally expensive. The loss function in Section 5.2 is based on the average distortion. That is, the distances over all pairs of points must be computed during training. Given that the computation of the Riemannian distance between two points on a Siegel space (see Section 2.1) is based on eigenvalue decomposition, our method is computationally expensive when it comes to learning on large graphs. Similarly, our method in Section 5.1 suffers from high computational cost in the setting of high-dimensional radar signals."}, {"review_id": "AKXAz1eQLt", "reviewer": "Reviewer_qfip", "summary": "This work focuses on designing neural networks adapted to Siegel spaces. It introduces two type of layer: 1) fully connected layers on the Siegel upper half space, and 2) multiclass logistic regression layers. The second type of layers rely on defining a suitable notion of hyperplane and of distance to the hyperplane. The authors propose two ways to extend hyperplanes to Riemannian symmetric space of non compact type, and in particular on Siegel spaces. First, they use gyrovectors operations, and show that they the distance to the hyperplane has a closed-form in this case. Second, they also propose to use vector values distances. Finally, they show on Radar clutter classification and node classification tasks the potential of their architecture to deal with data living on Siegel spaces.", "questions": "In Definition 3.12, it is stated that the distance between $x$ and the hyperplane is defined as $\\bar{d}(x, \\mathcal{H}_{\\xi,p})=\\langle d_\\Delta(x,p), a_\\xi\\rangle$. But in my understanding, it is only an upper bound, which is thus confusing. Mayber you could change the notation?\n\nIn Table 1, do you have an explanation why the AFC layers are always better than the DFC layers. Also, why aren't the vector valued layers compared on the radar data?\n\nComparisons in Table 3 are also done with Busemann layers. I guess it requires to compute the Busemann function. How is it computed on Siegel spaces? Why isn't it used in the experiment of Table 1?\n\nThere is an analysis of the complexity in Appendix 1. But it is not clear how it compares with the baselines used in the experiments sucha as the Busemann layers?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Few works have focused on Siegel spaces in Machine Learning. Therefore, it is nice to see a work focusing on defining tractable neural networks on this space. The proposed ways to extend notions of hyperplanes to Siegel spaces are very interesting, and pretty clear to understand. This results in layers that can be used to define neural networks, which are shown to work pretty well on real data, which is a very nice contribution.\n\n**Strengths**:\n- Derivation of 2 extension of hyperplanes on Riemannian symmetric spaces along with distances to hyperplanes\n- Two type of layers to define neural networks: MLR and fully connected\n- Application on real data, showing improvements over baselines\n\n**Weaknesses**:\n- The presentation of gyrovectors results is close to the one of [1], and it seems that the main contribution here is its specification to Siegel spaces.\n- The comparison between the proposed layers could be a bit improved.\n- For the vector-values distance, the distance to hyperplane is not known in closed-form.\n\n[1] Nguyen, X. S., Yang, S., & Histace, A. (2025). Neural networks on Symmetric Spaces of Noncompact Type. In The Thirteenth International Conference on Learning Representations.", "was_revised": true, "final_justification": "This work introduce the first neural networks on Siegel spaces. I believe that it is a very interesting contribution, as Siegel spaces are known to be more expressive than hyperbolic spaces or SPD spaces to embed complex data.\n\nSome concerns were raised about the mathematical contribution and on the applications of such networks. But the authors convinced me of the importance of their contributions. Thus, I recommend acceptance.", "rebuttal": "We thank the reviewer for their insightful comments and questions. \n\n*The presentation of gyrovectors results is close to the one of [1], and it seems that the main contribution here is its specification to Siegel spaces.*\n\nOur paper presents three new technical contributions with respect to [33] (the reference given by the reviewer). First, our method in Section 3.1 relies on a gyrovector space approach which is different from the approach in [33] based on Busemann functions. This allows us to derive a closed-form for the point-to-hyperplane distance in the setting of product spaces, which is not trivial for the approach in [33]. Second, our method in Section 3.2 is built upon vector-valued distances which do not appear in [33]. Third, the construction of our FC layers is fundamentally different from that of FC layers in [33] which is based on the point-to-hyperplane distance. From the practical aspect, to the best of our knowledge, we build the first Siegel neural networks for solving the tasks of radar clutter classification and action recognition (see Appendix 1.1.4).\n\n*In Definition 3.12, it is stated that the distance between $x$ and the hyperplane is defined as $\\bar{d}(x, \\mathcal{H}\\_{\\xi,p})=\\langle d\\_{\\Delta}(x,p), a\\_{\\xi}\\rangle$. But in my understanding, it is only an upper bound, which is thus confusing. Mayber you could change the notation?*\n\nTo avoid confusion, we propose to denote the upper bound of $\\bar{d}(x, \\mathcal{H}\\_{\\xi,p})$ by $\\tilde{d}(x, \\mathcal{H}\\_{\\xi,p})$ which is defined as $\\tilde{d}(x, \\mathcal{H}\\_{\\xi,p})=\\langle d\\_{\\Delta}(x,p), a\\_{\\xi}\\rangle$. We can then remove Definition 3.12 and in the experiments, we will state that the QMLR layer is built upon the upper bound of $\\bar{d}(x, \\mathcal{H}\\_{\\xi,p})$.\n\n*In Table 1, do you have an explanation why the AFC layers are always better than the DFC layers. Also, why aren't the vector valued layers compared on the radar data?*\n\nWe agree that the answer to this question would give one some insight on how to design better FC layers for dimensionality reduction on Siegel spaces. However, we do not think there is an obvious answer. A potential reason could be that learning our network parameters involves solving a highly nonlinear optimization problem, and all the parameters of the AFC layer are optimized on Euclidean spaces, while the parameter $b$ of the DFC layer is learned using Riemannian optimization on Stiefel manifolds.\n\nRegarding the radar clutter classification experiments, the input data lie on products of SPD and Siegel spaces. Our proposed VMLR layer is designed on Siegel spaces and cannot deal with data on product spaces (it is not trivial to adapt them to this setting using a similar derivation in Section 3.2, so we did not consider investigating this idea in our paper). For this reason, we did not use them in these experiments.\n\n*Comparisons in Table 3 are also done with Busemann layers. I guess it requires to compute the Busemann function. How is it computed on Siegel spaces? Why isn't it used in the experiment of Table 1?*\n\nThe key point in the computation of the distance [33] based on Busemann functions is the computation of the map $H: G \\rightarrow \\mathfrak{a}$ determined by $g_1 = k\\_1 \\exp H(g\\_1) n\\_1$ with $g\\_1 \\in G$, $k\\_1 \\in K$, $n\\_1 \\in N$, and $\\mathfrak{a}$ is the Lie algebra of $A$ (please refer to Proposition 4.9 and Corollary 4.10 in [33]). We recall here that any connected noncompact semisimple Lie group with finite center $G$ can be decomposed into subgroups $K$, $A$, and $N$, where $K$ is maximal compact, $A$ is maximal abelian, and $N$ is maximal nilpotent (the decomposition in question is called Iwasawa decomposition). The map $H(\\cdot)$ can be computed by Cholesky factorization [A9, A10] as follows. Let $g = \\begin{bmatrix} a & b \\\\\\\\ c & d  \\end{bmatrix} \\in \\operatorname{Sp}_{2m}$ and let $g^Tg = \\begin{bmatrix} a\\_1 & b\\_1 \\\\\\\\ b\\_1^T & d\\_1 \\end{bmatrix}$. \nLet $a\\_1 = q^Trq$ be the Cholesky factorization of the positive definite matrix $a_1$, where $q$ is a unit upper triangular matrix and $r$ is a diagonal matrix with positive diagonal entries. Then $H(g) = \\begin{bmatrix} \\frac{1}{2}\\log(r) & 0 \\\\\\\\ 0 & -\\frac{1}{2}\\log(r) \\end{bmatrix}$. We propose to add these details to Appendix 1.2.2. \n\nFor the same reason regarding the VMLR layer, the BMLR layer is not designed on product spaces and therefore is not evaluated for \nradar clutter classification experiments.\n\n*There is an analysis of the complexity in Appendix 1. But it is not clear how it compares with the baselines used in the experiments sucha as the Busemann layers?*\n\nThe BMLR layer has the same memory and time complexities as the VMLR layer, i.e., their memory complexity is $O(Mm(m+2))$ and \ntheir time complexity is $O(Mm^3)$. We propose to add this analysis to Appendix 1.2.3.\n\n**References**\n\n[A9] P. Sawyer. Computing the Iwasawa Decomposition of the Classical Lie Groups of Noncompact Type Using the QR Decomposition. Linear Algebra and its Applications, 493:573–579, 2016.\n\n[A10] T.-Y. Tam, Computing the Iwasawa decomposition of a symplectic matrix by Cholesky factorization, Appl. Math. Lett. 19 (12) (2006) 1421–1424."}]

Author rebuttal:

Meta-review: The paper proposes an approach for building discriminative neural networks on Siegel spaces, a family of Riemannian Symmetric Space (RSS).
In particular, the authors show how to build multiclass logistic regression (MLR) and fully-connected (FC) layers for Siegel neural networks, using the quotient structure of those spaces and the notation of vector-valued distance on RSS. Numerical experiments are provided to illustrate the proposed theoretical framework.

The paper has ratings: Borderline Accept, Accept, Borderline Accept, Accept.

Strengths: Reviewers all appreciate the theoretical contributions of the proposed framework.

Weaknesses: Reviewer ahAK would like to see a better motivation via practical applications, i.e. an application that can only be tackled by Siegel Neural Networks. As pointed out by Reviewers ahAK and qfip, the distance in Definition 3.12 is only an upper bound of the actual point-to-hyperplane distance based on vector-valued distance. The authors have said they would clarify this point.  

While the paper could certainly be improved, I believe the presented framework would be a novel and interesting contribution to the community. The final recommendation is therefore: Accept

The authors are expected to revise and incorporate the comments made by the reviewers into the final version, as promised in the rebuttal.
Decision: Accept (poster)