Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry
Authors: Ziheng Chen
Yue Song
Xiaojun Wu
Nicu Sebe
Venue: ICLR 2026 Poster (2026)
Keywords: Cholesky Decomposition
Symmetric Positive Definite (SPD)
SPD Manifold
Riemannian Metrics
SPD Neural Networks
Primary area: learning on graphs and other geometries & topologies
TL;DR: We reveal the product structure in the Cholesky manifold and propose two fast and stable SPD metrics, which enable the construction of classifiers and residual blocks in SPD neural networks.

Abstract: Recent advances in Symmetric Positive Definite (SPD) matrix learning show that Riemannian metrics are fundamental to effective SPD neural networks. Motivated by this, we revisit the geometry of the Cholesky factors and uncover a simple product structure that enables convenient metric design. Building on this insight, we propose two fast and stable SPD metrics, Power--Cholesky Metric (PCM) and Bures--Wasserstein--Cholesky Metric (BWCM), derived via Cholesky decomposition. Compared with existing SPD metrics, the proposed metrics provide closed-form operators, computational efficiency, and improved numerical stability. We further apply our metrics to construct Riemannian Multinomial Logistic Regression (MLR) classifiers and residual blocks for SPD neural networks. Experiments on SPD deep learning, numerical stability analyses, and tensor interpolation demonstrate the effectiveness, efficiency, and robustness of our metrics. The code is available at https://github.com/GitZH-Chen/PCM_BWCM.

Published as a conference paper at ICLR 2026
FAST AND STABLE RIEMANNIAN METRICS ON SPD MANIFOLDS VIA CHOLESKY PRODUCT GEOMETRY
Ziheng Chen', Yue Song?.), Xiao-Jun Wur & Nicu Sebe!
1 University of Trento, Trento, Italy
2 Shanghai Qi Zhi Institute, Shanghai, China
3 College of AI, Tsinghua University, Beijing, China
4 Jiangnan University, Wuxi, China
ABSTRACT
Recent advances in Symmetric Positive Definite (SPD) matrix learning show that Riemannian metrics are fundamental to effective SPD neural networks. Motivated by this, we revisit the geometry of the Cholesky factors and uncover a simple product structure that enables convenient metric design. Building on this insight, we propose two fast and stable SPD metrics, Power-Cholesky Metric (PCM) and Bures-Wasserstein Chhllesk Metric (BWCM), derived via Cholesky decom- position. Compared with existing SPD metrics, the proposed metrics provide closed-form operators, computational efficiency, and improved numerical stabil- ity. We further apply our metrics to construct Riemannian Multinomial Logistic Regression (MLR) classifiers and residual blocks for SPD neural networks. Experi- ments on SPD deep learning, numerical stability analyses, and tensor interpolation demonstrate the effectiveness, efficiency, and robustmess of our metrics. The code BWCM
is available at hhtps: heittps://qithbb..
github. www.oiittt-Chaeeeeeeeeeeeeee
INTRODUCTION
Symmetric Positive Definite (SPD) matrices play a central role in various machine learning applica- tions, including medical imaging (Chakraborty et al. 22200), eleetroencephalography (EEG) analysis (Li et al. 2025), brain functional connectivity (Dan et al. (2024). signal processing (Brooks et al. 2019), and computer vision (Huang and Van Gool 2017 Wang al. 2023 Chen al., 2025a Kang et al. 2025). As SPD matrices form a non nucliield manifold (Arsigny et al. 2005). traditional Eu- clidean methods are inadequate. To close this gap, several Riemannian metrics have been developed, including Affine-IIvariant Metric (AIM) (Pennec et al. (2006), Log-Euclidean Metric (LEM) (Arsigny cet al. 2005). Power-Euclidean Metric (PEM) (Dryden et al. 2010p,
Metrie (LCM) (Lin) 2019), iasses-Wasersttenn Metric (BWM) (Bhatia al. (019). and Generalized Cirrs-Waaseesteenn
Metric (GBWM) (Han et al. 2023). These metrics not only provide mathematically principled tools, but also enable the extension of core deep learning components such Multinomial Logistics Regres- sion (MLR) (Nguyen and Yangl 2023 Chen et al., 2024a c) Fully Connected (FC) and convolutional
layers (Chakraborty et al. 2020)) normalization (Brooks al. 2019 Chakraborty 2020; Kobler
ct al. 2022 Chen et al. 2024b |024b|[025b| Wang et al. 22225b c). attention (Pan et al. /0ng et al.
2025a), residual blocks (Katsman et al.] 2024). and pooling (Wang et al. 2020 Song et al. 2222
Chen et al. 2025a Gu et 2005), ). as well as emerging modules including graph & recurrent neural
networks Chakraborty et al. 2018 Cruceru al., (20TT) and generative models (Li et al. 2024
Chen and Lipman 2024). Therefore, designing Riemannian metrics that are theoretically convenient,
computationally efficient, and numerically stable is crucial to advance SPD matrix learning-
Due to the fast and stable computation of the Cholesky decomposition, LCM enjoys several advantages over the other metrics (Lin 2019), including simple closed-form Riemannian operators, efficiency, and numerical stability. LCM is induced by the Cholesky decomposition from Riemannian metric on the Cholesky manifold (Lin 2019). which is the space of lower triangular matrices with positive diagonal entries. We refer to this Cholesky metric the diagonal log metric. We reveal a simple product structure underlying the diagonal log metric: a Euclidean metric on the strictly lower triangular part together with copies of Riemannian metric on R++ for the diagonal part. This

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Metric
Cyno Suaaauess
Esisling Metric
Sec M
Duginal Meric
Meri
Bubling SPD Ncural Newwrrks
Cheluky
Cheificatzo
Cholesk Mariiidd
Sec 3
Product Coomany
Thm
Digiral Power
Deforned
Power Cholesky
Sec
Thes 3
Dagomii Burc-
Wassmahi Metro
Application
Chiloky Munilsid
SPD Mantisil
Figure
Overview of our theoretical framework. Here, L Cholesky matrix.
observation opens up a principled design space, as any metric on R can induce metric on the Cholesky manifold, and further yield corresponding metric on the SPD manifold via the Cholesky decomposition.
Building this product structure, introduce two Cholesky metrics, the diagonal power metric and the diagonal Sarrs-Wasseerscchn metric. which induce two SPD metrics via the Cholesky decomposi- tion: Power-Cholesky Metric (PCM) and saares-Wasersttenn- Cholesky Metrie (BWCM). Unlike the diagonal logarithm in LCM, our metrics rely on diagonal powers, improving numerical stability by avoiding exponentiation and logarithms. We further define the Cholesky factors of SPD matrices diagonal power deformation, which continuously connects existing and new metrics. As power 0 O, deformed metric converges to LCM, while at offering tunable trade-off. All proposed SPD metrics admit closed form Riemannian operators, including geodesics, logarithmic and exponential maps. parallel transport, as well as gyrovector operators (Ungar) 2022). which extend vector addition and scalar multiplication into manifolds. These operators make our metrics directly applicable to SPD neural networks. In particular, by substi- tuting these operators into prior formulations of Riemannian MLR classifiers (Chen al., (2024c) and residual blocks (Katsman et al.) 2024), we directly obtain SPD MLR classifiers and residual blocks under our metrics. Fig. [ summarizes the overall framework. We validate our metric with experiments on SPD neural networks, numerical stability analyses, and tensor interpolation, showing the effectiveness, efficiency, and robustness of our metrics. summary, our main contributions are:
recovers our proposed metrics, thereby
1. Revealing the underlying simple product structure in the Cholesky manifold;
2. Proposing two Cholesky metrics and their SPD counterparts, PCM and BWCM, which admit fast and stable closed-form operators:
3. Developing SPD classifiers and residual blocks based on our metrics for SPD neural networks.
2 PRELIMINARIES
This section reviews the necessary background. For readability. Sec. B provides summary of notation, while Sec. Cecalls gyrovector spaces and properties of Riemannian isometries.
2.1 PULLBACK METRICS
Definition 2.1 (Pullback metrics). Let M,, be smooth manifolds, g a Riemannian metric on N, and M N' a diffeomorphism. The pullback of by f is defined pointwise ( (Vi, 1/2) (F(P) (V.P))
where P E M, V1, V2 TpM, and P-)) denotes the differential of f at P. The metric f" is called the pullback metric by f. and is Riemannian isometry.
This paper only considers pullbacks by diffeomorphisms. Riemannian isometries generalize bijections in set theory manifolds while preserving their Riemannian structure. On the SPD manifold, pullback metrics are also known as deformed metrics (Thnnwerdas and Pennec 2022
Sec. 3).
2.2
SPD AND CHOLESKY MATRIX GEOMETRIES
SPD geometries. We denote n X SPD matrices as S and real symmetric matrices as St.
S is an open submanifold of the Euclidean space S", known as the SPD manifold (Arsigny et al.
2005). As mentioned in Sec. [ there six popular Riemannian metrics on the SPD manifol:: AIM
(Pennec et al.) 2006), LEM (Arsigny et al. 2005). PEM (Dryden et al. 2010). LCM (Lin) 2019).
BWM (Bhatia al. 2019) and GBWM (Han al. 2023 Note that PEM is the pullback metric of
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the Euclidean Metric (EM) by matrix power Powe(S) = S and scaled by 1/6² (# * 0) Thannerdas and Pennec 2022). while GBWM the pullback metric BWM by (TS) S with S, E S++ (Han et al. 2023). For clarity, we denote PEM and GBWM as 0-EM and M-BWM, respectively. We denote the metric tensors of AIM, LEM, 8-EM, LCM, BWM, and M-BWW as g² gttt ght g² 9 BW and g M.BW, respectively. Given SPD matrices P. Q. M E S+++ and tangent vectors V, W EpST4 in the tangent space at P, the metric tensors are defined as
of (V, = (P- ",,'",
LE (V, W = (mlog (V), mlog. mlog....W......... (W)),
(1)
(V W) 0z (Power, V, Powe,, W),
9P (V, W) = (LXX,, (YJ (L-1X,
gf M-BW (V, W) = (CP.M(V) (V), Wy.
When M is the identity matrix, M-BWM becomes BWM. The following explains the notation.
( is the standard Frobenius inner product.
Chol, mlog, and Pow, represent the Cholesky decomposition, matrix logarithm, and matrix power. Their differential maps at P are Chol,,p, Powa,, and mlog., respectively. Chol,,r(W)) and I,= (P).
X = Chol.,r(V),
is the strictly lower part of square matrix.
X. Y, and L are diagonal matrices with diagonal elements of X, Y. and L.
(V) is the generalized Lyapunov operator: M M. (VC..MMV)PPH PVP When M=1, [V] is reduced to the Lyapunov operator, denoted LP(V).
Cholesky geometries. The Euclidean space
lower triangular matrices is denoted L" Its open subset, whose diagonal elements are all positive, is denoted by C"... The Cholesky space CT
forms a submanifold of L" (Lin) 2019). For a Cholesky matrix I. E C" and tangent vectors X.Y E Tr.Ci the Riemannian metric on the Cholesky manifold, referred to as the diagonal log metric,
gr. (x.) (1x YY) (L-'X,L-'V).
LCM is the pullback metric of DPL by the Cholesky decomposition. As shown by Chen et al. 2024d Thm. III.1.), the diagonal log metric is the pullback metric, by the diagonal log map, of the Euclidean metric over L" which rationalizes our nomenclature.
PRODUCT GEOMETRIES ON THE CHOLESKY
We first unveil the product structure beneath the existing diagonal log metric on the Cholesky manifold. Based on this, we propose two novel Cholesky metrics.
3.1 DISENTANGLING THE CHOLESKY GEOMETRY
We denote n X n diagonal matrices with positive diagonal elements as D1 and n x n strictly lower triangular matrices SL" Then, SL" is Fuclidean space, and TD 2 [R is open submanifold of R Recalling Eq. (3), is defined separately on SL" and ID Besides, D can be identified as the product of copies of IR... The above discussions imply product structure. We denote the standard Euclidean metric over SL as g and define the Riemannian metric on R++ as
R++ (w, w) = vow,
Vp E R++ and v, w E T,R4+.
Then, LT is the product manifold of SL" and copies of R44:
(CT, = (SC", . [R [R++.g* x ... (R++.gR11) R113
(5)
3.2 PRODUCT GEOMETRIES ON THE CHOLESKY
The following definition characterizes the underlying product structure in Eq. 5
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Definition 3.1 (Product geometries). Supposing 9 is Euclidean inner product SL" and {g' hi= are Riemannian metrics on IR. then the weighted product metric g on D is defined as
= g [[X], + Σx=1 o:92., (Xu,, Ya), with L E _ X, Y. E TTL and a; > 0. Here, [X] and [Y] are strictly lower triangular parts of X and Y, while Lis X and Y are the i-th diagonal elements.
For simplicity, we focus on the case where gحh 9" is the standard Euclidean metric, all a, are equal to 1, and all g' are identical. Since +++ can be viewed as a 1-dimensional SPD manifold S the existing Riemannian metrics on the SPD manifold can be immediately used to build Riemannian metries on the Cholesky manifold. Simple computations show that AIM, LEM, and L.M coincide with Eq. 44 on S Therefore, the six metrics introduced in Sec. 2.2 are reduced three classes on S (1) LEM, LCM, or AIM; (2) 0-EM; (3) GBWM and BWM. When the metric on 2 is AIM (LEM or LCM), the resulting product metric on the Cholesky manifold is the diagonal log metric, and the pullback SPD metric via the Cholesky decomposition is exactly LCM.
Inspired by the above analysis, we obtain two new metrics on the Cholesky manifold by setting each g in Def. 3.1 0-EM and GBWM, termed Diagonal Power Metric (A-DPM) and Diagonal Mures-Wasserstein Metric (M-DBWM with M E D ). respectively. By product geometries (Lee) 2018), obtain the closed-form expressions of their Riemannian operators, such geodesIc, logarithm, parallel transport, and weighted Fréchet mean. These operators are significantly important in building concrete learning algorithms (Yuan et al. 2012
Lezcano Casado 2019 Brooks al.
2019 López et al. 2021).
Supposing L,K& E tT X,Y E TrAf4 and {Le E C' with satisfying wa 0 for all i and Lim1 wi 1, the Riemannian operators under 0-DPM with 0 # 0 are
Theorem 3.2 (0-DPM).
weights
(x, ([x), [Y]) 1,0,,,,
(L, [7] + + + 101.- PTz-x(() - [X] (L- 1-0 x
(K) = [K] - [4] + L (L-1K)- -1]
dL) [|LK]- - atz ||K L
WFM({u:); (Li)) = -, u. [. (Σ, wiL
where 11-1lp the Frobenius norm. X. V, K. and are diagonal matrices with diagonal elements from X, Y, L, K, and L4. (t)) denotes the geodesic starting at L, with initial velocity X.
PT PTL-K(()) (-) is the parallel transportation along the geodesic connecting L and K. Log, d, WFM are the Riemannian logarithm, geodesic distance, and weighted Fréchet mean. Note that x is locally defined in (1 E RR|L + 10% D++
Theorem 3.3 (M--DBWT) MI-DBWM with M D are
Following the notation in Thm. 3.2
the Riemannian operators under
sh M--RR (x, Y) = [[X], [ + ( M-
(t)) = [4] + + PTz-x((X) [ + (T. (1.
Togz (K) = [] [T]] + 2T.
[1_
dP(L, K) = [ILA] LIII
|MM (Kt )e
WFM(( (L.)) = Σ, wy w + (Σ, m.LL
where the geodesic XLLx)(1) is locally defined in (1 E RlI + 1 E D When M
I in
MI-DBWM, the resulting metric is denoted as DBWM.
On the SPD manifold, GBWM is locally AIM (Han et al. our M-DBWM is locally the diagonal log metric at E 4 gt-D6W (X,Y) = {[X], [X,,[[]] [Y])
2023).
Similarly, on the Cholesky manifold,
(L- L.-'', Y). Besides, GBWM on the SPD manifold generally has no closed-for form expression for the Fréchet mean (Bhatia al. 2019). Besides, the closed-form expression of parallel transportation under BWM is known only if two SPD matrices commute (Thanwerdas and Pennec 2023). In
contrast, all these operators have closed-form expressions under M-DBWWM on L"
3.3 DEFORMED CHOLESKY METRICS
On SPD manifolds, the deformed metrics by the matrix power can interpolate between given metric and an LEM-like metric (Thanwerdas and Pennec 2022 Sec. 3.1). Inspired by this, we define a diagonal power deformation on the Cholesky manifold. We denote the diagonal power as
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DPows + P - Ipt D++ We will show how our proposed metric is connected to the existing diagonal log metric by DPows.
Definition 3.4. Let ( g} {SL", g (D+ be product metric. We defined the
dingonall-ower-defformeddd metrics g as +. g6) {SC", ""c""."))"""" + कz DPowig):
The following lemma shows that 9" Def. 3.4 tends to be diagonal-log-lite like metric with 0 0.
Lemma 3.5. ! Given L 4 and X, Y T 9 in Def. 3.4 satisfies
9% (X,Y) (LX], + ØL 11--,,1---1
(LX), LY "L,
(6)
Now, we discuss the deformation on diagonal log metric, 0-DPM, and M-DBWM. Firstly, Eq. (6) indicates that the
9-DPM is the diagonal-power- power-deformed metric of diagonal log metric is itself. Secondly, metric of the Euclidean Metric on the Cholesky man- ifold. Besides, 0-DPM interpolates between the diagonal log metric (0 + 0) and Euclidean metric of M-DBWM, referred to as (X, Y) = (Lx YY) 1 (L. MY. When M I, the deformed metric of DBWM, i.e., B-DBWM, tends to be scaled diagonal log metric as 0 + of D.DDW( (X,Y) (X,,DW((,Y) = [[ [Y])
Metric (0 1). Thirdly, the (0.3))DBBW
(0, -D)DBDM,,,,,.. is 0i
(L X, L- YY). As (0, DDBM is the pullback metrie by di- agonal power and scaled by a constant, the Riemannian operators also have closed-fon fored-fommmoo expressions, which are discussed in Sec.
3.4 ALGEERAIC STRUCTURES
The gyrovector space forms the algebraic basis for hyperbolic geometry (Ungar 2005 as the vector space for the Euclidean space. It has recently been extended to matrix manifolds (Nguyen 2022).
Given P and Q in the Riemannian manifold M, the gyroaddition and gyromultiplication Nguyen
and Yang 2023
E and 2) are defined
Binary operation: P P Q = Expp (PTE-r(Log=(()),
Gyromultiplication:: @ P Expe (tLoge()))))
(8)
where is the origin of M. For more details on gyrovector spaces, please refer to Sec. C.I The gyro operations under diagonal log metric have been studied by Nguyen 2022). studies the gyro-structures over 6-DPM and (0, M)-DBWM.
This subsection
Let the identity matrix be the origin and C be 0-DPM or (6, MB-MBWM. We have the following. Lemma 3.6 (Gyro-structures),
For L, K E C++ and E R, the gyro operations are
L Le K= = [L] + [K] + (1 K - 11
t t L -= [| + (tL + (1 - t)]}*
(10)
where B 00 (-DPM, and B = =02 for for (0, -DBBM @e requires I. and to satisfy TLS + K IE D. while of requires (1 t)1 + tIL D++
Note that gyro operations can only be defined under the assumptions in Lem. 3.6 which comes from the locally defined Riemannian exp. In the following, we make these assumptions implicitly.
Theorem 3.7.
{CT,,455 { conforms with all the axioms of gyrocommutative gyrogroups (CT..00 & conforms with all the axioms of gyrovector spaces (Def. C.3
(Defs. C.1 and C.2
Corollary 3.8. The identity element of (C+ @C} is the identity matrix, i.e., VL C -I - 1 for L {L C |2 - E D.
C L L.
The inverses are e L
Remark 3.9. The gyro-structures on the Grassmann manifold have shown success in building Riemannian algorithms (Nguyen) 2022; Nguyen and Yang. 2023). Like our gyro-structure, the gyro-structures of the Grassmann manifold also require some assumptions for well-definedness (Nguyen 2022 Sec. 3.2). Therefore, we also discuss the gyro-structures under our metrics for completeness. In practice, such positivity constraints can be easily remedied by numeric tricks. Taking 2/ L e DT, as an example, one can use d, + max(d;, E) for each diagonal element d, with small constant > 0.
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Table Riemannian and gyro operators of different metrics on the Cholesky manifold. For diagonal log metric, log(-) and exp(-) are diagonal logarithm and exponentiation.
Operatons
Diagosal Log Metrie
4-DPM
( M.M))
sx(X.)
.X| + [X]+ "X-'x---'
[LX). - 1
(LX). (1'""X
70,81(0
[4] + [[X]+ + Lexp((L
[4] +[]+ + L t (3)
[4] + (I
(K)
[K] - [4] Llog(L-13)
[K] [4] + + (L-1x)*
[x] [4] + $L ((-11)
PT (T))
[X] (I- "K)
[x]] + (1 'k)"""x
[X] (L +8)
X)),,K)
[[X] - [4]1B + (og(2)) 1og(2)
- - + ||K - [[K]- L4116 + + | (x3 - 29)
WFM((m)).LL))
m. + Ox. (o)))
My OC, mLg))
L.KK
[1] + +
U++ [K] + (12 + K" -n²
14 K] + (1.3 +:3 1)
LGL
(4) +L'
(LJ++ (1-)1)
ULJ+ (ct (1=+(1-))))))
3.5
NUMERICAL, ADVANTAGES OVER DIAGONAL. LOG METRIC
Tab. [ summarizes all the Riemannian and gyro operators. The Riemannian operators under (0, J-DDBMM and 0-DPM are mostly calculated by the diagonal power function, while the ones under diagonal log metric are computed by the diagonal logarithm or exponentiation. This indicates that our 6-DPM and (0, M)-DBWM could have better numerical stability than the existing diagonal log metric, as logarithm or exponentiation might overly stretch the diagonal elements compared with the power function. The gyro operations under our 6-DPM and (0, MM)DBWM also have numerical advantages over the ones under the diagonal log metric. The former is based on linear operations combined with power and its inverse, causing relatively minor changes to the input magnitude, while the gyro operations under diagonal log metric are based on product or power, resulting in more noticeable alterations to the input magnitude.
4 GEOMETRIES ON THE SPD MANIFOLD
This section discusses the Riemannian metrics on the SPD manifold via the Cholesky decomposition. We first review some basic properties of the Cholesky decomposition, followed by the SPD metrics.
The Cholesky decomposition, denoted as Chol(.) : +S+ + CT is a diffcomorphism (Lin) 2019). Therefore, it can pull back the Riemannian and gyro-structures from the Cholesky man- ifold C to the SPD manifold S We denote the pullback metrics on S of 6-DPM and Power Cholesky Metric (6-PCM) and
(0, MD-DBWM on LT by the Cholesky decomposition Bures-Waan Wasserstein Cholesky Metric ((0, M)-BWCM), Then, the Riemannian operators under 0-PCM and (0, MWM can be obtained by the properties of Riemannian isometry (Sec. C.2)
Differentials. For P E S with Cholesky decomposition P p= LLT, the differential map Lin 2019 at P is Chol,,r(.) ST.ST_ - V + L (V.-V) E TT 4., with (V) [V] + V
Operators. Let C E (0-DPM, (0, M.M-DDDMMMM and S E (6-PCM, (0, BWCM)).... We denote LogS, Exps, 75, PTS dd WFMS, Φs, and O$ as the Riemannian logarithm, exponentiation. geodesic, parallel transportation along the geodesic, geodesic distance, and weighted Fréchet mean on (84,,g)) while Loge, Expe, S PTC, d...), WFM @ and Oc are the counterparts on (CT,,9"), J. For P,Q E S V, W E TrM, and (P. E MIN with weights [w: satisfying w; > 0 for all and ΣN w/ = 1, we have the following w.n.t. the Riemannian and gyro operators:
(t. Chol-1 (3 (2e,,) (t)
Log = (Chola,p)-+ (Logi(
Exp)(v) = Chol-1 [Exp] (v))
PT (Chola.)) ( (V)).
(II)
(P, Q) = df ddLL, K),
WFM ((P2), (wx)) = Chol-t - (WFM (1Lo), (w:)), t 8 Chol 1106 L),
8 Q Chol 'LL ~ K):

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Published as a conference paper at ICLR 2026
where == = LLT, KK", and P. L are Cholesky decompositions. Here, V Chol,,,(V) and Chol = LL Besides, when C' is the diagonal log metric, the above recovers the Riemannian and gyro-structures under LCM.
The above gyro operations, when well-defined, also conform with gyrovector space axioms.
Theorem 4.1.
! (S++, "S conform with all the axioms of gyrovector spaces.
Remark 4.2. As discussed in Sec. 3.5 the Cholesky metric 9-DPM and (0, -B)BBMM are more numerically stable than the existing diagonal log metric. As the pullback metrics by the Cholesky decomposition, (-PCM and (0, JBCMM therefore, preserve the advantage of numeric stability over the existing LCM. Besides, all the Riemannian operators have closed-form expressions and are easy to use, the differential maps of the Cholesky decomposition can be easily calculated.
APPLICATIONS TO SPD NEURAL NETWORKS
As discussed in Sec. 1 Riemannian metries enable the construction of basic building blocks in SPD neural networks. In this section, we apply our proposed 6-DPM and (0, MM)DBBM build Multinomial Logistics Regression (MLR) classifiers and residual blocks the SPD manifold.
MIR. typical classification layer in Euclidean networks MLR, Softmax (Ax + b), which computes the multinomial probability of each class. Given C' classes and input x E IR", MLR can be written by the point-to-hyperplate distance (Lebanon and Lafferty 2004, Sec. 5):
4k E {1,-. (1,...,C), C}, ply = k x)) x exp (sign (mx; pa)) d(x, Harp)))
(12)
where ag. Pk E R", and Har.p {Ir E R as the point-to-hyperplane distance. [Cen et al. (2024c) Eqs. 3-4) extended Eq. (12) into manifolds by Riemannian reinterpretation. Given input X M, the Riemannian MLR
(ak, x p&) 0} is hyperplane with d(z, Hay.p)
py X) x exp (sign((Ag, Logr, (X)) Al,,,,,, d(X, H Pw))
AX M
(13)
where Pk .M and A& Tp, M parameters. Following Chen et al. (2024c Thm. 3.3),
obtain the SPD MLR under our metrics.
Theorem 5.1. (0, BWCM are
( Given an input SPD matrix S E S the C-class SPD MLRs under 0-PCM and
0-PCM: ply = k
1 K SQ+) x exp (LK) - [L...LAA)
20 (KK" Lh, Ane)
1
(14)
(0, M0MM--WCM ply = k| E ST+) x exp (LKJ - LLxJ (Ax)) +
48 ( L 18,M-AAN
1
(15)
where S KK and Ps. = La Li are Cholesky decompositions. The parameters are PR E S and A E LTh for each class ,C.
Residual blocks. Euclidean residual blocks can be written as (x) = (6-11) + (z(i-11), where n; is a network. Katsman al. 2024) generalized this manifolds by replacing addition with the Riemannian exponential map 2
Expati 1) (es (x((-1))), where : M TM outputs vector field parameterized by the neural network. On the SPD manifold, the vector field generated by the eigenvalues. Specifically, the SPD residual block (Katsman et al. [2024] Eqs. 22-23)
Y Expx (Q diag (f(ssec(X))) Q²))
(16)
where X E ShL (X) contains all the eigenvalues. f _R + R is a neural network, and Q is an orthogonal parameter. Here, diag(-) returns diagonal matrix from the input vector. The only component that varies across different metrics is the Riemannian exponentiation. Therefore, we can readily build the residual blocks under our metrics.
EXPERIMENTS
6.1 EXPERIMENTS ON SPD NEURAL NETWORKS
We compare our metrics against the popular AIM, LEM, and LCM on building SPD MLR and residual blocks. Following previous work (Huang and Van Gool 2017 Chen et al. 2024c), adopt
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Table 2: SPD MLRs under different metrics on the SPDNet backbone. The best two results are
highlighted in red and blue.
Radar
(c) RPHA
Metric
Apc
Time
1-Block
2-Block
3-Block
Metric
AIM
94.53
0.95
0.80
Metric
Acc
Time
Aze
Time
AIM
85.57 = 0.50
7.14
LEM
93.55
1.21
0.76
$8,07
18.75
10:23
LEM
85,90 0
0.98
LCM
93.49
1.25
0.72
1JM
173
60.60 1:39
16,97 +061
1.83
02.61 1 1146
0069
2.92
7.40
00:24-009 (191
22.31=211 2.15
3.50
2.40
LCM
86.37 0,59
0.74
8PCM
9 0.38
64CM
1.5
66.78
2.06
6-PCM
6-WCM
93.93 0.79
0.71
WBBWCM
63:71
64.53 9..56
(7.40
2.87
6-BWCM
86.27 = 0.60
0.70
the Radar dataset (Brooks et al., 2019) for radar signal classification, and HDM05 (uüller et al. 2007) & FPHA GGarcia-Hernanl et al., 2018) datasets for human action recognition. For more details on datasets and implementation, please refer to Sec. E
6.1.1 RIEMANNIAN CLASSIFIERS
We compare the SPD MLRs under our metrics with those under AIM, LEM, and LCM (Chen et al.
2024c 4hm.4.2). Following Chen et al. (2024c) Nguyen and Yang 2023), we adopt SPDNet (Huang al. 2017) and GyroSPD (Nguyen and Yang 2023) two backbones, both mimicking feedforward neura networks. For simplicity, we set M in (8, M)-BWCM to the identity matrix.
SPDNet. On HDM05, we further evaluate architectures with up to three transformation blocks. Tab. reports the five-fold accuracy and training time per epoch, from which we draw the following
observations.
Effectiivnnes. Our metrics generally yield higher accuracy than their counterparts. Notably, they outperform LCM, although both originate from the Cholesky product structure. This improvement is attributed to the fact that the diagonal logarithm and exponentiation in LCM tend to overly stretch the diagonal entries, i.e., eigenvalues of the Cholesky factors, whereas our diagonal power transformation achieves a more balanced scaling.
Efficiency. Our metries substantially reduce computational cost compared AIM, remain faster than LEM, and achieve efficiency comparable to LCM. Together with their superior accuracy, these results highlight the dual advantages of our approach in both effectiveness and efficiency.
GyroSPD. Tab. 3 reports the results on the GyroSPD backbone, which consists of a single gyrotranslation layer followed by an SPD MLR. Similar to the SPDNet results, 0-PCM and 0-BWCM consistently outperform LCM across all datasets while maintaining comparable effi- ciency. They also deliver higher ac- curacy with lower runtime than AIM and LEM, particularly AIM.
Table 3: SPD MLRs on the GyroSPD backbone.
Radar
HDM05
FPHA
Metric
Acc
Time
Acc
Time
Acc
Time
AIM
96.80
0.59
1.23
66.05 + 1.80
21.65
11.48
LEM
96.58 0.27
1.18
66.42
2,02
85.87
0.79
1.22
LCM
96.29 0.53
1.12
68.37 1: 0.66
1.66
89.83 2 0,28
0.98
0.PCM
97.04 a 0.64
1.18
71.93 1.21
1.51
91.17 a 0.30
1.00
0-BWCM
96.21 a 0.25
1.05
72.74 a: 0.43
1.58
91.00 a 0.11
0.96
Deformation. In Sec. E.3 we perform a systematic ablation on the deformation factor 0 in 0-PCM and 0- 9-BWMM, sweeping from 2 to 1.5 on both SPDNet and GyroSPD. The results show that, across all datasets, suitably chosen consistently improves accuracy over the default 1, confirming the effectiveness of our diagonal deformation. The gains are modest on Radar and FPHA but much larger on HDM05. Sec.F..4r 4.44 further attributes this dataset dependence to the Cholesky diagonal statistics of each dataset. Radar and FPHA have relatively balanced Cholesky diagonals, whereas HDM05 has highly imbalanced diagonal entrics. In this case, adjusting 0 more effectively activates these under-representedd diagonal entries HDM05.
Scalability. Sec. E.5 evaluates the scalability of our metrics through SPD MLR. The synthetic experiments show that metries the most efficient ones. They perform par with LCM at small and medium dimensions and become clearly faster in hign-dimensional settings, while SVD-based metrics such AIM and BWM become prohibitively slow.
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Table 5: Failure probabilities (%) of geodesics under different metrics with small eigenvalues in L E +. An output matrix containing any INF or NAN is considered failure. Here, DLM denotes the diagonal log metric, while DPM and DBWM denote B-DPM and 8-DBWM, respectively.
for small
256 for large matriees
1.5
0.5
0.15
0
1.5
0.5
0.15
DLM
DPM
DBWM
DPM
DBWM
DPM
DBWM
DLM
DPM
DBWM
DPM
DBWM
DPM
DBWM
0.62
le 5.70
0
18.48
0
0
0
0
0
le-1
94.34
95.02
0
0
1em
99.39
99.17
0
0
1e 100
100
0
1e-15
100
100
0
O
1e 10-
100
0.02
1e-21
100
0
0.05
100
0
0
0.01
12-
100
0.25
100
00
0
0
0.23
le
100
2.26
100
0
2,42
1e-24
100
22.98
100
(1
0
23.13
1e-20
86.58
1e-30
100
100
100
0
100 0
6.1.2 RIEMANNIAN RESIDUAL BLOCKS
The backhone architecture of Riemannian ResNet (RResNet) (Katsman et al.] 2024) largely follows SPDNet. The key difference lies in the head: while SPDNet directly applies clas- sification layer, RResNet inserts residual block before the classification head. Following Katsman et al. (024)), we adopt Log, + FC + Softmax for classification under each metric. Since the Riemannian exponential similar for 9-PCM and (0, MMM-MBBMMMM. we focus on 0-PCM and compare it against AIM, LEM, and LCM in constructing RResNet. Tab. 3 reports the best results across three trials, showing that our metric consistently achieves superior accu- racy while maintaining comparable efficiency.
Table 4: Residual blocks under different metrics.
Radar
HDM05
FPHA
Metric
Acc
Time
Ace
Time
Acc
Time
AIM
96.4
1.02
$7.01
1.14
87.33
0.72
LEM
97.07
0.81
67.52
0.52
86.17
0.32
LCM
97.07
0.85
66.27
0.63
86.83
0.49
6.2 NUMERICAL EXPERIMENTS
0-PCM
97.87
0.85
68.05
0.63
88.33
0.48
Numerical stability. As discussed by Lin 2019
p. 16), LCM is more stable than AIM and LEM owing to the numerical advantage of Cholesky decomposition over SVD. Moreover, as highlighted in Rmk. 4.2 the essential distinction between our metrics and LCM lies the diagonal operations: ours rely on diagonal power, while LCM employs diagonal exponentiation and logarithm. This structural difference grants metrics stronger numerical stability and robustness compared with LCM, as well as AIM and LEM. To validate this, evaluate geodesics the Cholesky manifold. We generate 100,000 synthetic matrices L and tangent vectors X E C", where each entry is uniformly sampled from [0, 1], and we set the smallest eigenvalue (diagonal entry) of T. to c. We test representative sizes: x 3 matrices, commonly used diffusion tensor imaging ( (Arsigny et al. 2007). and 256 x 256 matrices, typical in computer vision (Li et 2018 Wang al. | 2023). The deformation parameter 0 set to 1.5, 0.5, and 0.15. For (0, M. MD-DBWW WM, we set MI = I. As shown in Tab. our 0-DPM and (-DBWM remain highly stable across wide range of e. For 3 3 matrices, the diagonal log metric already deteriorates at c = 1e with failure rates increasing rapidly as c decreases. For 256 x 256 matrices, instability emerges even earlier at le In contrast, our metrics remain stable down cases. The only exception occurs when 0 0.15, where failures appear around = 1e -20 This behavior expected since both 0-DPM and (-DBWM converge to the diagonal log metric as - O, thereby inheriting its instability in this limit. Overall, these results demonstrate the superior numerical robustness of our metrics.
Cholesky
1e -30 in most
Tensor interpolation. Geodesic interpolation of SPD matrices is widely used in applications such as diffusion tensor imaging (Arsigny et al.] 2007). Experiments in Sec. E.2 show that our 8-PCM mitigates the swelling effect in SPD interpolation observed under Power-Euclidean and Bures- Wasserstein metrics, while producing interpolation patterns visually similar to LCM, thereby retaining the practical potential of LCM.

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7 CONCLUSION
We identify the product structure in the Cholesky manifold and propose two new Cholesky geometries. Through Cholesky decomposition, these yield two fast and stable Riemannian metrics the SPD manifold, 0-PCM and (0, -BCMWCM. We provide a comprehensive analysis of their Riemannian and algebraic properties. Like LCM, our metries admit efficient closed-form formulas for key operators including geodesics, logarithmic and exponential maps, weighted Fréchet mean, parallel transport, and gyro operations. In addition to efficiency, they offer stronger numerical stability than existing approaches. Extensive experiments on Riemannian classifiers, residual networks, geodesic stability, and tensor interpolation confirm their effectiveness. Together, these results establish our metries as robust and practical alternatives to existing metrics for SPD matrix learning.
REPRODUCIBILITY STATEMENT
All theoretical results are established under explicit assumptions, with complete proof's in Sec. Experimental details are given for SPD neural networks (Sec. E. geodesic stability (Sec. [6.]] and tensor interpolation (Sec. E.2
The code will be released upon acceptance.
ETHICS STATEMENT
This work uses only publicly available benchmark datasets, which contain no personally identifiable or sensitive information. We do not identify any ethical concerns.
ACKNOWLEDGMENTS
This work was supported by EU Horizon project ELLIOT (No. 101214398) and by the FIS project GUIDANCE (No. FIS2023-03251), We acknowledge CINECA for awarding high-performance computing resources under the ISCRA initiative, and the EuroHPC Joint Undertaking for granting access to Leonardo at CINECA, Italy.
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APPENDIX CONTENTS
List of acronyms
15
Use of large language models
15
Summary of notation
16
Preliminaries
16
Gyrovector spaces
16
Riemannian isometries
17
Riemannian operators under (0, M) -BWM
17
Experimental details
17
Implementation details of SPD neural networks
17
E.I.1
Backbone networks)
17
E.1.2
Datasets and preprocessing
18
E.1.
Details of the experiments on SPD MLR
18
E.1.4
Details of experiments on ReesNet
19
Tensor interpolation
19
Ablations on Cholesky power deformation
Cholesky power deformation and diagonal statisties
21
E.5
Scalability
22
E.5.1
Asymptotic complexity
22
E.5.2
Empirical validation
23
E.6
Effect of M in BWCM
24
Hardware
24
Proofs
24
Proof of 3.2
24
Proof of Thm.
3.3
25
Proof of Lem.
27
Proof of Lem. 3.6
27
Proof of Thm.
28
Proof of Thm. 4.
29
Proof of Thm. 5.
29
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Table 7: Summary of notation.
Notation
Explanation
(M.g)
Riemannian manifold M with Riemannian metric g
(.,p(.) f'g (ogg()
Differential map of f at P Pullback metric by from Riemannian logarithm at P Riemannian exponentiatin P Parallel transportation along the geodesic connecting P and Q
Expr()
PT PTP-(()
(P))))
Geodesic starting at P with initial velocity V E 7'MM
d(..))
Geodesic distance
WFM(.)
Weighted Fréchet mean
S
SPD manifold of n n SPD matrices
S
Fuclidean space of n. n symmetric matrices
Euclidean space of n n. lower triangular matrices
Sc
Euclidean space of n n strictly lower triangular matrices
D
Cholesky manifold of n x n Cholesky matrices
IV R
Set of n x n diagonal matrices with positive diagonal
Real number field
Space of positive scalars
R"
n-dimensional real vector space
R"
Space n-dimensional positive vectors
Gyroaddition
Scalar gyromultiplication
TpS"
Tangent space at P E S Tangent space L C Tangent space
Ty
TEDe
D
Powa(-) or (-)
Matrix power Matrix logarithm
mlog()
Choll
Cholesky decomposition
Cra(()
Generalized Lyapunov operator
a(-)
Isometry pulling BWM back to GBWM
[-]
Strietly lower triangular part of square matrix
D-oow(+)
Diagonal power function
log(-)
Diagonal logarithu
exp(-)
Diagonal exponentiation
ge or g&t, gAl g gle gow gid gDL go DE g( My DOw g g gl (B.R).DDR
Standard Frobenius inner product
LEM, AIM, PFM, L.M,, BWM, GBWM
Diagonal log metric. 0-DPM, (0. MD-M 0-PCM, (0, MM-MBCM
LIST OF ACRONYMS
FC MLR
Fully Connected
1 Multinomial Logistics Regression
AIM
Affine-Invariant Metric 1
BWCM
Wasse-Wassesstein. -hoolesky Metric 2
BWM
Busss-Wassrsteiin Metric
GBWM
Generalized Bures- Wasserstein Metric
LCM
Log-Cholesky Metric
LEM
Log-Euce -uclideld Metric
PCM
Power-Cholesky Metric
PEM
Power-Euuliiean Metric
SPD
Symmetric Positive Definite
A USE OF LARGE LANGUAGE MODELS
Large Language Models (LLMs) were used primarily for language polishing and text editing. In limited cases, they also assisted in translating certain mathematical formulations into PyTorch code.
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All generated outputs were carefully reviewed and, where necessary, corrected by the authors. The authors take full responsibility for the final content of this paper.
B SUMMARY OF NOTATION
Tab. summarizes the notation used in the main paper.
C PRELIMINARIES
C.I GYROVECTOR SPACES
The gyrovector space extends the vector space in Euclidean geometry and serves the algebraic basis for manifolds (Ungar 2022), which have shown great success in various applications (Ganea et al. 2018: Skopek al., 2020 Gao al. 2023 Nguyen and Yang 2023). We first recap gyrogroups
and gyrocommutative gyrogroups (Ungar 2022). and proceed with gyrovector spaces (Chen et al.
2025c).
Definition C.1 (Gyrogroups (Ungar 2022 ) Given a nonempty set with binary operation @: G - G, (G, @) forms a gyrogroup if its binary operation satisfies the following axioms for any a, G
(G1) There is at least one element E G called left identity such that @
(G2) There an element @a G called a left inverse of a such that Oo @ a 6.
(G3) There is an automorphism gyrla, bj G + G for each b E G such that @ (b @ c) (a The automorphism gyr|a, b] is called the gyroautomorphism, or the gyration of G generated by a, (G4) gyr(a, bj gyrlu b, bj (Left Reduction Property).
@ gyr(a, bJc (Left Gyroassociative Law).
(17)
Definition C.2 (Gymocomuutativ gyrogroups (Ungar 2022 A gyrogroup {G, (G) is gyrocommu- tative if it satisfies
+ b gyrla, (b a)
(Gyrocommutativee Law).
(18)
Definition C.3 (Gyrovector spaces (Chen al. 2025ch). A gyrocommutative gyrogroup {G, 사} equipped with scalar multiplication o R x G - G is called gyrovector space if it satisfies the following axioms for s, and a, E G:
(V1) 110a=a
(V2)
(V3) (st) ① a 800=0
(V4) b| c) 1 gyr(a, bjc.
(V5) gyr(y ④ 0, ④ 0) = Id, where Id is the identity map.
Remark C.4. Nguyen (2022) presented similar definition, except that (V1) is defined O Ø Rmk. 5),
x, 1 r = 13 e, and
0 = 1 5 (-1) (-1) G = ex. However, as discussed by Chen al. (2025c Oz are redundant.
Gyrovector structures were first identified in hyperbolic geometry (Ungar 2022) and have recently been extended to matrix manifolds, such as SPD and Grassmann manifolds (Nguyen 2022). Given P and Q in the Riemannian manifold M, the gyro operations (Nguyen and Yang 2023 Eqs. and 2) are defined as
Gyroaddition: P e Q = Expp (PT (PTE-r((og,:()),
(19)
Gyromultiplication: ④ P Expg (LLogg())))
(20)
where E is the origin M, Exp and Log denotes the Riemannian exponential and logarithmic
maps, and PT is the parallel transportation along the geodesic connecting E and P. If Eqs.
(19)
and (20) conforms to the axioms the gyrovector space, (M, C) forms a gyrovector space. On
the SPD manifold, the gyro operations under AIM, LEM, and LCM construct gyrovector spaces
(Nguyen 2022). The gyro-structures on the Grassmannian (Nguyen 2022) and hyperbolic (Chen
et al. 2022c) are also defined by Eqs. (19) and 20
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C.2 RIEMANNIAN ISOMETRIES
This subsection reviews some basic properties of Riemannian isometries. For a more in-depth discussion, please refer Do Carmo and Flaherty Francis (1992); Gallier and Quaintance 2020
Let {M, g) and [W.g) be two Riemannian manifolds, and Φ .M- + W be a Riemannian isometry, i.e.g = o'g. We denote Log, Exp. 7, PT, d(:-), and WFM are the Riemannian logarithm. exponentiation, geodesic, parallel transportation along the geodesic, geodesic distance, and weighted Fréchet mean on {M, g). while Log, Exp, 7. PT, a)..)), and WFM are the counterparts on {NV g) For P, Q E M, V. W E M,,, and (P. E MM with weights {w;
} satisfying wi > 0 for all i
and -i=1 Wi:- 1, we have the following:
N
(().)()) - (()6))))).)())((()),
(21)
Logr(Q)= (d),p) (ossrr)) (0()))
(22)
Exp =6- ((XP
(23)
PTP-QQ (V) )-- (0-.)) (PTOP))))))) ((),)())).
(24)
d(P,Q) = ((((P))((Q))
(25)
WFM(( (w:)) Φ-1 (WTM ((o(P))) (w::))
(26)
where VP,.v (t) is the geodesic starting at with initial velocity V E TpM, and p., TpM +
TopPyN is the differential map of ø at P. Eq. (26) is the direct corollary of Eq. (25).
D RIEMANNIAN OPERATORS UNDER (0, MMM-BM
We first define a map @g : C++ c as
Φo(L) = [L] + L 7L E c++
(27)
Its differential at L E 4 is given as
Ø L (X)= - [X] + A1 8-1 1x, VX E T
(28)
Let g M-DBW and g (R. be M-DBWM and (0, MD.D respectively. Since constant scaling of a Riemannian metric preserves the Christoffel symbols, the Riemannian operators such as Rie- mannian logarithm, exponentiation, and parallel transportation under g (0,3)).DB is the same as the pullback metric oggght DBW Following Sec. C.2, these Riemannian operators under (0, -D)DBBM can be obtained by M-DWW Besides, as constant scaling does not affect WFM, the WFM under as the one under M-DBW The latter can be calculated by the properties
(0, M)-DBWM is the of isometries presented in Sec. C.2 Therefore, by the properties of isometry in Sec. C.2 and Eqs. (27 and 28 we can obtain all the Riemannian operators.
E EXPERIMENTAL DETAILS
E. IMPLEMENTATION DETAILS OF SPD NEURAL NETWORKS
E. BACKBONE NETWORKS
SPDNet (Huang et al. 2017) consists of three basic building blocks:
BiMap: Sk W 5* W
ReEig: S = 1 max(=== cI, .CI,)U
(29)
LogEig: s = mlog(55-1)
where 5k-1 - is the eigendecomposition, and W is sollmn-wiseele orthogonal.
BiMap (Bilinear Mapping) and ReHig (Figenvalue Rectification) the counterparts of linear and
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ReLu nonlinear activation functions in Euclidean networks. LogEig layer projects SPD matrices into the tangent space at the identity matrix for classification. However, LogEig might distort the innate
geometry of SPD features. Recently, Nguyen and Yang (2023); Chen et al. (2024a c) generalized the
Euclidean MLR to SPD manifolds for intrinsic classification.
GyroSPD (Nguyen and Yang 2023) substitutes the BiMap layer in SPDNet with gyrotranslation:
S W e S where W Is an SPD matrix parameter. Following Nguyen and Yang (2023), ) we
use the AlM-based gyrotranslation:
Po P PAQP!,
VP,Q S
(30)
RResNet (Katsman et al. 2024) on the SPD manifold follows the same architecture as SPDNet, except that it inserts an SPD residual block before the LogEig layer.
E.1.2 DATASETS AND PREPROCESSING
The Raday" Brooks et al.
2019 dataset consists of 3,000 synthetic radar signals. Following the
protocol in Brooks et al. (2019). each signal is split into windows of length 20, resulting in 3,000
SPD covariance matrices of 20 20 equally distributed in 3 classes
The HDMO Muüller et al. 2007 dataset contains 2,273 skeleton-based motion capture sequences executed by various actors. Each frame consists of 3D coordinates of 31 joints of the subjects, and each sequence can be modeled by 93 x 93 covariance matrix. Following Brooks et al. (2019), we trim the dataset down to 2086 sequences scattered throughout 117 classes by removing some under-represented classes.
The FPHA GGariia-errandee et al. 2018) includes 1,175 skeleton- first-person hand gesture videos of 45 different categories with 600 clips for training and 575 for testing. Each frame contains the 3D coordinates of hand joints.
E.1.3 DETAILS OF THE EXPERIMENTS ON SPD MLR
We follow the official PyToorc code of SPD MLR (Chen al. 2024a) for implementation. Due the lack of an official code, the GyroSPD backbone carefully reimplemented in PyTorch following the original paper (Nguyen and Yang 2023). For simplicity, set M in (0, MMCC as the identity matrix.
SPDNet (Huang and Van Gool 2017). We replace the vanilla tangent classifier (LogEig + FC + Softmax) in SPDNet with SPD MLRs induced by AIM, LEM, LCM, 0-PCM, and 0-BWCM. We use Riemannian AMSGrad (Bécigneul and Ganea 2019) with learning rate of 1e batch size of 30, and a maximum of 200 epochs. The architectures are denoted as [do, di...., .......,,].. where d, is the output dimension of the i-th BiMap layer. Following prior work (Huang and Van Gool 2017 Brooks et al. 2019 Chen et al. 2024b), we adopt [20, 16, 8] on Radar, (63 33) on FPHA, and for 1-, 2-, and 3-block variants on HDMO5. As suggested by Chen et al. 2024b), matrix power improves the performance of Cholesky-based metric on FPHA.
93, 30J. 193, 30, [93, 50, 30
We therefore apply power -0.25 before SPD MLR layers under LCM, 0-PCM, and O-BWCM.
For (-PCM FPHA, we further adopt weight decay of le The deformation factor is reported
in Tab. g
Following Nguyen and Yang (2023), the backbone consists of one gyrotranslation layer followed by an SPD MLR. We compare LEM-, LCM-, AIM- and our metrics under the same settings: Riemannian AMSGrad with learning rate of 1e and batch size of 30. The training epochs are up to 100, 100, and 50, respectively. Similarly, a matrix power of 0.25 applied on FPHA for LCM and Cholesky-based metrics. The deformation factor 0 is reported in Tab. 8
GyroSPD Nguyen and Yang 2023).
SPD input of SPDNet. Following Chen (202cc App. G. 1.2), we model each sequence into global covariance of 20 x 20, 93 x 93, and 63 X 63.
https://qithub..c httaaa///itbbbb.ccoom///chhhhhha
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Table 8: Hyperparameter 0 in 0-PCM and 9-BWCM. is selected from Tabs.
10 and
11
Backbone
Metric
Radar
HDM05
FPHA
#-PCM
-1.5
-0.5
0.75
SPDNet
0-BWCM
-0.75
-1.5
0-PCM
-0.75
-0.75
-0.75
GyroSPD
0-BWCM
-1.5
-1.5
-0.5
SPD input of GyroSPD. For Radar, the input is the same SPDNet. For HDM05 and FPHA, we follow Nguyen and Yang 2023) to model each sample into a multi-channel covariance tensor [c, n, n]- Specifically, we first identify the closest left (right) neighbor of every joint based on their distance to the hip (wrist) joint, and then combine the 3D coordinates of each joint and those its left (right) neighbor create feature vector for the joint, For given frame compute its Gaussian embedding (Lovrić et al. 2000):
Ex + me (1a)² (14)
Y, (det Mx) nt
1t
(31)
where Ar and Σ, the mean vector and covariance matrix computed from the set of feature vectors within the frame. The lower part of the matrix log (Y,) is flattened to obtain a vector Ür All vectors within time window [1, 1 c- 1], where e is determined from temporal pyramid representation of the sequence (the number of temporal pyramids is set to 2 in our experiments), are used to compute a covariance matrix as
1 M
Σ = Σ (o: - (0) (0: vz)²
(32)
where D, = - 2 + 6. The resulting (E,) are the covariance matrices that need. On FPHA, generate the covariance based on three sets of neighbors: left, right, and vertical (bottom) neighbors. After preprocessing, the input correlation matrices [3, 28, 28), and [9,28, 28] on HDM05 and FPHA, respectively.
E.1.4 DETAILS OF THE EXPERIMENTS ON RRESNET
Implementation. The backbone architectures of RResNet (Ktsman al. 2024 similar to those of SPDNet: [20, 16, 8) on Radar, [93, 30) on HDM05, and [63, 33| on FPHA, respectively. The classification layer, whereas RResNet attaches a residual block before the classification head. Following RResNet, we adopt Log, + FC Softmax for classification under each metric. We use the official code to re-implement the AIM- and LEM-based RResNet. For the RResNets based on LCM and our 8-PCM, we adopt the following settings: a learning rate of 1e 2 batch size of 30, and maximum of 200 epochs. We use the standard cross-entropy loss the training objective and optimize the parameters with the Riemannian AMSGrad optimizer (Bécigneul and Ganea 2019). The Cholesky diagonal power is set to 1, 0.5, and -0.5 on the Radar, HDM05, and FPHA datasets, respectively. Additionally, on the FPHA dataset, a matrix power -0.25 is applied before the residual blocks to activate the latent geometry for both LCM and our metric.
only architectural difference lies in the head: SPDNet directly
SPD input. The input SPD matrices are the same as those used in SPDNet.
E.2
TENSOR INTERPOLATION
As shown by Arsigny et al. 2005 2007 geodesic interpolation of SPD matrices is important in diffusion tensor imaging. This subsection illustrates the geodesic interpolation under different SPD metrics.
https://github..ce emanni dual Neura Networks
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Figure 2: Geodesic interpolation of SPD matrices under different Riemannian metrics. Each 3 x 3 SPD matrix can be visualized as an ellipse (Arsigny al. 2005). The two endpoints are fixed across all metrics.
Table 9: Swelling effects of geodesic SPD interpolations. Deeper greens indicate greater swelling.
The determinant of i-th interpolation
Metric
4
7
9
L.-EM
3.07
104.86
182.09
234.38
261.35
262.64
237.86
186 64
108.61
3.38
0.5-EM
3.07
18.67
39.93
59.53
71.96
73.79
64.14
45.01
21.73
3.38
0.1-EM
3.07
4.25
5,42
6.38
6.96
7.05
6.62
5.75
4.6
3.38
LEM
3.07
3.1
3.14
3.17
3.2
3.24
3.27
3.31
3.34
3.38
AIM
3.07
3.1
3.14
3.17
3.2
3.24
3.27
3.31
3.34
3.38
BWM
3.07
15.32
32.04
48.14
59.22
62.09
55.33
39.93
19.98
3.38
LCM
3.07
3.1
3.14
3.17
3.2
3.24
3.27
3.31
3.34
3.38
0.I-PCM
3.07
3.15
3.23
3.29
3.34
3.37
3.39
3.4
3.4
3.38
0.5-PCM
3.07
3.35
3.59
3.79
3.91
3.97
3.94
3.83
3.64
3.38
1.0-PCM
3.07
3.6
4.07
4.46
4.72
4.83
4.76
4.49
4.03
3.38
Suppose P P= LL and -KT as the Cholesky decomposition of P,Q E S the geodesics that connect P and Q under 0-PCM and (0, M.)-BWCM
(L t(K - LD)) #1
8-PCM: Chol
L
+ +t[[
[
[L])
(33)
t(k3 L8) 8
(0, MBMM) Chol
[L] + +[[[
[L])
L
(34)
As the geodesics under 0-PCM and (0, -BMM have similar expression, we focus on 0-PCM. visualizes the geodesic interpolations on S under different metrics, including 0-EM, LEM, AIM. BWM, LCM, and 0-PCM. 9abb/preeeentt the associated determinant of each interpolated SPD matrix. We can make the following observation.
Fig. 2
1. The standard Euclidean metric (1-EM) exhibits significant swelling effect, where the maximal determinant of interpretation is extremely larger than the determinants of the starting and end points. Although matrix power mitigate the swelling effect, 0-EM still suffers from swelling.
2. We find that BWM also demonstrates a clear swelling effect. In contrast, LCM, AIM, and LEM show swelling effect.
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Table 10: Results of SPD MLR on the SPDNet backbone under 0-PCM and 0-BWCM with different deformation factors 0.
Rador
ABBWCM
6.PCM
66.7
91.90 6.331
92.91 LB5
95.7
036
1.10
1:3
9499
1..79
1.01
94.51 =0.75
9016::139 1.30
+1.75
66 00.5
8
02:68:0000
93.87 0.40
91.77
1.0.6
10.75 0
0..52
HDM25
6-WWCM
64.20-7601
67.40 (L90
63.32 136
68.29
2.33
6
2.90
6467
15.1
0.75
63:31
68:001
PMIA
6.BWCM
6-RCM
86:20 6:22
0.66
85.674006
68.3020.4
15.09 2007
959
0..30
59.08 4.22
2.87
88.97
96.00
2015
29.00
20.75
89003 2..16
0.:33:0033
Table 1I: Results of SPD MLR on the GyroSPD backbone under 0-PCM and 0-BWCM with different deformation factors 0.
Dutuett
Meuic
-1.5
4.75
40.5
.0.25
0:25
0.5
0.75
I-WWCM
96 16 0.90
96.31
$5.64
0.44
95.00
95.94
94 0.
45.55
8-PCM
0..98
25
6:24
1.1
111
69.44 0668
07.08
6 0.28
56.02 11.0
67 1
1.69
181M91
R33WCM
73.64
1.34
09.34:1.14
0434
68.10=1.00
07.72
07.72
07.45:0.55
6.PC3
0.55
91.72003
912
91.F200300
9:0020030
2012
90.97 20.16
90.9
#012
20.7
00:00:005
98.97 2:221
6-BWCM
90.87+80.16
90.80-000
$0.97 8.10
VLJB
8L11
9017-0013
46,07
8.13
90.91
90 9B 008
46.07 +0113
3. The trivial PCM (0 1) considerably mitigates the swelling effect compared to the Euclidean metric, but it still exhibits some level of swelling. However, by introducing Cholesky power deformation, 9-PCM effectively reduces the swelling effect. Notably, the swelling effect of 6-PCM is significantly weaker than that of (-EM under the same 0.
4. Our PCM shows visually similar interpolation as L.C., This suggests that our metric retains some practical potential of LCM but with better numerical stability (as demonstrated in Sec. 6.2
E,3 ABLATIONS ON CHOLESKY POWER DEFORMATION
To better understand the Cholesky power deformation factor 0, we sweep & from -2 to 1.5 for both 0-PCM and 6-BWCM. We focus on the MLR experiments. For SPDNet, we reuse the architec- tures described in Sec. E. 1.3 and adopt the 3-block variant on HDM05. For GyroSPD, we follow the backbone in Sec. E. with one gyrotranslation layer and an SPD MLR. All other training hyperparameters are kept fixed.
Tabs. 10 and II summariie the 5-fold results on SPDNet and GyroSPD backbones, respectively. We make the following observations.
Effectiveness. On both backbones, most tested values of 0 already match or surpass the AIM, LEM, and LCM baselines, and suitably chosen consistently improves accuracy over the default 0 = 1. confirming the effectiveness of our deformation factor. Besides, the magnitude of this improvement is dataset dependent. The gains are modest on Radar and FPHA, whereas HDM05 benefits much more from tuning 9. Sec. EE.4] relates this behaviour to the statisties of Cholesky diagonal entries of the input SPD matrices on each dataset.
Relative stability of BWCM. On HDMO5, when 0 takes small negative values (for example, = -1 1.5 or 0 = 2)), the accuracy of PCM deteriorates, whereas BWCM remains comparatively stable. This behaviour is consistent with Thm. [5.1] Since the SPD MLRs under PCM and BWCM differ mainly in the diagonal powers (.)0 versus (-)0/2, BWCM experiences a milder deformation
for the same 0.
E.4 CHOLESKY POWER DEFORMATION AND DIAGONAL STATISTICS
According to Thm. 5.1 the deformation factor 0 influences the SPD MLR only through the diagonal powers of the Cholesky factors. Therefore, the effect of f is inherently tied to the distribution of the Cholesky diagonal entries. To further interpret the detaset-dependent sensitivity to 0 and, in particular, the preference for negative values HDM05, we analyse the diagonal distribution in the Cholesky domain. For each dataset, we compute the Cholesky factors of the input and output SPD matrices in SPDNet and collect their diagonal entries. For HDM05, focus the 3-block architecture. Fig. reports the resulting distributions, from which we make the following observations.
Balanced diagonals on Radar and FPHA On Radar and FPHA, both the input and output Cholesky diagonals are relatively balanced, with most entries lying in moderate range. In
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Racar (Inроt)
Radar (Outpul)
2.0
1.0
0.5
0.5
11
15
Diagoral
Diggonal Index
HDMOS (Inpul)
HDMOS (Qutput)
1.0
-
0.5
Diagoral
Diagonal Indix
FPHA (Inpul)
FPHA (Oulput)
0.20
3 0.10
0.10
0.05
0.06
100
030
AF
10
000
3
30
Diagoral Index
Dagonal Indox
Figure 3: Distributions of the Cholesky diagonal entries of the SPDNet input and output covariances on Radar, HDM05, and FPHA.
this case, applying power deformation with different 9 mainly rescales the diagonals without drastically changing their relative order. This explains the results in Tabs. [0 and [1 performance is relatively stable across wide range of 0 these Imbalanced diagonals on HDMO5. In contrast, HDM05 exhibits a highly skewed distribution for both input and output covariances, where most Cholesky diagonal entries very small and concentrated close to zero, while only a few entries are much larger. Therefore, varying 0 induces much more pronounced reweighting. Moreover, a negative effectively acts as an activation: taking negative power turns these small diagonal entries into larger values while relatively shrinking those larger than 1. As result, many previously under-represented directions in the Cholesky spectrum are emphasised, which accounts for the noticeably larger gains of negative 0 in Tabs. TO and
where the
datasets.
E.5 SCALABILITY
In this subsection, we investigate the scalability of our PCM and BWCM, comparison with five existing SPD metrics, namely AIM, LEM, LCM, PEM, and BWM. We use SPD MLR representative application. We first analyze the asymptotical complexity of each SPD MLR. We then complement this analysis with synthetic experiments that measure the actual runtime of single SPD MLR training step across different matrix dimensions.
E.5.1 ASYMPTOTIC COMPLEXITY
As shown by Chen et al. (202cc Thm. 4.2) and our Thm. 5.1 the C'-elass SPD MLRs under different metrics for the input ST,
[og
1p-ssp )).A).))) )A
AIM: ply k S) x exp
(35)
LEM: p④y = k S) exp (log((S) log(Pe), As)]:
(36)
[>.] + log(K)) log(Le), [Ac] + 1 _/xe)
LCM: ply k () exp
[K]
(37)
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Table 12: Number of matrix functions required per sample for C-class SPD MLR. Spectral matrix functions include matrix logarithm, matrix power, and the Lyapunov operator.
Metric
Num. spectral matrix functions
Num. Cholesky decomposition
AIM
+
2C
LEM
1+C
LCM
0
1+C
PEM
1+C
BWM
3C
C
PCM
1+C
BWCM
14C
Table 13: Asymptotic per-sample complexity of C-class SPD MLR for an n x n input SPD matrix.
Metric
Asymptotic complexity
AIM
( + 2
LEM
((9(++))n))
LCM
0 ((4+
PEM
((9(++))n)
BWM
((9(1 3C) O 3 1 0 3
)n
PCM
BWCM
1+C
5 Ph A1)
PEM: ply= = k 5) x exp
(38)
0
BWM: ply = k 9) x exp
1
(Pr5)) + (SP))
2P,, Lr. (Lx (Lx
(39)
2
0-PCM: ply = k 5) (LKJ [Le,, [Ax +
1
K LLe, kxx
(40)
20
().M) ply = k () exp
(LKJ [L,), [Ax)) + 40
k L M- As
(41)
where Pk ST and A₂ E S are MIR weights, log(.) is the matrix logarithm, and Cr(V) is the solution to the matrix linear system [V + PLp V, known the Lyapunov operator.
Analysis. Tab. 12 summarizes the number of spectral and Cholesky matrix functions required by each SPD MLR. Cholesky decomposition requires ((/3n))) flops. while eigendecomposition costs O(9n²) flops (Golub and Van Loan 2013 Algs. 4.2.3 and 8.3.3). Combining these counts, Tab. 13 reports the resulting asymptotic per-sample complexity for each metric. Cholesky-basee metrics (LCM, PCM, BWCM) asymptotically more efficient than the eigen-based metrics (LEM, PEM, AIM, and BWM), with AIM and W being the slowest among the considered methods. addition, PCM and BWCM be practically more efficient than LCM, since diagonal powers cheaper to compute than diagonal logarithms.
E.5.2 EMPIRICAL VALIDATION
Setup. To validate the asymptotic complexity in Tab. 13 we measure the average wall-clock time of a single forward-backward training step an SPD MLR classifier as the matrix dimension increases. The model consists of single SPD MLR layer with 50 output classes followed by cross-entropy loss. For each dimension n (32, 64, 128, 256, 512]. we randomly generate a batch of 30 n n forward and one backward pass and record the total
SPD matrices. In each run, perform runtime of this step. For PEM, we set the matrix power to 0.5.
Results. As reported in Tab. 14 our PCM and BWCM the fastest metrics across all tested dimensions, and the gap becomes particularly pronounced in the sigh-dimessional case. For small and medium scales (32 and 64), LCM, PCM, and BWCM have very similar runtimes and all are
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Table 14: Average runtime (in seconds) of one SPD MLR training step under different dimensions.
Dim
AIM
LEM
L.M
PEM
BWM
PCM
BWCM
0.2380
0.0077
0.0046
0.0076
0.2377
0.0040
0.0040
64
1.0139
0.0395
0.0303
0.0473
1.1205
0.0251
0.0225
128
3.6256
0.1832
0.1490
0.1844
4.0674
0.1013
0.1019
256
14.5142
0.7793
0.5833
0.7853
16.5918
0.3848
0,4077
512
60.1918
3.2948
2.5030
3,4357
70.8647
1.7553
1.7526
Table 15: Ablation on the diagonal matrix M. in BWCM. We compare BWCM with a variant that learns the diagonal of M (MI-WCCM on SPDNet and GyroSPD.
Backbone
SPDNet
GyroSPD
Dataset
Radar
HDM05
FPHA
Radar
HDM05
FPHA
M--WCM
95.15 0.61
66.71 0.96
86.27 0.63
96.93 + 0.40
72.62
91.00 + 0.11
BWCM
93.93 # 0.79
67.40 = 0.90
86.27 + 0.60
96.21 # 0.25
72.74 0.43
91.00 = 0.11
clearly faster than AIM, LEM, PEM, and BWM. When the dimension increases 512, AIM and BWM require about 60 and 70 seconds per training step, whereas PCM and BWCM remain within roughly 1.7 seconds. In this setting, PCM and BWCM are even faster than LCM. This behaviour can be explained from three perspectives.
E.6 EFFECT OF M BWCM
Setup. To examine the effect of the diagonal matrix M in our (0, MCCMM, we compare BWCM with MI against variant that learns M E D (M-BWCM). We follow the SPD MLR configurations on both SPDNet and GyroSPD backbones. The value of f is the same for both BWCM and M-BWCM and set as in Tab. [ In M-BWCM, we optimize an unconstrained diagonal vector that the diagonal of M remains positive. We initialize e to zero, corresponding to M at the beginning of training. The size of e matches the input dimension of the SPD MLR, which is 8, 30, and 33 SPDNet and 20, 3 28, and x 28 GyroSPD for Radar, HDM05, and FPHA, where and are channel dimensions.
and set Mi diag(exp(())
Results. Tab. 15 reports the five-fold results. On both backbones, M-BWCM slightly improves accuracy on Radar, is marginally worse on HDM05, and matches BWCM on FPHA. Since the additional parameters introduced by M. relatively small, its impact the overall model is marginal. These observations suggest that we can simply fix M = / in BWCM in practice, which is the setting in our main paper.
E.7 HARDWARE
The experiments the GyroSPD backbone executed single NVIDIA A6000 GPU. Other experiments require SVD and Cholesky decompositions on relatively large SPD matrices, which are more efficiently performed on CPUs. Therefore, these experiments conducted on Intel Core 19-7960X CPU with 32GB RAM.
F PROOFS
F.1 PROOF OF THM. 322
Proof. As 0-DPM is the product metric of {SL", " and n copies of [R__, We first show the Riemannian operators (R++.g" ), and then on (. DDD DED by the principles of product metrics.
readily obtain the Riemannian operators
As shown by Thanwerdas and Pennec 2022
O.F. is the pullback metric of g by power function Pow F Besides, constant scaling does
Powe(.) and scaled by æz. expressed g" E
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not change the Christoffel symbols, the geodesic, Riemannian logarithm & exponentiation, and parallel transportation along geodesic remain the same under g and Powb g". These Riemannian operators under Powe g can be obtained by the properties of Riemannian isometries (Sec. C.2) Specifically, given p. E R++ and w, v E R we have the following:
Powe..(1) 0pp v
(42)
A. 9p (v, w) = 02 F. (Powse.p()) Power, = (00 (0-',, =
=121001111 =2200-1)
(43)
(t) = pow,'' (Powo(p) - pPower.(u))
= (p" + 10pp~11))
p(1 + 1:00-11) with & E {t R t0p R++3.
(44)
Logp(1) (q) = Powge.p (Powa(g) Powe(D))
( p
1 = 8 (a -p²)= 8º
(45)
1-8
PTp-+g()) = Powg- 0r.9 Po
U.
(46)
The geodesic distance between p and q under 0 is given
d' ②) = -
8-B (Log,(() Logp(1)) = 02 (1" - pp0?
(47)
The weighted Fréchet mean of (p: RR+ J with weights {w:) satisfying w; 0 for all and
E; wi = 1 under E is defined as
N
WFM (wi}) = argmin
Σ wy (p.p.). p
(48)
Obviously, the WFM of {p:} under of is the same the one under Pow` gB. Due to the isometry of Pow dh to g". the WFM of {pr} under g can be calculated as
WFM((p:), (w:)) = pow (WFM (WPoWo((P))) (w:))) (wi))) = (<
1=1 wip?
(49)
where WFM in Eq. (4 is the Euclidean WFM, which is the familiar weighted average.
So far, we have obtained all the necessary Riemannian operators on [R++.9 FF., Combined with the Euclidean space SL", one can obtain the results in the theorem.
F.2 PROOF OF THM.
Proof. Similar the proof of Thm. 3.2 we only need show the Riemannian operators of [I++,g mBB3 } with m E 1 The expressions of Riemannian operators under GBWM can be found in Han et al. (2023). Here, we further simplify the associated expressions for the 1-dimensional case. Specifically, given P.q E 14+ and w, u E T. we have the following:
(v) 2mp'
(50)
g BBW
(v, w) (Fp.m())) (v), = 4mp"
(51)
(tv)²
p(1 + to
Vp,oo) (1) =p p + tv (lv) (m))mm mm2 = p + tu
(52)
4p
2p
Log, (9) = (m(m 2 =pq) 2 -p) ((pg)² -p) 2p (9) p
- 1
(53)
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Published as a conference paper at ICLR 2026
(p( 9, 9,""BW (Log, (1), Log, (q)) [ogg(())=
1
4 (pq) pp)²
4mp
pq 22 + +
mpp
(54)
m-1
(q 2p 2z +
m (o p² )²
As shown by Han al. )2023), GBWM S++ is the pullback metric of BWM by x(S)
M M-3SM' à for all S E S++ with M E S4+ For the specific R++ S++, the isometry is simplified
T (p) = m p. vp E R with m E R
(55)
The geodesic ui (t) under BWM over R++ exists in the interval (t R| +Cp(t) E R++}
(Malagò et al. 2018), which can be simplified as { { R| 1 E RR+ Therefore, the geodesic
Knw) (t) under g gm exists in the interval:
E R| + (t)
R + t E B l E
R++
(56)
(T(
2p
According to Thanwerdas and Pennec 2023 Tab. 6), the parallel transportation {R++.9 RW) is
1
PT p-g (v)
(57)
Therefore, the parallel transportation on ( mBBW is
[q)) 3 (pp)
PTp->o()) = =xaig ( = =
Te,p(()
(58)
y/2
v.
Lastly, we show the WFM on (I4+,gm.BW) BW Given {pr E R J with weights [w;} satisfy-
ing wi >> 0 for all and Σ N
i== wy the WFM (R++.9 mnW is
WFM((P))) = argmin
Σ wc dl (p. PER11
Wim -1 (pt -p= 11
argmin +
wi (p² -pt))
argmin +
(59)
argmin pER++
Wi p 1p22p 2p
N
argmin
2p
wap
pER++
Let f(p) =p- 22p² = wip Then, the 1st and 2nd order derivatives are
df p-
w.p
(60)
dp
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Published as a conference paper at ICLR 2026
d
1 2
Σ u.p > 0, Vp E R+++
(61)
dp²
Therefore, the optimal solution can be obtained by setting Eq. (60) equal to 0:
d
0 p"
E, wpp
(62)
dp
Combining the above results with the Euclidean geometry on SL", one can readily obtain the results.
F.3 PROOF OF LEM. 3.5
Proof.
The differential of DPows at L D is
Doowe., (V) AT. 1T,",V, WET.L,
(63)
Putting Eq. 663 into Def. 3.4 one can easily get the results.
F.4 PROOF OF LEM. 3.6
We first present a useful lemma.
Lemma F.1. The Riemannian exponentiation, logarithm, and parallel transportation along the geodesic are the same under (8, MD)-MMM and 8/2-DPM.
Proof. This can be directly obtained by Thms. 3.2 and 3.3
Now we begin to prove Lem. 3.6
Proof. According to Lem. F.1
we only need to show the expression of 9/2-DPM. We omit the
superscript for simplicity.
For X E Tr Tr we have the following:
Log, (L. [.] 1 [T. -),
(64)
PT)+2(X) |X] + L L1-0x 2x
(65)
Exp, X [X] + (1 + 0X)
(66)
For the binary operation, putting Eqs. (44 and 65
into Eq. 7 we have
L 2 K = Exp,, (PT,-,,. (Log,(K)))
[K] + (K. - -1)
ExPL
PT
Expr,
[K] + 0
1-1 (K" - -1)
(67)
(7L-- (K* - -1))
[L] + [K] + IL
+ OI
L K (L KD 1)²
In the 3rd row of Eq. 667 the well-definedness of exponential map requires
1--- (K" - n) 1
L 0
DJ e L K - D D
(68)
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Published as a conference paper at ICLR 2026
For the gyromultiplication, injecting Eqs. (44 and (66 into Eq.
8
have
t L = Exp, (oLg
Expr [BLL] _ 0 (1--1))
(69)
-tL4] (I + (LL - 1))
= =[[A] + (IL + (1 t))
In the 2nd row of Eq. 66 the exponential map requires
@ 1 (I
. (1--1) E D++ + tI + (1 - t) E D
I+0
(70)
F.5 PROOF OF THM. 3.7
Proof. In this proof, we assume L, K, J E 2 and s,t R. Note that as stated in the main paper, we assume all the gyro operations satisfy the required assumption presented Lem. 3.6 As indicated by Lem. F.1 we only need to prove the case of 6-DPM. For simplicity, we omit the superscript C.
Axiom (G1):
Eq. implies that the identity element the identity matrix.
Axiom (G2): We define the inverse element of L as
- (2)
(71)
Simple computations show that @T 4 T.
Axiom (G3); Gyroaddition in Lem.
3.6 indicates that
L @ (K + (L
K + JJ=
[ +
KJ +
[J] + (LL + K + J - -22
(72)
Therefore the gyroautomorphism is the identity map, i...grr L, K] id.
Axiom (G4): This is a direct corollary of (G3).
Gyrocommutative law: Gyroaddition in Lem, 3.6 indicates that
L K==Ko
(73)
Axiom (V1): This can be obtained by gyromultiplication in Lem. 3.6 and Eq. 71
Axiom (V2):
+ ((s + (1 (s+ t))1)* 1
(s+ +
=>[L] + [L] + (>IL +(1 8) - 1L + (1 1)I
(74)
(s () (
Axiom (V3):
(st) o L = (st) + (stLL + (1 st)r)
[st) + (s (ILL (1 t)1) + (1=s)1)*
(75)
= =so [1L] + (IL + (1 (1- (1-+)))**) $1
= =
Axioms (V4) and (V5): These two axioms can be directly obtained, as gyroautomorphisms are all identity maps.
28

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Published as a conference paper at ICLR 2026
F.6 PROOF OF THM. 4.1
Proof. According to Nguyen and Yang (2023 Thm. 2.4), the gyro vectors can be preserved by Riemannian isometry. Besides, the Cholesky decomposition is the Riemannian isometry:
Chol: +5 9 {..+
(76)
F.7 PROOF OF THM. 5.1
Proof. Putting the associated operators in Sec. into Eq. (19) in Lem. H.1 by Chen et al. (2024c).
one can directly obtain the results.
29


Reviews:
[{"review_id": "GJOSrNP0mO", "reviewer": "Reviewer_V5WE", "summary": "The authors proposed two new metrics for the Cholesky manifold inspired by the \nLog-Cholesky Metric (LCM) [Lin, 2019] and Bures–Wasserstein Metric (BWM) [Bhatia et al., 2019]. They show that the Cholesky manifold admits a simple decomposition into a linear space of strictly lower triangular matrices (SL) with the Euclidean metric and the product of positive numbers for the diagonal part. \nThe specific choices of metrics for the later results in a novel \n$\\theta$-Diagonal Power Metric ($\\theta$-DPM) and $\\mathbb{M}$-Diagonal\nBures-Wasserstein Metric ($\\mathbb{M}$-BWM). The authors  develop closed formulas for geodesics, Riemannian logarithm, vector transport and weighted Frechet mean, which allows for \nderiving gyro-structures on the Cholesky manifold. Finally, the authors suggest \nusing their metrics for SPD manifold as pullback metrics via Cholesky decomposition, which was proved to be a diffeomorphism in [Lin, 2019].\n\nThe authors claim that new metrics (both on the Cholesky manifold and SPD as a pullback) are demonstrating better numerical properties since \nthey are based on linear operations and inversions instead of exponential functions. The authors have conducted two experiments for the Multinomial Logistics Regression (MLR) classifiers on Radar dataset [Brooks et al., 2019] (signal classification), \nHDM05 [Muller et al., 2007] and FPHA [Garcia-Hernando et al., 2018] datasets (human actions classification). The proposed approach is appears to be more reliable in the SPD interpolation task.", "questions": "- Have you conducted experiments for failure probabilities estimation for pullback metrics on the SPD manifold? \n\n - How can I estimate $\\theta$ parameter for a dataset in advance? Why is it negative in Tab. $8$ (lines $918$-$924$)? Is there any analysis for negative $\\theta$?\n \n - What does the resulting asymptotical complexity of the proposed metrics equal to?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- Rigorous analysis of proposed Riemannian metrics for the Cholesky manifold: from geodesics to pullback metrics for SPD manifold.\n - Stability experiments: empirical demonstration of failure probability for the derived metrics.", "weaknesses": "- I am not sure if the composition of logarithm and exponentiation from LCE is not possible to be done using some standard numerical tricks like logsumexp or smth in this direction. Could you give the exact formula for the unstable part of LCE and reason why there is no any standard workaround? I will be ok with raising the score if this part is properly addressed, because it seems to be one of the key advantages of choosing your method.\n- Given that $\\theta$ is a central contribution of this work, an accurate ablation study is important to isolate its effect. Similarly regarding the matrix $\\mathbb{M}$ across the $\\theta$-DPM, $\\mathbb{M}$-BWM, and $(\\theta, \\mathbb{M})$-BWM models.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "t95N9k0oQ7", "reviewer": "Reviewer_SYJp", "summary": "This paper proposes two new SPD metrics called Power–Cholesky Metric (PCM) and Bures–Wasserstein–Cholesky Metric (BWCM). The proposed metrics are used to develop Riemannian Multinomial Logistic Regression (MLR) classifiers and residual blocks for SPD neural networks. The authors conducted experiments on radar signal classification, action recognition, and tensor interpolation to show the effectiveness of the proposed metrics.", "questions": "This paper is heavily based on the works in [Nguyen and Yang, ICML 2023; Nguyen et al., ICLR 2024; Chen et al., NeurIPS 2024]. The proposed Riemannian metrics are formed by introducing small changes to existing ones.\n\nFor the experiments, while some improvements are shown on the chosen datasets, I do not think it is due to a better design of the proposed metrics compared to the well-known ones such as AIM, LEM, and LCM. This is because the performance of the proposed metrics depends on additional parameters (e.g., $\\theta$) and one can easily figure out good settings for these parameters to get enhanced performance on some specific datasets. The main advantage of the proposed Riemannian metrics w.r.t. AIM and LEM is computation time, but as can be seen from the experiments, they have similar computation times as LCM. \n\nOverall, I think the paper presents marginal contributions to the literature of SPD neural networks which are not good enough for publication in ICLR 2026.\n\nQuestion:\n\n1. In terms of technical contributions, what is the novelty of this work w.r.t. those in [Nguyen and Yang, ICML 2023; Nguyen et al., ICLR 2024; Chen et al., NeurIPS 2024] ?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- This paper aims at improving SPD neural networks which have applications in many fields. \n- In general, the exposition is clear and the paper is easy to follow.", "weaknesses": "- Contribution is marginal.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HO65zQAXJU", "reviewer": "Reviewer_cJuQ", "summary": "This paper revisits the geometry of the Cholesky manifold and identifies a product structure that decomposes the Cholesky factor into a Euclidean component. Leveraging this insight, the authors introduce two new Riemannian metrics on SPD manifold: PCM and BWCM. The authors apply the proposed metrics to build Riemannian MLR classifiers and residual blocks. The authors evaluate the two metrics and applications with 3 datasets: Radar, HDM05 and FPHA.", "questions": "1. Can the authors provide guidance on how to choose the $\\theta$ values\n2. The authors presents the stability of the proposed metrics, but I didn't see how it is validated with experiments\n3. As this paper is on Bures–Wasserstein–Cholesky metric, I think the authors should compare with othe works Bures–Wasserstein geometry, e.g.,\n         1. Learning to Normalize on the SPD Manifold under Bures‑Wasserstein Geometry (Wang 2025)\n         2. Learning with Symmetric Positive Definite Matrices via Generalized Bures‑Wasserstein Geometry (Han 2021)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Replacing logarithms with power transforms to address numerical instability in LCM is well-motivated and empirically validated. The proposed metrics maintain closed-form. This paper is in general sound, the usability of the two proposed metrics is critical for deep learning integration.", "weaknesses": "This paper is in general well-wriiten, but there are still some concerns:\n1. Several Riemannian and gyro-operators used in this work are only local defined. In particular, certain expressions are well-defined only under positivity constraints such as $L^{\\beta} + K^{\\beta} - I \\in D_{n}^{++}$\n2. 𝜃 power is a main contribution, however, the authors don't present how to choose 𝜃\n3. Recent SPD learning works with GBWM-based classifiers are not experimentally compared\n4. The stability of the DPM and DBWM are not empirically validated", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Qel3KRttAC", "reviewer": "Reviewer_QVyo", "summary": "The paper revisits the Cholesky geometry of SPD matrices and exposes a product structure: a Euclidean metric on the strictly lower-triangular part plus (n) copies of a 1-D positive-reals metric on the diagonal. Leveraging this, the authors define two new Cholesky-side metrics -- Diagonal Power Metric (θ-DPM) and Diagonal Bures-Wasserstein Metric ((θ,M)-DBWM)--and pull them back through Cholesky to SPD metrics: Power-Cholesky Metric ($\\theta$-PCM) and Bures-Wasserstein-Cholesky Metric (($\\theta$,M)-BWCM). These metrics admit closed-form operators (geodesic, log/exp, parallel transport, Frechet mean) and gyrovector operations, while replacing diagonal log/exp by diagonal powers for improved numerical stability. Empirically, they integrate the metrics into Riemannian MLR classifiers and Riemannian residual blocks, showing accuracy/runtime advantages over AIM/LEM/LCM on Radar, HDM05, and FPHA; and they present large-scale stability tests (failure rates under tiny diagonal entries) favoring the proposed metrics.", "questions": "1. Invariance & trade-offs. How do $\\theta$-PCM/($\\theta$,M)-BWCM compare to AIM’s affine-invariance in practice? Are there scenarios where losing/altering invariance harms performance, and can deformation ($\\theta$) mitigate this? \n2. Baseline expansion. Can you include PEM and BWM/GBWM (even via numerical solvers) as end-to-end baselines on at least one dataset to calibrate geometry choice vs. closed-form convenience? \n3. Domain of validity. Please quantify the practical parameter/step-size ranges ensuring (L^\\beta) feasibility during training; do you observe violations, and how are they handled (clamping, retraction)? \n4. $\\theta$ sensitivity. Provide systematic sweeps of ($\\theta$) (and (M)) across tasks; when does the limit ($\\theta\\rightarrow 0$) (LCM-like) help/hurt in practice? \n5. Scaling & large-scale tasks. Any results with high-dimensional SPD in modern deep nets (e.g., covariance pooling on ImageNet-scale features) to validate computational/storage gains? \n6. Code & seeds. Please clarify #seeds, hyperparameter parity, and release a minimal reference implementation for log/exp/transport under your metrics to facilitate adoption.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Originality: Product-geometry perspective on Cholesky leading to $\\theta$-PCM/($\\theta$,M)-BWCM with closed-form operators and gyro-structures. \n* Quality: Mathematical development is careful; operator summaries are comprehensive; empirical evaluations show consistent accuracy/runtime gains and robustness under tiny eigenvalues. \n* Clarity: Pipeline overview and tabulated operators (Table 1) facilitate adoption. \n* Significance: Practical drop-ins for SPD MLR and residual blocks; mitigates numerical brittleness of diagonal log/exp while retaining LCM-like utility.", "weaknesses": "1. Baseline breadth. Experiments primarily compare to AIM/LEM/LCM; PEM and (G)BWM baselines as end-to-end learners are not presented, leaving open how much improvement stems from closed-form convenience vs. intrinsic geometry. \n2. Assumption transparency. Gyro-structures and some exp/log formulas are locally defined and require ($L^\\beta$) feasibility (e.g., ($L^\\beta+K^\\beta-I\\in D_{++}$)); these constraints aren’t prominently quantified for training dynamics. \n3. Reproducibility details. While mean$\\pm$std are reported, seed counts, episode splits (for cross-dataset parity), and tuning parity across metrics could be clearer; code is not yet available (this is highly preferred). \n4. Invariance discussion. The paper could explicitly contrast affine-invariance and other desirable properties of existing metrics vs. the proposed ones, and discuss trade-offs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)