Special Unitary Parameterized Estimators of Rotation
Authors: Akshay Chandrasekhar
Venue: ICLR 2026 Poster (2026)
Keywords: optimization
special unitary
rotation estimation
quaternion
learning rotations
machine learning
Wahba's Problem
Primary area: optimization
TL;DR: Solving Wahba's problem with special unitary matrices, leading to new optimization formulations as well as representations for learning rotations in neural networks.

Abstract: This paper revisits the topic of rotation estimation through the lens of special unitary matrices. We begin by reformulating Wahba’s problem using $SU(2)$ to derive multiple solutions that yield linear constraints on corresponding quaternion parameters. We then explore applications of these constraints by formulating efficient methods for related problems. Finally, from this theoretical foundation, we propose two novel continuous representations for learning rotations in neural networks. Extensive experiments validate the effectiveness of the proposed methods.

Published as a conference paper at ICLR 2026
SPECIAL UNITARY PARAMETERIZED ESTIMATORS OF ROTATION
Akshay Chandrasekhar
ABSTRACT
This paper revisits the topic of rotation estimation through the lens of special unitary matrices. We begin by reformulating Wahba's problem using (U) to derive multiple solutions that yield linear constraints on corresponding quaternion parameters. We then explore applications of these constraints by formulating effi- cient methods for related problems. Finally, from this theoretical foundation, we propose two novel continuous representations for learning rotations in neural net- works. Extensive experiments validate the effectiveness of the proposed methods.
INTRODUCTION
3D rotations are fundamental objects ubiquitously encountered in domains such as physics, aerospace, and roboties. Many representations have been developed over the years describe them including rotation matrices, Euler angles, and quaternions. Each method has specific strengths such as parameter efficiency, singularity avoidance, or interpretability. While special orthogonal matri- ces 50(3) are widely used, their complex counterparts, special unitary matrices (U(,), are less explored in areas like robotics and machine learning. This paper showcases the utility of special unitary matrices by tackling rotation estimation from different perspectives.
1.1 WALIAAS PROBLEM
Wahba's 's problem Wahba 1965) is fundamental problem in attitude estimation. The task refers to the process determining the orientation target coordinate frame relative to reference coordinate frame based on 3D unit vector observations. More formally, it is phrased as seeking the optimal rotation matrix minimizing the following loss:
min (ESO(3)
Σ We bb; Ra
where a; are the reference frame observations, b, the corresponding target frame observations, and w; are the real positive weights for each observation pair. The problem can be solved analytically by finding the nearest special orthogonal matrix (in Frobenius sense) to the matrix B below:
=
Σ w
(2)
Today, this solution typically computed via singular value decomposition (Markley 1987
Alternatively, the solution be estimated unit quaternion. Davenport (1988) introduced the first such method 1968 by showing that the optimal quaternion q is the eigenvector corresponding the largest eigenvalue of 4x4 symmetric gain matrix K, which can be constructed as:
Tr(B)
ZT B BT - (TBB
K
(3)

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where I is the identity matrix, Tr(B) = Σ, Bu, and z Σ, w,a x b,. The solution via eigen- decomposition relatively slow as it solves for all the eigenvectors of the matrix which are not needed. Later solutions improve upon this by calculating the characteristic equation of K and solv- ing for only the largest eigenvalue (Shuster and Oh 1981 Mortari 1997 Wu et al. 2018). For an overview of major algorithms, see Lourakis and Terzakis 2018).
1.2 REPRESENTATION FOR LEARNING ROTATIONS
In recent years, there has been great interest in representing rotations within neural networks, which often struggle with learning structured outputs. Directly predicting common parameterizations such as quaternions or Fuler angles has generally performed relatively poorly (Geist et al. 2024). In fact, was shown that any 3D rotation parameterization less than five real dimensions is discontinu- ous, necessitating non-minimal representations for smooth learning (Zhou et al.) 2019). Addition- ally, challenges like double cover some representations can further hinder learning. Two leading approaches. Levinson et al. (2020) and Peretroukhin et al. (2020), essentially interpret network out- puts as B and K matrices (Eq.. (2) and (3) respectively). mapping them to rotations via solutions to Wahba's problem. Thus, the two tasks can be linked. For more in depth overview of the task, see Geist et al. 2024).
1.3 CONTRIBUTIONS
This paper establishes new theoretical results rotation estimation by utilizing special unitary matrices within the framework of Wahba's 'ss problem. We explore several applications of these results, with particular emphasis on our two novel representations for learning rotations in neural networks.
We highly recommend the reader to first review Appendix A to become f'amiliar with the relevant mathematical background and notation used throughout the paper.
SOLUTIONS TO WAHBA'S PROBLEM VIA SU(2)
Transferring Wahba's 's Problem to complex projective space, solve for the optimal rotation as a special unitary matrix.
2.1 STEREOGRAPHIC PLANE SOLUTION
First, establish the proper distance metric in complex projective space corresponding to the spherical chordal metric in Eq. (1) For points a, b E S and their stereographic projections (aaa) = z = [21, =2]" 1" and (b) p Pr. p2 T we can show that the metric be expressed in the
following way (derivation in Appendix B.I.I.:
bl 4|2122 22P11
(4)
We now seek find the rotation parameterized by corresponding special unitary matrix U in complex projective space that minimizes the objective in Eg. (1) Applying our derived metric and Eqs. (32) and (34) we can construct for each weighted input correspondence z; and p;:
wal b - 22,ll2 = Awa (arl(-8al (-32,,1 + 022 Pi.1 - (a2,,1 + P
(la:2,11 + B22, 2 _21,1 22, P)] |p:
4wy1(-B24,1 224,2 Pr,1 (a.2,11 0z, Pr.2
||zz,
|2:||
where a, are the complex parameters defining U from Eq. (31) By definition of unitary matrices,
|2||22 = ||z|
Thus, we can rewrite our expression as the following target constraint:
4w I2g,1 (z212)).
(a:2,11 1 + (2,,2)).,22
(5)
|2|||||||2
2/w
32g, + @2122)P..111 (a22,1 + S24,2)Pi.2)
(6)
/
/p..1 1p:2 ?
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The expression is now just linear function of rotation parameters. It is a general constraint it handles the entire complex projective space (proof in Appendix B.1.2). However, in practice, our inputs are more commonly given projection coordinates on the complex plane. As such, we have:
= 3i + yid, Pi, 14 - m, + n4d, 21,2 - Pi.2 = 11
for each point correspondence (Ii, Ys. mi, ng E R). This simplifies the constraint to the following:
2 wi B2g P; az; B)
2 2 -
We can rearrange the equation to the following linear form with u
3 [ 3]
4m;
w/ (12/2 + 1)(++(+))
(8)
2 -1
P1z. u= A/A,u=
(9)
Each imput point pair gives us a complex constraint A, Stacking A, together and multiplying the weights through, we can write the relation succinetly Au 0 (A is complex n points). With noisy observations, the constraints do not hold exactly, so we aim to find the best rotation that minimizes the least squares error |ull. It is nontrivial to solve for the minimizing vector u while ensuring the result will form a valid special unitary matrix (u, = ū3 U2 = û4 u1 + Ugll 1). To more effectively solve this, we use Eq. (35) to transform the vector u to a corresponding quaternion q = [W, Zg Bg zy] that has simpler constraint (q must he unit norm). We carry out the complex multiplication for each A, and break the constraint into two constraints, one for the real and imaginary parts respectively:
matrix for
4w;
w
(10)
(1+ v2)
m3 +
T4 my
Vi nh + m; m,y: + n4:2 1 --,44 + n:yl
1 MTJ; 1433
milki + nAT;
9 u=vuDDqq0
(11)
n2
Multiplying the weights through again and stacking together D, for each correspondence into D (real 2n matrix), we can arrive the following constrained least squares objective:
= D" q"d ( D.(D.D.)q=aa - qpG
min q" ''' s.t. ||و|| =1
(12)
The formulated objective in (12) equivalent the original problem statement, and the solution is well known as the eigenvector corresponding to the smallest eigenvalue of Gp. Using [gq (35)
again, we can map q back to a special unitary matrix U giving a solution to the problem. Note that -q is also solution since eigenvectors are only unique up to scale. However, the sign is irrelevant as q and q map to the same rotation due to the double cover quaternions over SO(3) in Eq. (36)
For further theoretical details this solution, see Appendix C
2.2 APPROXIMATION VIA MOBIUS TRANS TRANSFORMATIONS
We can approximate the previous solution in the complex domain by first estimating an optimal Möbius transformation M and mapping it to special unitary matrix. Relaxing the special unitary conditions in Egq. (9) we can treat u as a flattened form of M, leading to a modified constraint A that holds when M aligns stereographic point pair:
m = vec(M) = To
E 7
87"
2; p;z;
Ps m A/m= =0
(13)
Note Eq. (13) does not preserve the metric in Eq. (4) between p4 and transformed point (M())))
We can stack each A into matrix A' (n x complex matrix) and similarly estimate the best (in a least squares sense) Möbius transformation aligning the points
GM = A Σ A
min m # Garmm s.l. |m|| =
(14)
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The constraint in Eg. (14) is necessary to prevent trivial solutions, but the choice of quadratic con- straint on m is arbitrary. With our constraint choice, the optimal m is the complex eigenvector corresponding to the smallest cigenvalue of G Since G positive semidefinite and Hermitian (GM G M)) by construction, the eigenvalues are real and nonnegative, facilitating straightforward be obtained directly from the kernel of A'. Either way, the solution is
ordering. If 4, not unique as cigenvectors and kernel vectors can be scaled arbitrarily, particulary by phase en0 However, by Eq. (42). scaled Möbius transformations are equivalent, so our result properly defines the transformation.
Given m. we can reshape into M and scale M M (Mt à M (allowed since the scale of M is arbitrary) so that det(M² = 1. It is known that the closest unitary matrix to M in the Frobenius sense can be computed by UV# (Keller 1975). where U and V are from the singular value decomposition M UEVH Since det M 1, the nearest unitary matrix to M is special unitary (proof in Appendix B.3.T) and in fact the approximate solution. Note that this matrix not necessarily the nearest special unitary matrix to M itself. By normalizing the determinant, prevent the rotation mapping from being affected by arbitrary phase scalings of m.
2.3
3D SPHERE SOLUTION
If our inputs are given as unit observations in 3D, we could project them by s and use the earlier solution. However, through Eqs. (37) and (38) we see that we can act directly on 3D vectors with special unitary matrices which suggests alternative formulation. Upon examining the structure of the matrices that maps to, one can show that Eq. (1) can be equivalently expressed as:
xa,) + Zi, x(b.) H P; Σ we lol Ra, 2
1 Σ P. UZ
(15)
where ||- ||r denotes the Frobenius norm and U is the special unitary matrix that maps to R. The Frobenius norm is unitarily invariant, so we may multiply the inside expression on the right by to obtain new target objective and corresponding constraint:
Σ 0z P.U UZ
(P.U- =0
(16)
We arrive at a linear constraint again via special unitary matrices. Inspecting the matrix within the Frobenius norm reveals that the loss contribution from the top row elements is identical to that the bottom row elements. Consequently, we only need to compute the loss from single row, allowing us to eliminate the factor of 1 can write the following complex constraint:
2 from equation Eq. (16) With a; = (2,, 24) and b, = (m, n,, Pt),
(m=
) z (m)i
U1 zvi
P.d u= V=
(17)
n4 + P,i
C, has a rank of at most if a and have the same magnitude. We reformulate the constraint, once again breaking the complex terms of into their real components. This yields the following linear constraint in terms of quaternion parameters:
0
mI
V/
11;
2i pi
11
my 24
B4
y1
+
0
0
ps
y: +
-t
m;
q
(18)
pi
-Vi
74
I; + m,
Note that Q is a 4x4 skem-symmetric matrix and has at most rank and have the same magnitude. As a result, optimization now becomes:
Σ min G s.t. ||و|| =
w. Q. Q, = Σ w. W. = Gs
(19)
The solution again the eigenvector corresponding to the smallest eigenvalue of G.g.
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3
OPTIMIZATION METHODS FROM LINEAR QUATERNION CONSTRAINTS
Our previous general solutions notably distinct from other methods they allow for the prin- cipled construction of linear constraints (Eqs. (11) and [T8) on quaternion parameters. We discuss some applications and desirable properties of these results.
3.1 RESIDUAL BASED OPTIMIZATION
While Wahba's problem admits direct solution, many related rotation estimation tasks require it- erative methods. These often involve repeatedly evaluating per-observation losses for a candidate quaternion. Examples include alternative loss functions like the absolute chordal metric (L1 dis- lance) or robust approaches such as iteratively reweighted least squares (IRLS). In these settings, our linear constraints serve as drop-in, efficient method for residual computation. The stereographic formulation in Eq. (IT) is especially appealing as is far more compact (8 elements versus 12 for Eq. (18) while avoiding branching construction, especially in the general case of Appendix C.2
3.2 CONSTRAINED OPTIMIZATION
When the constraints for an observation pair hold exactly, our formulas yield a convenient analytical characterization of all rotations that align the pair. A practical use case for this is rotation estimation with an axis prior (e.g. gravity vector measurement from an IMU). Traditional methods rely on sequential rotations intermediate coordinate frames to simplify the problem (Magner and Zee 2018 Chandrasekhar 2024). In contrast, because both Eqs. (11) and (18) reduce to rank this setting, we can linearly express two quaternion parameters in terms of the other two and solve directly and efficiently a reduced space, climinating the need for intermediate frames.
3.3 TWO-POINT CASE FOR WAIIBA'S PROBLEM
More generally speaking, when the constraints hold exactly for one more observation pairs (i.e. noiseless scenarios), we can obtain the solution from the kernel of those constraints in closed-form. For example, with two noiseless 3D sphere observation pairs, the aligning rotation can be given by:
(a2 b2) aa. b, (a= - b.)
a1 +
b1)
(20)
where q denotes the unnormalized form of rotation q. Appendix Djdescribes our methods to robustly and efficiently construct these rotations of exact alignment. These simple kernel formulations are key to enabling our solutions to the case of Wahba's problem when = 2.
Weighted Wahba's 's problem for the two-point case well known have closed-form expres- sions (Shuster and Oh, 1981, Mortari 1997, Markley 2002). We propose an alternate solution which given by the weighted average of the two (unnormalized) rotations that each noiselessly align the cross products of the reference and target sets, along with one of the two corresponding observation pairs (proof in Appendix B.4.T). Using the average rotation definition from Markley et al. (207)) (i.c. in Frobenius sense for (O(3)), the solution is:
|aj
ag
(a, + b.) (n, n12) (a: b:) x (nj n2)
n1 a1 a2, n2
(bı x b2).
q:
|
b
(wi
w2))
( |
12 11: w 2w/ |22 2 Q2)
2w
( -2),
1 = T v 72 ww
μqn + vQ2
q
(21)
11 q42 22² 2(q) qu)
where q. d2 denotes the usual vector dot product between qı and dp. See Appendix B.4.3 for derivation and additional details.
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by
by
by
R
R
(a) Gram-Schmidt
(b) 2-vec
(c) QCQP/SVD
(d) QuadMobius
Figure 1: (a)-(b) Illustration of difference between Gram-Schmidt and 2-vec in 2D. bx by are predicted axes directions from the model, and Rx, Ry are the orthogonalized coordinate axes from each mapping. Gram-Schmidt favors bx aligning Rx with it greedily while 2-vec uses bx by in a balanced way. (c)-(d) Conceptual illustration of QCQP, SVD, and QuadMobius maps in context of Wahba's problem in 3D. QCQP/SVD can be interpreted as direct projection of target points (red) to an orthogonal frame. QuadMobius first maps those points to an intermediate representation Möbius transformation, defined by three points (blore-effore projecting to SU(2) rotation.
Unweighted In the case of w) = W2, the optimal rotation simplifies to the rotation which exactly aligns a + a2 b + b2 and a a2 to b b2 (proof in Appendix B.4.2). This is given by:
+ a1 a2
$1 = a1 + a2,
$2 V + b1 b2
(b1 + b2)
a1 a2 (b1 - b2)
dj = a1 a2. d2 1 b1 b
($) + 82) (d) - -d2)
q
($1 82) x (d) d2)
(22)
The aligning rotation formulas are given in the form of equation Eq. (20) for simplicity, but in practice we use the approach described in Apppeddxx D.2 for robustness. that case, singular cases only arise when x or bj xb2 where no unique solution exists, and particular one may be obtained via the special unitary constraints in equation Eq. (17) (see (ceeendxxx B.4.4). Notably, the two solutions above optimal in the sense of Wahba's 's problem and simplified compared to existing two-point methods, especially for the unweighted case (see Table 5).
An example use case of these methods is estimating the orientation of camera given image of a rectangle. Under a pinhole camera model, the image of a 3D rectangle adheres to the rules of perspective geometry. Since the rectangle's 's opposite edges are parallel in 3D, their projections in the image converge at vanishing points that represent the direction of these lines in the camera's frame. Because the two sets of parallel edges in the rectangle are orthogonal in 3D, the corresponding vanishing points should also be orthogonal. However, due to measurement noise, this orthogonality often violated. Our two point solutions can recover the best estimate of the camera's orientation in these cases.
4 REPRESENTATIONS FOR LEARNING ROTATIONS
Based on previous formulations, we introduce two higher-dimensional representations for learning rotations. See Appendix B.2) for derivation details and Appendix F for further theoretical support and explanation of both representations.
2-vec The first is based on our formula for the optimal rotation from two unweighted observations and is denoted 2-vec. Similar to the output vector from model as target 3D x and axes (denoted by. by)) Unlike the Gram-Schmidt method which greedily orthogonalizes the two vectors by assuming the x-axis prediction is correct, 2-vec maps the two vectors rotation optimally in the sense of Wahba's 's problem, balancing error from both axis predictions (Fig. 4p. Eq. (22) could be used, but since the reference points are the xr coordinate axes, we can instead obtain rotation matrix in simpler fashion through the same
map in Zhou et al. 2009). 2-vec interprets a 6D

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principle:
|bx ||byl] by: b
by + by
b= bg - b/y
by
b|b + byll" ||b b/
R
ve (b+ b-),
v2 (b+ b-), b- b+ E SO(3)
(23)
This method has a similar singular region and computational complexity as that of Gram-Schmidt.
QuadMobius A second parameterization is based on the approximation from Section 2.2 involv- ing Möbius transformations. Taking inspiration from the approach in Peerttrokkhin et al. (2020), (real) 16D network output @ = [0: : 1... 16) arranged into the unique complex elements of G as below:
01
02 + 0zi
04 + 0g4
0g + 0şi
02 0gi
0s
0g
01o
011 012
Gm(
(24)
04
0gi
0g
io
8 13
14 + f15
Oo
07i
011
014
isi
0
(O) Hermitian with real (and assumed distinct) eigenvalues where we can select the eigen- vector m corresponding to its smallest eigenvalue. After reshaping to a Möbius transformation M. we can map to a rotation by the approximation procedure in [Section 2.2) The procedure can be
performed via singular value decomposition (M UEV to obtain special unitary matrix Q:
Vdd(UVV))UV" SU
(25)
Alternatively, we can algebraically solve for Q as follows:
det(M
M (2M + Tr(M"M) M
Q M + (d) E (U(
(26)
where Tr(-) denotes the trace and a.j denotes the adjugate. In both cases, Q is mapped to quaternion via Eaq. and) (45) and M is assumed to nonsingular. We denote the SVD method QuadMobiusSVD and the algebraic method QuadMobiusAlg With these maps and our assump- tions (observed valid in practice), we define a mapping from to q that has a defined numerical derivative for backpropagation (see Appendix for derivative formulas). We remark that this map motivated by ideas from Levinson et al. (2020) and Peretroukhin et al. (2020), inheriting many of their properties (e.g. interpretation Bingham belief (Kent) 1994). differentiability (Magnus 1985 Wan and Zhang 2019)) while offering a potentially more flexible (kigher-dimensional, complex) learning representation.
EXPERIMENTS
5.1 WAHBA'S PROBLEM
Synthetic experiments performed to validate the proposed methods for Wahba's problem. For each trial, a ground truth quaternion rotation qot is randomly sampled from Sd, and n reference points are randomly sampled from S The reference points are rotated by qgt to obtain target ob- servations. Gaussian noise is added to each component of each target observation, and the target observations are subsequently re-normalized afterward. Weights are randomly sampled between and 1. Accuracy is measured by the angular distance Perr cos ( (qeee 9gr)² 1) in de- grees between the estimated rotation qest and 4gt, where (-.)) denotes the usual vector dot product. Numerical results shown in Appendix.
We first test our solutions to Wahba's problem for both 3D and stereographic inputs (Egs. (12) and (19). The input for the latter is created by projecting the 3D points by th. We also test the approximate solution in Seectionn 2.2 The solutions to all three are obtained by eigendecomposition using Jacobi's eigenvalue algorithm. For validation, compare against several quaternion solvers

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introduced over the past decades. For the two-point case, also compare against the closed-form solutions in |Mrkley] (2002) and Shuster and Oh (198I). All solutions were reimplemented and optimized similarly in C++17 and compiled with the flag -03. We perform one million trials for each configuration.
Table 4 confirms that our optimal solvers match the results of Davenport's Q-method in the general case. In contrast, our Möbius approximation demonstrates a sensitivity to noise (potentially a benefit in the learning context of next section). We note that this approximation could likely be improved with a normalization step common in real homography estimation (Hartlyy and Zisserman 2004).
Table 5 similarly confirms that our two-point methods achieve the same optimal results as existing solvers. By utilizing unnormalized rotations, our weighted algorithm minimizes normalization costs, streamlining the compute. Most notably, in the unweighted case, our tailored solution only requires roughly a third of the multiplications of other methods, marking significant gain in efficiency.
Chair
Sofa
Toilet
Mean
Med.
Accs
Acc10
Mean
Med.
Accs
Acc1o
Mean
Med.
Aces
Accio
Euler
21.479
10.777
0.129
0.457
22.033
9.462
0.153
0.529
14.495
8,375
0.197
0.604
Quat
23.640
12.664
0.083
0.350
23.426
10.778
0.128
0,452
14.959
9.913
0.128
0.511
GS
13.606
6.320
0.350
0.738
15.015
5.469
0.441
0.801
6.586
3.708
0.682
0.915
QCQP
13.131
5.786
0.416
0.773
13.916
5,476
0.436
0.795
6.070
3.452
0.730
0.929
SVD
13.061
5.815
0.412
0.773
14.967
5.812
0.406
0.774
6.135
3.502
0.710
0.930
2-vec
12.544
6.100
0,380
0.751
15.077
6.217
0.364
0.753
6.069
3.483
0.713
0.926
QMAlg
12,604
5.696
0.425
0.783
14.336
5.657
0.419
0.793
6.079
3.590
0.714
0.930
QMSVD
13.157
6.211
0.366
0.748
13.683
5.421
0.443
0.799
6.026
3.601
0.699
0.926
Table I: Perr mean/median and accuracy (subscript indicates threshold) 3D shape alignment for different O-SSSO3 categories (Liao et al., 2019). Bold indicates best, underline indicates
second best.
5.2 LEARNING EXPERIMENTS
We conduct several experiments evaluate our proposed rotation representations. The primary loss function is the squared Frobenius norm |Rrredd - Rg which we refer to as Chordal L2, where Reredd is the predicted rotation and Rgt is the ground truth. For quaternion outputs, Chordal L2 is computed same as Peretroukhin et al. (2020). We compare our representatiioss-22 vec, QuadMobiusAlg (QMAlg) and QuadMobiusSSD
ler angles (Tait-Bryan YXZ), Quat (quaternion). GS (OMSVD)-againn several baselines: Eu- (Zhou et al., 2019), QCQP (Perctroukhin et al., 2020), and SVD (Levinson et al. 2020). In both QuadMobius variants, we use the algebraic method in the forward pass to avoid SVD computation and isolate differences to the backward pass. This section presents results on three public benchmarks. Additional synthetic ex- periments exploring different learning conditions are included in [pppeddix G.2.2] and full training details are provided in Appendix G.I
ModeINet1O-SO3 We first evaluate the representations on the 3D shape alignment task from Liao et al.] (2019) using the ModelNet1O-SC dataset. This dataset comprises of images of 3D CAD models under uniformly sampled rotations with multiple object models per category. The task is to predict the object's orientation directly from its image. Table I reports the results on three object categories, chosen for their low rotational symmetry following the choice in Levinson et al. 20200.
Inverse Kinematics Next, we test the representations an implicit learning task, applying them to the inverse kinematics task from Zhou et al. (2019). Given 3D human pose joint locations (from real-world motion capture data), network predicts the joint orientations relative to reference pose and uses a fixed forward kinematics function to obtain predicted joint locations. The distance loss is applied between the predicted and given joint locations. In this task, the rotations used as implicit representations through which the gradients must flow rather than direct prediction targets. Fig. 2/compares the results of the different learning representations on this task.
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Percentile Error Distribution Relative QMSVD
GS
1.20
Mcan
25th
50%
75%h
SVD
2vec
Euler
2.653
1.447
2.062
3.146
Joint Emoo Ratio
QMAlg
Quat
2.945
1.529
2.356
3.543
1.10
QMSVD
GS
1.629
0.8767
1.256
2.015
1.06
QCQP
1.511
0.7729
1.188
1.85
SVD
1,55
0.7647
1.16
1.855
2-vec
1,574
0.809
1.174
1.854
0.96
QMAle
1.51
0.8633
1.182
1.757
0.90
QMSVI
1.509
0.7421
1.13
1.81
0.86
20
60
80
100
Percentile
Figure 2: Results of implicit learning for Inverse Kinematics task (Zhou et 2019), Left: Mean and percentile L2 distance error (cm) of predicted joint locations. Bold indicates best, underline indicates second best. Right: Ratios of joint errors relative QMSVD across error percentiles (Euler/Quat omitteed due to large ratios).
Camera Pose Estimation Finally, we replicate the experiment from Walch al. (2017) which utilizes an LSTM to directly regress camera's 's pose from real world images. Training requires simultaneously optimizing over both the camera's ' orientation and translation. Data comes from the Cambridge Landmarks dataset (Kendall et al. (2015) which includes labels estimated from tradi- tional structure from motion pipelines. The results are seen Table 2irr training on select scenes,
following the choice of Chen et al. (2022).
Results Overall, the proposed representations demonstrated strong performance and versatility across the three benchmark tasks. Despite its lower dimensionality, 2-vec proved competitive, oc- casionally achieving the best result. Notably, it typically outperforms Gram-Schmide positioning itself as attractive alternative. The QuadMobius approaches showed their potential by achieving the top result in nearly all experiments over favorites like SVD and QCQP.
King's College 25th
Shop Facade 25th
Old Hospital
Mean
50th
t
Mcan
50
75th
Mcan
25th
50
75th
Buler
4.192
2.403
3.684
5.509
6.826
4.129
6.050
9.305
4.748
2.204
3.247
6.162
Quat
2.759
1.367
2.251
3.499
6.604
3.762
5.339
8.153
4.570
2.486
3.377
5.546
GS
3.298
1.764
2.583
4.137
6.559
4.376
5.660
8.343
4.295
1.897
3.070
5.698
QCQP
3.204
1.540
2.537
4.129
6,802
3.901
5.797
8.539
4.454
2.156
3.304
6.267
SVD
3.292
1.589
2.624
4.110
7.117
4.157
5.647
8.370
4.574
2.420
3.485
5.961
2-vec
3.085
1.536
2.371
4.014
7.118
3.789
5.762
8.957
4.294
2.085
2.950
5.292
QMAlg
2.631
1.337
2.052
3.267
6.317
4.050
5.268
7.758
4.426
2.035
3.238
5.640
QMSVD
2.706
1.391
2.177
3.345
6.715
4.074
5.710
8.947
4.409
2.077
3.146
5.744
Table Mean and percentile Our predicted rotations from direct pose prediction on different scenes in Cambridge Landmarks Dataset (Kendall et al. 2015). Bold indicates best, underline indi- cates second best.
CONCLUSION
This paper demonstrated the utility of special unitary matrices for rotation estimation. Several new formulas and algorithms were presented from this perspective for the real and complex domains, tackling Wahba's problem and rotation representations in neural networks. Various experiments con- finmed the potential of these approaches. Future work may include further solidifying the theoretical and empirical foundations of our rotation representations and applying special unitary matrices to other tasks such analytical camera pose estimation.

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Published as a conference paper at ICLR 2026
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Special Unitary Parameterized Estimators of Rotation
Appendix
A MATHEMATICAL BACKGROUND AND DEFINITIONS
The mathematical background for special unitary matrices and related concepts is briefly reviewed. The formulas are all established and generally known. A complex square matrix is defined as unitary if:
UU U²U = [det(U)) =
(27)
where H denotes the conjugate transpose, | denotes complex magnitude, and del(.) denotes deter- minant. The matrix is special unitary if it has the additional restriction that det UU) = 1 exactly.
Stereographic projection ei an invertible mapping of the sphere S {(x'. yo: 2e) 2 + }} from the point p* (0,0, 0, 1) to the complex plane and is given by:
Iz
Y
vc(aa
+
Ip
+ yp =
(28)
1 Za
1 23
ve' (1 (1+ x7 + y
2%p
2tp
1
x p
y
(29)
1 x2 + 83
1 x2 + Vイ
where 2 and C. This projection is visualized in Fig. 3 Note that dhe is undefined when p". To overcome this, the map is extended to the complex projective space CP which includes the point at infinity so we can define vier (p*) = co. The projection now redefined below with equivalence relations:
a a
1
Mer a)
(30)
00
a= pp'
AEC, X 0, &e (Ver(a))
In this paper, our use of v generally refers to t'cp. From the above definition, (aaa) can be arbitrarily scaled, and bijectively maps the entire sphere the complex projective space. Note that this mapping is not unique, particularly since choice of p" is arbitrary (any point on 5 is valid). We will use the specific projection defined above for this paper as is convenient for image processing.
A special unitary matrix U E S2) can generally be written as:
8
U
(31)
3
á
aã + 33 = 1, a,
where the bar denotes complex conjugation. U transforms a complex projective point z = [21, 22) and complex plane point by:
8 21
U 2 d' Un
(32)
-3
22
0z + B
32
+
ā
0
(33)
-3z + à'
These transformations are of importance as they act analogously to rotations of the unit sphere in R3. Specifically, for a 3x3 rotation matrix R E S3(3) that rotates a unit vector v E S2 as v' Rv, there exists some U such that:
v = ( (v)
(34)
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The exact relationship between (2) and SO(3) is made clearer by their relationships with unit quaternions E which also act as rotations in R The isomorphism between SU(2) and unit quaternions is given as:
q Wy + Ligh yad + kn w + x + ,/g + =/ =
Wiy Ligs Dq: Fq
a - w + Ligh B yy + zqd
(35)
and the mapping of unit quaternions to special orthogonal matrices is given by:
1 2y²
2
2gggyy 2w 1 2x
2z 2 + 2w M/g
Rg
2xqU4 + 2
2
2zy 2
(36)
20 2wayl
2g + 2
1 2x 20
Eq. (36)iis the well-known 2-to-1 surjective mapping between quaternions and rotation matrices. By their isomorphism in in/Eq. (35) (U( also has similar surjective mapping with SO(3), linking the three rotation representations. Note that the mapping given by (35) not unique. Furthermore, special unitary matrices have the ability to act as rotations in R directly by first mapping points to 2x2 complex matrices. For point X
(x, y, 2) E R
++
X: X
(37)
1 + -ii
( ) X x(x2) + X2: X1 E R
X = UX, Ue (U
(38)
Note if |X|| = 1, xx) E (2)... Also note that the map x is not uniquely defined either.
Relatedly, Möbius transformations are general 2x2 complex projective matrices, characterized sim- ilarly by:
E
M
(39)
7 8)
(Mt 0, o,E, C
M: z +2 Mz
T
21
(40)
7
22
o2 + 8
Φ 12 12 2'
72 +
(41)
72 + 8"
M AM, E C, X
(42)
Möbius transformations conformally map the complex projective plane onto itself. They uniquely determined (up to scale) by their action on three independent points, and SU( elements constitute subset of them.
B PROOFS AND DERIVATIONS
B. PROPER METRIC IN COMPLEX PROJECTIVE SPACE
B.1.1 DERIVATION OF METRIC
Complex projective rays are equivalent if they are linearly dependent. We can test this condition by setting up the following constraint on complex vectors [21, 2y.2?] and =[prP22] for 21. 22, PI, P2 E C:
21 (nl) det
21P2
22P1 =0
22 P2
Ves Ne. (z,: 22,, = (m,, ny. Pa) E S (assume a * pp, b + p²) whose projections via db [Eq. (30) correspond and p respectively, can show that testing the linear independence
For vectors a
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thus showing that the two formulas yield the same quantity. is easy to see that Eq. (4) is symmetric, so the same result would hold if p* and b + p. If b '=. we see that ||a bll² is clearly 0. At the same time, the numerator of Eq. (4) would be 0 while the denominator is nonzero as Published as a conference paper at ICLR 2026
Figure 3: Visualization of stereographic projection from the sphere (S²) the complex plane. The projection is performed by taking the line between p" and each point and intersecting that line with the plane through the equator. The point p" itself mathematically mapped infinity.
of complex vectors is in fact related the chordal distance sphere:
yo
m. nz
A
X2
X1 C, A1 # 0, #0
1
2
1+ P.
del
21
22
Ax 2 Aa [1( + p3) + - (1 + 2, (m, + n1,))2
P1
122
|^ 1112 2 (11 23)
v (1+ 21+ + +
- pb) (1 + 2 2x) + yens))
An
A2 (1 (1) 2. ((1 +,, - (1+ p.) 2(r,, + ysn,))
1 (1 (1+ 2(x.m, yonu + 2.P+))
= | A2 + p1)) + a
+ and |2 + p.) 11 Substituting this into our expression and rearranging, arrive at the final expression for the equivalent distance metric in complex projective space as:
Notice that |^ı (1 + 2.)
2
2
2 P2 22P1 2
aa b//2 (22122 + 222 (2 1p2
The last substitution may unnecessary at first: however, this form is more useful it gener- alizes the metric to hold even when = p or b p' (proof below). It also gives an intuitive interpretation that the spherical chordal distance related type of "cross product" magnitude between the two projective rays' unit directions.
B.1.2 PROOF METRIC FOR POINTS AT INFINITY
Proposition Ifa = p* or p² in Eg. (4) the proper metric is still valid.
Proof The The squared distance between unit length points a (I.,, Us. 2s) and b = (0, -1) is:
|a b/1a 2 2a b- 2(1 + 2x))
Using vectors z (aa A [z, yai, 1 + 2 2= and a o p", we can calculate the same quantity via the formula in
T. p <'pp)
122 0 with nonzero \1 Ae E C
Eq. (4)
4 4|2122 - 2
4| A2 2.)12
2(1 + 2.)
2| 2 12 +z))
the projective scalars X, * 0 for any valid complex projective point. Thus, both quantities equal
in that case as well, so the formula gives the spherical chordal distance between any two points on the sphere via their stereographic projections.
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B.2 REPRESENTATION DERIVATIONS
B.2.1 DERIVATION OF 2-VEC
For 3D vectors ba, by extracted from model output representing predicted target z and y axes respectively, we apply the method from [ecctinn 3.3 in the unweighted case to arrive at an optimal rotation matrix (in the sense of Wahba's problem). We assume by x by 7 O. First, b. and by must
have the same norm for the method be unweighted, so we transform by via by / l1,,1 by.
Since the reference points are constant (al = (1,0,0) ized sum and difference vectors are at <2 (1, a = (1, (1,-11 11,0). Similarly, we create
(0, (0.1,0),, we know that their normal-
normalized sum and difference vectors for the target points as b b +b;
lbb_ +b]1 and b b,-bi b bbb, Il
The optimal rotation aligns at to b+ and a to b noiselessly. This can be achieved because all the vectors have the same magnitude (normalizing to unit norm was found to be more stable than matching magnitudes like by) and because the sum and difference vectors are always orthogonal. Since rotation matrices naturally encode how an orthogonal coordinate frame transforms in their columns, we can construct the aligning rotation by joining the two rotations R, and Rs which ro- tate the coordinate frame to the reference sum/difference vectors and target sum/difference vectors respectively:
Ra = [at, a,, at x RB = [b+, b, b x b-] R = R = (b+ bb),, (b+ bb), b xb+
Because the sum/difference vectors are orthogonal and have unit norm, Ra Rb. R E SO(3). Given the natural representation coordinate transformations in rotation matrices, using the rotation ma- trix formulation was more appealing for the map than the quaternion formulation in Eq. (22) It also provided a more direct comparison with the Srhmm-Shhidtt map. Nonetheless, the core sight was derived from the original linear constraints on quaternion parameters. The unweighted method was chosen for its geometric and computational simplicity, but weighted version of the map incorporating the magnitudes of b by can be similarly formulated from Eq. (21)
B.2.2 DERIVATION OF QUADMOBIUS FORMULAS
Following the algorithm in Section 2.2
normalize 2x2 complex projective matrix M by its determinant and find the nearest unitary matrix, which by Appendix B.3. is special unitary. The following are two different approaches impelement this. We assume M has full rank.
Linear Algebra Instead of normalizing M directly, we take more streamlined approach by uti- lizing the properties of polar decomposition and determinant. We express det(M) in polar form as =0 with = [det E R,r > 0 and et = [de[[M]] lying on the unit circle. For polar decomposition M = QP with unitary matrix Q and positive definite Hermitian matrix P, we have (M) = det( (P). Because Q is unitary, [de((Q) = 1, and because is positive definite Hermitian, det(P) ) is real and nonnegative. It follows then that det(Q)
det(M)
e and det(P) r. To normalize M, we typically multiply it by a nonzero scalar À E C. For polar decomposition to
(1)) Q) (|2PP), meaning that only the phase of À affects the unitary factor. Since the unitary factor Q is the nearest unitary matrix to M in the Frobenius sense, the final solution is just A Q such that det( 2 [x] Q) = 1 to be special unitary. [X]
remain valid under this scaling, À must distribute as AM
We can therefore reverse the order and first compute Q before normalizing its determinant. We find scalar X' such that det (XQ) X²2 Q) (since Q is 2x2) for |X| 1. We can easily solve X' del(() Since Q = UV from SVD (M- UEVH) and |det Q)) = 1, we can rewrite our expression simply as / (U If M is singular, there is unique solution as SVD is no longer unique. This formula may still be used in practice with specific SVD.
Algebraic First, we normalize M M' det(M) M such that det(M') = 1. Next, we can utilize the isomorphism between (U and quaternions in Eq. (35) to algebraically solve for the nearest special unitary matrix. It's easy to verify that the unitary matrix Q that minimizes
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the Frobenius distance to M' maximizes R(Tr(M"Q) Q)) where R(-) denotes the real part. From Appendix B.3.1] we know that Q will be special unitary. Thus, we can express the optimization problem (using symbols from Eqs. (31) and (39))
max (2)
3RTTrMM' Q)) R(Fa -B da -38)
max (R(o) || (3)) + (RR(E) - R(()) +
for quaternion q = Wig + Lgi + Dqd + z,kh and 3(.) denoting the imaginary part. For q to be a valid rotation, must have unit norm. Thus, the optimization problem can be rephrased as finding the unit norm vector whose dot product with the coefficients of the quaternion parameters above maximized. The solution is trivially obtained by the unit norm vector in the direction of those coefficients. Using Eq. (35) again, we can express the solution as:
q (M) R(()) + (3()) 3())) (R(()) R()) (3(E) 3())){ ā = a + 6, - Q M' + )dj(M))
where tilde denotes unnormalized parameters and a-j denotes the adjugate. We can nor- Vlo + o =
malize the parameters by dividing ā and 3 by V |a) MT" + ()(de((M)) (Tr(M"MM') + Since that factor is real and distributes linearly through ā and the elements of M', tor into the original normalization factor of (M) - in the first step. The combined normalization factor can be written as:
efficiently combine this normalization fac-
1
v MM) /T Mr +2
MM)
Tr(M" MJ
+
[d]
[M)
(M)
(M + 2(d))(MM))
[det(M)[[I((MMMM M) + 2)det(M)))
Applying this normalization factor to M obtain M will ensure that M adj(M))" e SU(2).
B.3 NEAREST UNITARY MATRIX
B.3.1 PROOF OF NEAREST SPECIAL UNITARY MATRIX
Proposition 2 Möbius transformation M has det(M) the nearest unitary matrix to M in the Frobenius sense is special unitary.
Proof M has singular value decomposition given M UEVH where U and V unitary matrices and Σ a diagonal matrix with singular values. The determinant of M can be expressed
(M) = det (et((U)de(())eee()))
(43)
by product rule of determinants. Multiplying both sides by their complex conjugates, obtain:
(2) |det (e){ det) (E) [elelet(t))))
Since U and I unitary matrices, the magnitude of their determinant is the expression simplifies to:
(de) 2 [2)
[det(M)) |det (E)
because the determinant magnitudes are real and nonnegative. Since Σ a diagonal matrix with real, nonnegative elements, its determinant is simply the product of its diagonal entries and is in turn real and nonnegative. If de((M) = 1, then [de((E) = ( = 1. Coming back to the first
expression, we can now write:
det(M) = det (ed(Uedet(HH) det dvt(UU
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It is known that closest unitary matrix to M in the Frobenius sense is the unitary part of polar decomposition (Keller 1975) which can be computed by UV" From above, we can see that det(UV") which means that UV" is special unitary by definition.
In noiseless situations, Σ observed to be the identity matrix det(MM = 1. As noise is added, the diagonal elements of Σ drift from 1, E encodes notion of how close Möbius transfor- mation's s action is to rotation or how much noise the problem contains, making it a candidate for optimization.
B.3.2 DERIVATION OF NEAREST UNITARY MATRIX DERIVATIVE
The nearest unitary matrix in the Frobenius sense to complex square matrix M is given by the unitary factor Q of its polar decomposition matrix (Keller 1975). We can find the derivative of Q with respect to the elements of M by taking the derivative of both sides of the polar decomposition:
QP where is positive semidefinite Hermitian
dM = d(PP)
dM [
Q" " (dM) = Q ed (dQ)P +
Taking the conjugate transpose both sides and subtracting the two statements:
(dM Q p (dQ" dP"
n (d(MM) - (dM" " Q - Pd (dQ")Q (dP-dP")
P
We observe that because P is Hermitian for all values of M, dP must also be Hermitian, so the last term cancels out. Furthermore, we can deduce the following from definition of unitary matrices:
Q"Q= =I
(dQ Q (dQ) =0 (dQ H
Q Q" (dQ)
implying that (dQ#)Q is skew-Hermitian. Denoting X Q (dQ) and C =Q (dMM) (dM")Q, QQ we can now write:
XP
which takes the form of Sylvester equation. Since P is Hermitian, it admits diagonalization YAYH, where Y is unitary and is diagonal matrix of cigenvalues P:
= --XAAY" + YAY"X 'C - A(Y
The right hand side has the same term Y multiplied the left and right respectively by diagonal matrix A. As such, we can equivalently express the result as follows in order to solve for X and ultimately dQ:
y = =ddiag(A A) @ diag(A) ) @ (Y")Y)
-'
(A) @ (A
Y#CY
x Y
Y
(A) (
dQ QY (Q" (dM) (dM QQYY Y
diag(A) + Aiag(A)
where 2 denotes an outer sum operation, 3 denotes Hadamard multiplication (element-wise), the division is Hadamard division (element-wise), and diag(-) is vector formed from the diagonal elements of the matrix Note that this solution is only properly defined M is nonsingular (i.e. A has full rank). Otherwise, the polar decomposition is not unique and neither is its derivative. In practice, we choose to replace any instances of division by 0 in the result above with multiplications by O as specific solution.
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B.4 TWO-POINT SOLUTIONS
B.4.1 PROOF OF WEIGHTED CASE
Proposition 3 Let and b; represent the reference and target points respectively and K2 a1 and k2 b1 b2. For n. = 2 points, ka + O, and kú + the optimal rotation to Wahba's problem given the weighted average (in the Frobenius sense) between two rotations RA and R2 defined by R,a, = b, and R.
k, [all
ks 'Ik]ll
Lemma: If all points lie in the plane z=0 and k. + 0, kb. + O, and k.. ko 0, the optimal rotation a rotation around the z-axis.
Since all points lie in the plane result, the last column and row of BB" and B"B are also zero, so they both have a kernel vector of (0, 1). For the SVD of B given UEVI, the optimal rotation R (via Markley (987)) can
0, the last column and row of B (Eq. (2) zero, As
take the form:
1 0
0
0 O
07
R
1
0
0
det det(U)det(V) v)
where det(U))et(( V) is either 1 or since U and V are orthogonal matrices. Thus, the last column and row of R are both 0, or (0, 0, -)). In order for R be valid rotation matrix, the remaining upper 2x2 submatrix must be an orthogonal matrix which can be generated by a single parameter 6. Furthermore, the sign of the bottom right corner element of R must be the same as the determinant of the upper 2x2 submatrix for del(R) 1. These conditions reduce R to one of the two general forms:
(cos(0) 01)
s0n 0
coo((02) sin(
sin)
0
(1)
0
(O)
0
We denote the former as Rso and the latter Ro. The optimal solution to Wahba's problem
maximizes the gain function Tr Tr(RB) (RT)) Lourakis and Terzakis 2018 This quantity for both forms can be expressed below:
Tr(RsoB") )1,1cos(01) (1,2sin(11)
Tr(RoB") = (22,co0((02) 22,
\1. B1,1 B2,2, 21,2 B2,1 B1,2
22,1 = B1,1 - B2,2, 22,2 = B2,1 + B1,2
The gain function in both is the dot product between (Mr,1 Av.2) and sin) Its maximum value (subject to the constraint cos(f:)2 sin(0.)2 1) is obtained by the unit vector aligned with (Ai,1, 21, i.e.:
cos(O.) = A sin(O.)
A
V AA X2
/x2, + X2
Substituting this back into the gain function, we see that the optimal value is simply the magnitude of A
Tr Tr(Rso v Ai, + + Tr(RoBT) X2,1 + X2
Since the square root function is monotonically increasing, the larger of the two radicands corre- sponds to the larger gain value. We can compare them directly by taking their difference:
(A + + - (22, x222) = 4w w2 (k, ks)
where w; are the weights. Since the weights are positive and the cross products are assumed nonzero, the quantity above is positive when k, and k; point in the same direction and negative otherwise. Thus, when the cross products of the reference and target sets are aligned, Rso corresponds to the larger gain value and is the optimal rotation. It takes the form of rotation about the z-axis.
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Proof We assume that all points lie in the plane 0 and that the eross product of the reference and target sets are nonzero and are aligned. This will be generalized later. We construct rotations RJ and R2 be rotations about the z-axis that align a to b and a2 to b2 respectively. Since the input points have unit length and the vector norm is rotationally invariant, we can rewrite the loss function
W1 b1 Ra W2 b2 Ra |
w1 a1 R Raj 2 + w22 a2 R Ray
w1 [1 R R.)aja | || P + wyy w2|l([ R. R)
wraa (I R "R R. R R|ai + w2a (I- - RR- R) (I R R)aa2
2w + w22 2w al R Rai - 22w2a3 R Raz
using the fact a RRRRRa, a R.a. Under our assumptions, the lemma establishes that the optimal rotation R. is rotation about the 7-axis. Since both R. and R2 are also rotations about the 7-axis, we can easily verify that the products R R. and R are rotations about the z-axis as well. Using Rodrigues' rotation formula,
expand the term below as follows:
af R Rai Rai (c-(co(())) (in( k a (1 cos(o)) a))k) cos( + sin ())()) (o) (a) (k x ai)) cos(o)
where & is the angle of rotation of R and k [0, 0, 1|" is the axis of rotation. The simple result due to the fact that aı is orthogonal to the axis of rotation and has unit length. On the other hand, note that the Frobenius norm between R₁ and R computes the following:
R1 R P Tr(RR R) (R1 R))
= 6 2 ( (R.
6 - (Tr((co( sin (6) ) [k| + (1 cos (0) )kkk
6cos(Ø) 2(1 cos( (d)) - 4cos(
cos(Ø) = R. R |
The expansion of R R, above is due to the axis-angle formula for rotation matrices where [k] denotes the traceless skew-symmetric matrix formed from k representing vector cross product. Deriving a similar result for a Rí Ray and plugging both back into our reformulated loss function, rewrite it
2(w) + w2) 2wi (1
1 4
R R.IA.) 2w2
1 | R
2 R RI/ + - R₂ - R
Through this expression, we can see that the rotation R which minimized our original loss is exactly the rotation that represents the weighted average the Frobenius sense between R, and Rg as specified in Markley et al. 2007). The uniform factor of 2 is irrelevant to the optimization.
Now we generalize the result. Starting from the assumed configuration. we can extend it to general configurations by applying arbitrary rotations R,, and R, to the reference and target points respec- tively, transforming them into a& and b'. In this new coordinate frame, the rotation matrix R' related to the original optimal matrix R shown below:
Σ wi b. Ra
Σ my R&b, - R. Ra,
wi R,,; R&R(R R. Jail?
Σ wwillby (R,RR?))a(
R'= R&RR
Because the vector norm invariant under rotation, the optimal loss value remains unchanged across all coordinate frames. Since the optimal value from the original coordinate frame is preserved
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above, R' represents the optimal rotation in the new frame. Furthermore, the Frobenius norm is also Inaatin-invariint, so we can apply the required rotations to estimate R' as follows:
Σ
w R R.l
W R.R.R R.
Σ R.R.R. - R
R = R R - R
Thus, the general case, the optimal rotation given by the weighted average rotation between Ri and R22 We can uniquely identify those rotations with at least two linearly independent points they transform. Starting with the reference and target sets:
R, b;
R, R. = R,b;
R
Each rotation still aligns their respective reference point to their target point. Furthermore, in our original coordinate frame, k2 and k; are aligned and are parallel or antiparallel to R, 's axis of rotation (z-axis), so they are unchanged by R. As a result:
ko
kb
R
kal
[kb
kn
kb
R,R,,R,, R.,)
Rs
[kal
|
Ra (a. X (22)
||aa Rbb x b2)
R
RR, (a) a2)l
b2)ll
R a1
x ab
bj x b/
|a
| bl
due to rotations distributing over the product. Thus, that align their corresponding reference point to their target point along with the cross products of the reference and target sets. As the cross products assumed nonzero and are orthogonal to their respective point set, the two points aligned by each rotation are always independent and therefore uniquely define the rotations. As shown, the optimal rotation is the weighted average in the Frobenius sense between them.
identify R2 and R'S the rotations
B.4.2 PROOF UNWEIGHTED CASE
Proposition 4 Let a,, b,, and w; represent the reference points. target points, and weights respec- tively. Given 2 points, Wi W2. a1 X a2 # O. and bı b2 * 0. the optimal rotation to Wahba's 's problem is given by the unique rotation R defined by R( a [[b,+bal| b.
b and
R(
b.-ba
Proof For two 3D unit vectors v1 and we introduce the following notation and easily verifiable results:
Vi
V2
V1 + V2
V
|-
V v- =0
V1 v = =V +
V x 4 = =(( X V2)
VI V V2 0 = v + 0, v+ 00
vi X V2 * 0, then the two vectors v and + well-defined and form orthonormal basis for the plane spanned by v, and v2. Consequently, v and + created from one pair of linearly independent unit vectors can be perfectly aligned with those created from another pair.
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With a1 x a2 b1 b2 # O, we initially assume that the points are configured such that they all lie in the plane 2 = 0 and that at = bxt and a = b This is generalized later. For this configuration, we note the following:
a1 X a2
(a
X a
x aa)
=
a
11(b x b+)
2
2
||
|
(b x b*)
(b1 x b²)
2 b- b
[b b+1
[a a2)
(b. x b2) >
0
Thus, the cross products are aligned in this configuration, and from the lemma in the general case proof, the optimal rotation is rotation about the z-axis.
From the dot product equality above, we can deduce that at is equidistant from ai ag. The dot product caleulates the cosine of the angle between linearly independent unit vectors measured in the plane spanned by the vectors (2 = 0 in our case). We know from the proof in the general case that the dot product of a unit vector in the plane the cosine of the angle of rotation. That angle is measured in the plane perpendicular to the axis of O. Thus, constructing rotations Raj and Raz which rotate at
with itself after rotation about the z-axis is
rotation, which is also the plane about the z-axis to and a2 respectively, we can write the following:
ai- a a2 a a = a+ (Ra, a+) a (Ra,a*) = cos(o)
where e denotes the angle of rotation of Ray. making أल) (canonically positive) the angle between a1 and at. In general, Rai j Rag otherwise and a would be identical. In order for the above to still hold, the angle of rotation of Ra, must have the same magnitude but opposite sign of o. A similar statement can be made for the target points.
Let Rp, and RD, represent rotations about the z-axis that align bt with b and b2 respectively. Recall a = bt. We construct the rotations R, = Rb, R and R₂ Rb, R which are also about the z-xxs to align a₁ with bj and ag with b2 respectively. If 1 is the rotation angle of R.,, then the angle of rotation for R₁ is - + vo since Ra, and RD, share the same axis of rotation and transposing a rotation matrix negates the rotation angle. For R2, the rotation angle is Φ - as Ra, rotates by ő and Rb; by =tb Thus, the rotation angles of R₁ and R2 have equal magnitudes but opposite signs.
From the proof in the general case, the optimal rotation R is the weighted average in the Frobenius sense between the rotations R2 and R2 recently constructed. The weighted average rotation max- imizes the quantity Tr(RB(T) where B' = E, W,R, Markley al. (2007) Given the previously made statements and the fact that W = W2, we can calculate B' as:
cos( - + 1/ sin Φ + v))
sin( (- o (b)
coos ( - 1)
-sin( - v))
0
R1
COS(
+)
R2
sin ④ +b)
cos( v)
0
0
1
Ccos( - ò + w)
B' = W + w2 R2 2wn
cos(
+))
0
due to the fact that sine is an odd function and cosine is an even function. Since R is a rotation about the 7-axis, we can directly compute Tr(RBB") as 2wn (2c0s(-0) + ())) + 1) where R's angle of rotation. We can trivially see that 0 must take on a value of 0 or (mod 2m) to be optimal, depending on the sign of cos(-o) o++ + v) as wi is positive. That sign can be determined considering a and b are aligned:
ā b > 0
Rm,, a - Rn (Rb, b- -R,, b-) 0
a (Ra, Ra, 1T (RBb -Rb, )at) >
cos( - + 1b cos( (p00-000 1/ - cos(Φ + v)) + cos(Ø v/ > 0
2c0s( (ooss-0+00 16) 2cos(o + 1))
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Since a and b* are also aligned, can similarly derive +c) 2cos ( both inequalities together (valid since they are positive quantities). we find that cos( + w)) 0. Thus, I must be 0 to maximize Tr(RB"), resulting in R. being the identity matrix and indicating that the current aligoment the optimal one.
0. Adding
To generalize this, again apply arbitrary rotations Rag, R₂ the reference and target sets respec- tively, transforming them into ax. bj. From the proof in the general case, the new optimal rotation R' = R.RRR. = R.R. Now, we simply verify below that this rotation aligns a b and a' to b' (combined + notation for convenience):
+
b + b2 [
a b+
|a
+ a2
b2
R + a22)
bj + b/2
|ai + a2
||b
bgll
R. R (a) + ab))
b' + b/2
||a + aL)
||b] + bbll
R'a = b/#
Since a and a'- are orthogonal, they are also linearly independent, and their transformation uniquely defines the rotation R', thereby completing the proof.
B.4.3 AVERAGE Two UNNORMALIZEI QUATERNIONS
In Markley al. (2007). was shown that the average rotation matrix in the Frobenius sense can be calculated via the quaternion which optimizes the following:
M
Σ w.q:q.
max 7 Mq s.l. ||و|| =
q
Where qe are the unit norm quaternions corresponding the rotations being averaged (sign of q is irrelevant). The solution is the eigenvector corresponding to the largest eigenvalue of M. In the two point approach Wahba's problem proposed previously, we need to construct two quaternion rotations and average them. The formulation above assumes all quaternions have unit norm. How- ever, would be computationally advantageous (see Table if we did not have normalize the constructed rotations, thereby avoiding two square root and division operations. From Markley et al.) (2007), known that the average rotation in the two rotation case is simply linear combination of the rotations being averaged. To average unnormalized quaterions q and q2. can express M and q as:
| |
M
11
2
TT 91 q1
+ u2u2q?
q
مور + v
where μt, scalars. The above takes advantage of the fact that scaling M does not change its eigenvectors. Thus, we reduce the problem from estimating a unit quaternion to estimating two scalars. As result, rewrite the objective as:
qn
/4
T
q1
Q2
92
A1,1 = wi
w2 q )2 |1
A1,2 = A2,1
(w) + w2) |221
(q: 42)
Wi (ql qo )
2 || W2 |2
|
max v Av s.t. vTV== =
where denotes the usual vector dot product. is the quadratic constraint ensuring that the linear combination of q and qa has unit norm, and A is the new 2x2 objective optimize over. Using
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the method of Lagrange multipliers, we find that the solution to the above takes the form of a gen- eralized eigenvalue problem Av XIv. Note that the scaling constraint I is positive semidefinite, generally representing the equation of an ellipse. Assuming I is invertible and well-conditioned (it is discussed later when this is not the case), the solution is the eigenvector of r-1 A corresponding to the largest eigenvalue. Through simplification and scaling, we can express the matrix similarly as:
||q ا2 (p)
wille willaal W2 1
r A
W2 q (q)
which maintains its eigenvectors from before. Since the matrix is only 2x2, the eigenvector v corre- sponding to the largest eigenvalue can be expressed closed form. Scaling the eigenvector by the constraint I'v and substituting it back into the original linear combination of qı and qz, obtain the average quaternion as:
udn
vd2
q
q P2 +
42 222 +
2( q2)
where the values μ and v can be expressed equivalently in two ways:
,(1) (wi - w2) 11 (2) 2wi [ ( a2), wa) 2w / [i -2)
a = -(1) + (((a))2 + w()))(1)
or
,(2) = -w'2- wi (mm)l/@h11 || (2)) = 2w2 ( |1 q2), 12 = 2w /q2l 42)
w = =17)) (+(22) + (())
Both yield the same result except when qi q = 0 in which case the rotation corresponding to the larger weight chosen. If wi = W2 in that case, then there is no unique solution and either of the rotations can be selected. The former solution set is used when W1 S W2 as to approach the correct value as qi q2 +
W2 and the latter is used when
Note that the denominator in the expression for the average quaternion simply VvTTv. Previ- ously, P was assumed non-singular and well-conditioned, but there are two cases in practice where this fails to hold. The first is when qı and q are linearly dependent, i.e. they represent the same rotation. If we choose the solution constants above by the previously described strategy and examine the expressions for ja and v, then it can be seen that TT is fact strictly positive for nontrivial solutions v and nonzero weights/magnitudes. Furthermore, it can also be seen that μÂ and v42 share the same direction in this case and thus cannot cancel out. The second case occurs when the magnitudes of q and/or q2 are small, causing I be ill-conditioned. This using the strategy described in [pppendix D.2 to only obtain quaternions of sufficient magnitude or by simply scaling/normalizingg the rotations when necessary.
be avoided by
B.4.4 DEGENERATE CASE SOLUTION
The degenerate case occurs when either of the cross products of the reference or target points vanish, and the previous approaches for the two point case cannot be applied. This is because the solution no longer unique. A particular one can be efficiently found through the following approach.
We assume without loss of generality that the target points are collinear (the reference points may or may not be) and the first target point is aligned with the x-axis (i.e. b (1,0, 0)). In this case, the last two columns of the constraint C, (Eq. (17) vanish. We can thus write our optimization as:
(m- x)i
y zi
a B
C
y zi
(o + m))
u
Σ w.C C.
min u Zu s.l. n u=
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This optimization is simpler than before and can now be solved directly over the special unitary parameters. Since Hermitian and positive semidefinite, the solution is the complex eigenvector of Z corresponding to the smallest eigenvalue. For reference points a; (2,, Mis 2i). this can be expressed in closed form as:
w 2 + W222 + |, + 2222
ú
w1 + W222 (1 w23/a)
or
W1 22 + |mua 2222
u =
21 22 (118/1 W232)/
where is the unnormalized eigenvector and the correct solution depends on the target points' configuration. If the dot product of the target points is positive, then the first expression is correct. Otherwise, the second is correct. Note that cigenvectors are only unique up to scale, so even after normalizing the solution so that u u 1, corresponds to a rotation about the x-axis and parameterizes the family of optimal solutions.
still apply arbitary unitary scaling of e This
For arbitrary collinear target points, we simply need to find any rotation aligning the x-axis to the first target point by and then compose it with u. If the reference points were collinear instead, we swap the reference and target points in the above approach and invert the rotation afterwards. In practice, we would choose the more degenerate (i.e. larger dot product magnitude) of the two sets treat collinear.
Examining the solution closer, it can be seen that u represents a rotation aligning weighted com- bination of the reference points we refer to as the "weighted average" with the x-axis. The weighted average takes the form of sum (W1 a1 + W2az) or difference (W1a: Wyay) depending on the sign of the dot product between target points. This suggests that more straightforward approach in practice would be to simply calculate the normalized weighted average of the reference points and align with b, directly. This generalizes to the case when the reference points are collinear similarly to before. If the weighted average is zero, then any rotation is optimal.
C ADDITIONAL STEREOGRAPHIC SOLUTION DETAILS
C.1 RECOVERING R
The solution U obtained precisely satisfies the relation in Eg. (34) However, using the maps laid
out in Eqs. (35) and (36) directly will lead rotation Ru that is not necessarily equivalent to the desired R in Eq. (1) This because our choice of p" and choice of isomorphism between
quaternions and special unitary matrices can each add implicit orthogonal transformation their map. Since their combined transformation V and its inverse are applied before and after estimation respectively, the relationship between U and R is characterized by the conjugate transformation:
R RUR
(44)
For our definitions, find that & is simply 90 degree rotation about the y-axis. When applied directly to the resulting q from the algorithm, the transformed quaternion is given as:
q" wy zai + Na. + I,kk
(45)
which is just permutation/neeaatio of the elements of q. We can verify that mapping q" to R via Eq. (36) indeed gives us the true optimal solution to the problem.
C.2 GENERAL STEREOGRAPHIC CONSTRAINT
The generalized constraint between complex rays [21, 22 and [2i, where 21 20 + yid.
22 22 + yzi, P1 = mı + nii, and p2 = m2 + nz2 is given by:
4w;
w/ (12) 1
|22 (pT
2 (22)2)
A [-21P2
-2222
P122
-11
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for complex inputs and below for real inputs:
m221
m2 + M1y2 nath
-22th m1y2 12211 822
Di,0
m2z/1 - m1132 + 122 - 11
M221 + m1 - 111/2 72
D
mIx1 + m2222 11 7/21/2
11
1222222 + 21 722
mayi m22222 + 21 + 12222
m2232
11111 + 11y/1 7/2./2
D = Di.0 D
We can verify that with 22 and P2 1, we obtain the original results in Eq. (9) and Eq. (TT)
Furthermore, we can use 22 0 and P2 = O to calculate results involving the projective point infinity. Thus, there are no singularities using the general constraint. From this, we can derive similar formulas and algorithms for the one and two point cases those proposed earlier.
Similarly, the following is the general constraint for estimating Möbius transformation from stere- ographic inputs:
A,m = [-21P2 -2222 P121
P1 m
D ROTATIONS OF EXACT ALIGNMENT
The equations this section are derived from the constraint in Eq. (18) for 3D points. However, we can easily derive similar equations for stereographic points using Eq. (1I)
D.I ONE-POINT CASE
Finding a rotation that aligns two unit vectors (i.e. b = Ra) a special case Wahba's problem where Since aligning pair of points constrains two out of three rotational degrees of freedom (D; and Q; have rank 2), there are infinite solutions in this case. The rotation whose axis the cross product of the points often chosen for geometric simplicity and can be calculated efficiently as:
2 b)
a b
q = 2'
(46)
Instead, we may choose another convention where we constrain an element of the quaternion to be O. Since the points can be perfectly aligned, q Gsq O, so q NN,l(Q,.). Leveraging this fact, we can simply take two linearly independent rows from Q and set them to 0 explicitly, imposing a rank 2 constraint. Given the homogeneous nature of this system, we can disregard the weight and determine the rotation using straightforward linear algebra techniques. Each row below is member of the kernel that has quaternion element equal (note only two rows are linearly independent):
0
+ m
2 +
x + m
0
2 -
n
ker <kr(Qi))
(47)
y
p
0
+ p
y n
m r
Normalizing any nonzero of Eq. (47) gives optimal rotation. Compared Eq. (46) this
approach has several advantages. First, the rotation is simpler to construct. Second, one of its elements is guaranteed be so composing rotations and rotating points requires fewer operations and memory accesses. This is particularly true for the first row of [Eg. (47) as it represents a 180 degree rotation whose action on a point can be more efficiently computed as reflection about an axis. Finally, [Eq. (46) has singularity when the cross product vanishes. Although each row of Eqq (47) has own singular region, straightforward to systematically select another row that well-defined in that region.
D.2 NOISELESS TWO-POINT CASE
With two independent sets of correspondences, we are able to fully constrain the rotation to a unique one. If we assume that the two sets can be aligned perfectly, then we recover optimal rotation
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from the intersection of the constraint kernels. Two independent rows of Eq. (47) can be basis vectors for the kernel of Q1- We can determine the optimal rotation by finding the member of ker( Q1) (represented as linear combination of basis vectors) that is orthogonal to independent row of Q2. For example, with the last two rows oq. (47) as a basis of Q and the first row of Q2, can solve for the linear combination weights a, b (note scale is arbitary):
21 P1
1/1 + n1
I2 m2
(a
y1 R1
+
P1 21
) =
1/2 n2
m1
x1
P2
0
21
a = (I2 (2) - Pi) + (22 p2) - x11)
(x2 m2) n1) (V2 m2) m2)
Substituting a and back into the linear combination and dividing by my 1 I1 gives the result from Eq. (20): This result is equivalent to the simple estimators found in Markley (1999); Choukroun (2009). However, an issue with this approach is that the singular region of this estimator is not simple, and the equation fails to produce valid rotation under several conditions (see |eng and Choukroun| (2024). Rather than checking each condition with threshold or applying sequential rotations to avoid these cases like other kernel methods, we can more systematically select the three vectors in our computation to guarantee a valid result.
In general, we observe that for point pair, either + b - b will have least significantly nonzero element. We can select the two rows from Eq. (47) corresponding to a nonzero element from these vectors for the first point pair to ensure linearly independent kernel vectors. We then choose one of the two rows of Q2 corresponding to a nonzero element of a + b or a - b for the second point pair to solve for the rotation. For instance, if 2 + mi # 0 and 8/2 + n2 + O, we can choose the first two rows of Eq. (47) and the last row of Q2 to produce another equation for the rotation:
kj [pI 21
1/1 m1
31 m. j
k2 [2] + Pı
V1 n1
m1 21
kg = [p2 - 2/
1/2 72
22 + m2
k1 ks
q
(48)
k2 k3
Though the dot and cross products are in different indices from before, the formulation is equally simple to compute. We select the nonzero elements by largest magnitude for robustness. At least one of the two rows we select from Q2 will yield valid rotation for a₁ X * 0. Otherwise, the rotation any kernel vector of Q1- We verify row validity by checking if either coefficient a or b for the relevant constraints is nonzero. Those coefficients are always reused in the final rotation calculation (e.g. a and b are the second and first elements respectively in Eq. (48)). This process therefore covers the whole domain and only requires handful of operations and comparisons even in the worst case.
E BACKPROPAGATION DERIVATIVES
For simple complex square matrix G. the derivative of an eigenvector v G with respect the elements of G can be computed as Magnus (185)):
VV vHy (dG)y
dv = (AI G)
where À is the eigenvalue corresponding to I is the identity matrix, and denotes the Moore- Penrose pseudoinverse. Typically, v = 1 by convention for most eigenvector solvers. our original problem (Eq. (14)). G is Hermitian as opposed to general matrix, so the elements of are repeated in the matrix through conjugation. Using complex differentiation conventions consistent with many deep learning frameworks, the loss derivative can be written as:
dL
(/(600 dv dL
dl dvv
(
d(G 2 dG dv
dv
dG,,
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10a
10a
10
10
14,
1..
1
1 c
10a
10
10-1
10
3
(a) Gram-Schmidt
(b) 2-vec
Density plot of loss gradient ratios for Gram-Schmide and 2-vec. The x-axis represents the loss L. and y-axis shows the ratio of loss gradient magnitudes Vb. LI| for the predicted rotation axes b, and by. See Appendix F for details. 2-vec exhibits noticeably lower variance, suggesting more stable gradients during learning.
Figure 4:
where ( denotes the complex inner product and C is the scalar loss. dC de can be extracted from the upper triangular portion of 2GM (after reshaping to 4 x 4), multiplying by 2 for the off-diagonal parameters to include the lower portion contribution. This method avoids the need for the other eigenvectors eigenvalues of G that weren't used in the forward pass.
dc
For QuadMobiusSVD (Eq. (25)), the backpropagation must go through the SVD operation M
UΣV well known that the nearest unitary matrix corresponds the unitary component Q of the polar decomposition of M = QP, where P a positive semidefinite and Hermitian matrix Keller (1755). Thus, instead of backpropagating through the SVD components individually, we can backpropagate through Q in more direct manner. [Appendix B.3.2 outlines the details of the derivative of Q with respect the elements of M. Given the well-known relationships between the polar decomposition and SVD (Q = UV# P = VEV#), we can reuse the SVD elements from the forward pass to calculate the derivative more simply as:
S (Eag(E) diag(E) U (MM (HM JU
dQ =
V
where ① denotes an outer sum operation, and the division is Hadamard division (element-wise). From this equation, the numerical complex derivative can be expressed as follows (note the indices, F is x
J. &
U,A
JC
FH J.m
dc
dc
F jam
U
VH
U
V
dMj.m
dQ
dQ
where (r denotes the complex Frobenius inner product.
The remaining operations in the maps are algebraically straightforward to differentiate through. We observe that the previous formulas compute the same gradients as PyTorch's automatic differentia- tion through complex functions torch. linalg. e eigh and torch. 1 minalgg......mmmm...mmmm....a.gg svd but in more streamlined manner.
THEORETICAL INVESTIGATIONS OF REPRESENTATIONS
2-vec The core idea behind 2-vec lies in leveraging more optimal projection (in the sense of Wahba's problem) than Sram--Schmidt to improve learning performance without increasing com- putational cost dimensionality. To theoretically support this, replicate the gradient analysis
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1.8
Benchmarks Synthetic
GS/ G5/2-vec
1.4
0.8
10
12
2-vec Loss
Figure 5: Visualization of loss ratio between Gram-Schmidt (GS) representation and 2-vec represen- tations for all reported figures in this paper (accuracy converted 1-Acc maintain directionalityy. Gram-Schmidt performs around 10% worse average than 2-vec with some experiments showing large discrepancy. 2-vec performed better on 41/52 reported metrics.
3.5
QCQP
SVD
QuadMotius (9)
QuadMotiis
Mean Loss
0.5
0.0
Dropout Rate (p)
Figure 6: Plot mean loss (Chordal L.2) against dropout rate of map representations. θ and M denote whether dropout applied to map inputs intermediate representation for QuadMobius.
2024) which evaluates how learning signals propagate through the representations. We first generate thousand random 6D vectors, each with components sampled uniformly from [-2, 2]. Each vector split into two 3D components, b, and by, representing predicted target æ, y coordinate axes. These are then mapped to a rotation matrix using both the Gram-Schmidt and 2-vec methods. For each mapping, we compute the Frobenius norm loss L be- tween the resulting rotation and the identity matrix. We then calculate the gradient magnitudes of L with respect to b and by and analyze their ratio. The results are plotted in Fig. We can see that the gradient ratios for 2-vec more tightly concentrated around indicating relatively balanced gradient flow between the two vectors. In contrast, the Gram-Schmide method exhibits a wider distribution with significant skew, often yielding ratios in the range of 10-100 which highlights its disproportionate focus on by. These results support the hypothesis that 2-vec facilitates more stable gradients for optimization.
experiment from Geist
QuadMobius In our experiments, QuadMobius has consistently shown strong performance as a learning representation. To better understand why. conduct two experiments to probe its be- havior. We begin by generating one thousand realistic map inputs for each representation using trained models from synthetic Wahba's problem (trial #15 in Appendix G.2.2). All models fed the same noiseless inputs on which they perform equivalently for fair comparison. In the first experiment, test how resilient each map corrupted inputs by applying dropout. Fig. shows
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0.00014
0.010
0.00012
0.088
000010
0.006
000008
19c4
0.00006
0.004
0.00004
0.002
0.00002
QCQP
SV13
QuadMobius
QuadMebius
0.00000
0.000
1.5
12
E
Figure 7: Distribution plot of loss gradient magnitudes against loss L (Chordal 1.2). The left shows the gradient with respect the map inputs O, while the right shows the gradient with respect the Möbius transformation M estimated from eigendecomposition in QuadMobius.
the results of applying increasing dropout probability θ mean loss. For QuadMobius, also test applying dropout to its intermediate Möbius transformation M instead (real and imaginary parts treated independently). While might expect the sensitivity to dropout to decrease with di- mensionality, this is not necessarily the case as seen with QCQP. Notably, QuadMobius appears to be the most resilient dropout on O but is also the most sensitive when applied M. For the second experiment, we replace 10% of the model inputs with outlier points from another rotation, simulating
inference. Fig. plots the distribution of loss gradient magnitudes against loss. Gradients with respect to θ are similar across all maps. consistent with their equivalent perfor- mance on the task. In contrast, gradients with respect to M in QuadMobius are both significantly larger and more tightly concentrated, following square root trend. Together, these two experiments suggest that QuadMobius's 's eigendecomposition step enables the learning of a stable intermediate representation that is buffered against poor inputs, while its subsequent (U) projection ensures predictable, high-fidelity gradient flow, leading to its strong empirical performance.
SU(2) A natural question whether we can just directly predict an SU) representation and project it onto the manifold. This approach is simpler than QuadMobius and still provides an overparameterizee representation (8D). However, like quaternions, SU(2) suffers from the issue of double cover. Both Möbius transformation predictions M and -M map to the same 3D rotation, introducing ambiguity in learning. Furthermore, one might hope the rows of M offer two differ- ent estimates of quaternion rotation (similar to theoretical arguments of information averaging in SVD and QCQP). However, in (U) the rows encode the same information, so independence is not enforced during learning. Empirically, (U) prediction performed much worse synthetic experiments than QuadMobius (often close to quaternion) and was thus not included in results.
To further validate the QuadMobius approach, we conducted a toy ablation experiment in Table 3 We took 10k random map inputs and mapped them to quaternions. We then calculate the squared quaternion loss (accounting for sign) against a set of random ground truth quaternions and com- pare the loss gradient magnitudes of the inputs for the different map variants. The variants include SVD projection only (8D -i SU (2)). Eigendecomposition only (16D -i. Möbius transformation M, taking the first row of M as quaternion with and without normalization), and QuadMobius. The percentiles the gradient distributions and their subsequent percentile ranges are shown the table below. The QuadMobius approach yields significantly tighter distribution and lower amount of large outlying values than the other isolated components, suggesting that it provides more stable gradients for learning with both eigendecompossiion and projection.
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Method
10%
25%
50%
75%
90%
25-75%
10-90%
Projection
2.04e-5
2.66e-5
3.23e-5
3.65e-5
4.02e-5
9.90e-6
1.98e-5
Eig. (mo norm)
2.06e-5
2.87c-5
3.41e-5
3.88c-5
4.08c-5
1.00c-5
2.02c-5
Eig. (norm)
1.43e-5
2.07c-5
2.51e-5
3.01e-5
3.20c-5
9,41c-6
1.78c-5
QuadMobius 1.46c-5
1.79c-5
2.20c-5
2.49c-5
2.64c-5
6,98e-6
1.17e-5
Table Toy ablation experiment showiinng gradient magnitude distributions for isolated components of QuadMobius algorithm. Bold indicates lowest for spread quantities.
G EXPERIMENTS
G.1 EXPERIMENT SETTINGS AND DETAILS
These are the specific experiment settings used to obtain the results in our learning experiments.
ModeINet10-SO3 ADAM optimizer, learning rate 5e-4, NVIDIA L1 GPU, batch size 100, Chordal 1.2 loss, 300/400/800 epochs respectively for chair/sofa/toiler to train for roughly equal it- erations given dataset size differences. Architecture is ShuffeNetV2211. backbone Ma et al. (2018) (used for its quick training) pretrained on ImageNet weights followed by two fully connected lay- featuring ReLU activation and dropout applied before the layers with probability 0.4 and 0.25 respectively, Models saved by best average rotation error.
Inverse Kinematics Original author source code and settings Zhou et al. (2019) were utilized. Trained on NVIDIA LI GPU for million iterations. Epoch with lowest median rotation error was chosen for results.
Camera Pose Estimation Training code and settings obtained from Chen et al. (0222). Model initialized from pretrained GoogleNet weights recommended by original paper. Used NVIDIA L1 GPU and beta values 500/100/1500 for King's College/Shop Facade/Old Hospital. Trained for 1200 epochs with batch size 75, Models saved every 5 epochs, and models from last 300 epoch were used for testing (batch size in testing). Epoch with lowest median rotation eтror was chosen for results.
G.2 ADDITIONAL EXPERIMENTS
G.2.1 WAHBA'S PROBLEM
n 100
Algorithm
= le
= 0.1
Timings
1e
c 0.1
Timings
Q Q-method Davenport
1968
7.4676c-4
7.4868
3.583
1.2487c-4
1.2551
5.375
QUEST Shuster and Oh 1981
7.4676c-4
7.4868
0.250
1.2487c-4
1.2551
1.875
ESO Mortari 1997
7.4694e-4
7.4869
0.375
1.2487e-4
1.2551
2.000
FLAD Wu
2018
7.4676e-4
7.4868
0.333
1.2487e-4
1.2551
1.875
OLAE Mortari al.
7.7118e-4
7.8639
0.208
1.3120e-4
1.5952
2.167
Ours (Gp,
Eq. (12)
7.4676e-4
7.4868
4.084
1.2487e-4
1.2551
9,917
Ours (Gs.
Eq. (19)
7.4676e-4
7.4868
3,625
1.2487e-4
1.2551
6.500
Ours (GM Eq. (14)
1.2614e-3
12.608
0.917
3.5870e-4
3.7782
41.875
Table 4: Results of various Wahba's Problem solvers against varying noise levels with n = {3, 100). Accuracy values reported are median 0grr, and timing values median runtimes in microseconds. Timings taken with 00.1..=0.1. See|Seccion 5. for more info.
G.2.2 LEARNING WAHBA'S PROBLEM
To evaluate rotation representations robustly across various conditions, replicate the synthetic learning experiments from Peretroukhin et al. (2020); Levinson et al. (2020): Zhou et al. 2019),, using fully-connected neural network from Peretroukhin al. (2020) to learn the solution
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Algorithm
X
5th
50
95th
QUEST (Shuster and Oh 1981
89/ 99
1/1
3/3
3,3082/3,4111
9.1727 / /9.3970
27.0520 / 27.1371
Fast 2 Vee (Markley 2002)
72/ 78
3/3
4/4
3.3082
3.4115
9.1727 / 19.3970
27.0520 / 27.1371
SUPER (Ours)
74
3/2
3/3
3.3082 3.4115
9.1727 / /9.3970
27.0520 27.1371
Table 5: Operation counts and Per percentiles (Cnosse = 0.1) for two-point Wahba's problem solvers. Values given for wnweighted//eiighee algorithms without edge case handling. Bold in- dicates best.
LR
Loss|
Dom
Euler
Quat
GS
QCQP
SVD
2-vec
QMAlg
QMSVD
3
le-4
1.2
R
9.009/0
8.964/1
1.761/0
1.676/141
1.641/696
1.701/1
1.658/51
1.689/110 10
le-4
L22
C
119.364/0
13.632/0
5.768/0
4.237/1
4.264/1
5.781/0
3,823/109
3.761/889
Se-4
L2
R
12.154/0
9.618/0
1,583/5
1,518/143
14491582
1.560/0
1.501/217
1.527/53
3
Se-4
L2
C
119:403/0
12.238/0
4.016/0
3.586/2
3.735/6
3,917/0
3,447/751
3,408/241
3
le-3
L2
R
14.693/0
9.159/0
1.575/1
1.497/170
1.509/245
1.578/2
1,486/87
1.499/495
le-3
L2
C
119.397/0
11.212/0
3.290/24
3,289/190
3.253/384
3.269/0
3.250/110 10
3,232/292
1c-4
LI
R
8.063/0
4.120/0
1.603/0
1.445/135
1.421/622
1.570/2
1.469/164
1.459/77
le-4
L1
C
119,388/0
9.812/0
4.734/0
3,259/0
3.238/1
4.663/0
2.835/492
2.786/507
3
5e-4
L1
R
8.687/0
4.355/0
1,459/0
1.15/175
1.322/279
1.416/0
1.303/418
1.306/128
10
3
5e-4
L1
C
119.334/0
7.500/0
3.290/0
2.760/3
2.857/3
3.113/0
2.750/921
2.807/73
11
le-3
LI
10.833/0
4,436/0
1.434/0
1.312/53
1,301/338
1.427/0
1.317/337
1,291/272
le-3
LI
C
119.483/0
6,930/0
2.916/0
2.475/92
2.447/25
2.874/0
2.478/211 2.472/446
13
100
le-4
L2
R
3.784/0
3,277/0
0,569/0
0.253/138
0,243/389
0,313/0
0.255/169 0,251/304
14
100
le-4
L2
C 18.175/0
4.988/0
1.400/0
0.638/254
0.637/136
0,850/0
0.625/281 0.634/329
15
100
5c-4
L.2
R
5.395/0
3.712/0
0.547/0
0.249/121
0,247/175
0.303/0
0.247/368 0.242/336
16
100
5c-4
L2
119.370/0
5.009/0
1.586/0
0.831/682
0.866/223
0.940/0
0. 0666//6
0.848/29
17
100
lc-3
L2
R
6.608/0
3.269/0
0,537/0
0,243/292
0,272/112
0,297/0
0,261/299 0,253/297
18
100
le-3
L2
C
118.381/0
5.056/0
1.480/0
0.845/121
0.836/499
0,887/0 0,859/71 0.826/309
19
100
le-4
L1
R
2.249/0
1.794/0
0.356/0
0.269/293
0,261/327
0.332/0
0.264/130 0.265/250
20 100
le-4
LI
C
109.217/0
3.209/0
0.927/0
0.665/268
0.667/469
0.889/0
0.669/196 0.669/67
100
5c-4
L1
R
2.666/0
1.055/0
0.355/0
0.275/83
0.284/339
0.316/1 0.289/209 0,272/368
22
100
5e-4
L1
C
119.299/0
1,954/0
0.938/0
0.883/780
0.877/101
0,956/0
0.873/73 0,878/46
23
100
le-3
L1
R
3,867/0
1.384/0
0.366/0
0,280/167
0.280/316
0.331/0
0.277/346 0.291/171
24
100
le-3 L1
C
83.623/0
2.184/0
0.952/0
0.830/466
0.835/61
0.919/0 0.826/366 0.849/107
Table 6: Trial results for learning Wahba's problem with different rotation representations. n is number of points, LR is learning rate, Loss is type of chordal loss function, Dom is the domain, specifying whether the network is real-valued or complex-valued. Results are shown as /Lrr pairs where Our average rotation error on validation set, and Ldr. is the number of epochs where that representation was leader, i.e. had the lowest ferr overall as of that epoch. Bold indicates best value, underline indicates second best.
to Wahba's problem. Problem points and rotations are generated according to same procedure de- scribed in [Section 5.1] Each epoch, we dynamically generate 25,600 training samples and validate on fixed set of the same size (Enoise = 0.01 added to all samples). The models are trained for 1000 epochs with ADAM optimizer on an NVIDIA T4 GPU. In addition to Chordal L2, we also define the loss function Chordal L1 analogously as the sum of absolute differences between the elements of Ryred and Ryt. F.nalilll given our complex representations, we also evaluate training complex-valued networks |iao| (2023); Barrachina et al. (2023) of equivalent size for the task with stereographic complex inputs (Eq. (30). For real-valued representations, we take the real part the model output in this case.
As expected, the compact representations (Euler, Quat) performed relatively poorly. Overall, the best performers (QCQP, SVD, QuadMobiusAlg, QuadMobiusSVD) were all quite competitive with each other, having similar results and convergence rates. However, the QuadMobius representations together demonstrated edge, leading most of the epochs and having the lowest error in majority of trials. Although mathematically equivalent, the two approaches produced different results with neither approach consistently outperforming the other. On the other hand, 2-vec outperformed the
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other non-eigendecomppsitton representations (including Gram-Schmidt, beating them most trials, at times by a large margin. Although sigmificant differences for the complex cases were not observed among representations, some of the complex-valued trials featured the highest leader counts overall by our representations (e.g. trial #2, trial #10). The leader count gives a sense of the converenne/dominance of the learning well how cherry-picked the results may be based number of training epochs. See Fig. 8 for sample training/validatie curves which illustrate the advantage of noncompact representations and the competitiveness of our approaches.
G.2.3 REPRESENTATION TIMINGS
Euler
Quat
GS
QCQP
SVD
2-vec
QMAlg
QMSVD
Training
0.2123
0.0691
0.4903
0.5223
0.4904
0.4447
1.2231
1.6247
Inference
0.0401
0.0056
0.1050
0.2435
0.2737
0.0803
0.4298
0.6221
Table 7: Comparison of timings of different representations run with batch size 128. Measured on Apple MI Silicon CPU. Values reported are median measurements of 10000 runs in milleseconds. Training includes forward and backward passes (PyTorch train mode), and Inference includes only forward pass (PyTorch eval mode).
Table 7 shows the compute timings of the representations. 2-vec has notably fast inference tim- ings. QuadMobius representations are slower than others as they involve complex arithmetic and more compute steps overall. However, training time differences were observed to be negligible be- tween them and QCQP/SVD bottlenecks are typically present elsewhere the pipeline (e.g. data loading, network compute).
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Figure Progression of average Per over the training and validation sets for learning Wahba's 's problem Appendix G.2.2 for trial #15 in Table 6 Orange is training, blue validation.
1000
1000
020
000
600
QuadMMbiusSSV
Epoch
2-vec
400
200
Lonk
200
05
5000
1000
000
Gram Schnidt
QuaddMoblusAlg
600
400
200
200 ..a M
10.0
1.0
0.1
0.1
1000
1000
800
000
000
Qustemnion
Eperhh
400
200
200
10.0
0.1
011
5000
1000
800
800
Epech
Fuler
OCGP
4003
250
200
16.0
I6.0
C1
33


Reviews:
[{"review_id": "kApi324KXN", "reviewer": "Reviewer_Xg3U", "summary": "This work presents a comprehensive and theoretically grounded exploration of 3D rotation estimation, introducing Wahba's problem's solution via the special unitary group SU(2). This work proposes two continuous representations for learning rotations in deep neural networks:\n1. 2-vec: different from Gram-Schmidt's greedy approach, 2-vec maps these vectors to an optimal rotation in the sense of Wahba's problem.\n2. QuadMobius:  A 16D representation based on the paper's Möbius transformation approximation. It maps the network output to a Hermitian matrix, solves for its smallest eigenvector, and then projects onto SU(2) to get the final rotation.\nThe authors demonstrate these contributions through extensive experiments, including 3D shape alignment, unsupervised inverse kinematics, and real-world camera pose estimation.", "questions": "I suggest the authors highlight the intuition why SU(2) derived rotation form performs better early on in the paper. Though the theoretical in-depth derivation is presented beautifully, it's unclear whether the true source of improvement of QuadMobius is from the properties of SU(2) or Mobius transformation. I hope the authors could clarify this.", "limitations": "", "rating": 10, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1. The reformulation of Wahba’s problem via SU(2) presents a rigorous mathematical derivation. Translating theoretical derivation to practical representation (2-vec and QuadMobius) of deep rotatiton estimation is novel and intriguing.\n2. The paper tests its representations on a diverse set of tasks on ModelNet10-SO3, Inverse Kinematics,  and Cambridge Landmarks, providing strong evidence of their robustness and general effectiveness of the proposed representation.", "weaknesses": "1. In Appendix F, the author tried to directly predit an SU(2) form but it works poorly due to the double cover issue like quaternions. It raises the question that the true improvement stems from the intermediate Möbius transformation rather than the property from SU(2). The paper might want to clarify its narrative to highlight the key contributing factor.\n2. The QuadMobius has noticeably slower inference speed, shown in Tab. 7, but I believe this is a minor issue as running the neural network inference would dominate the slight increase of arithmetic computation from QuadMobius.", "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": "wtSeuLy9cX", "reviewer": "Reviewer_JA2J", "summary": "This paper proposes new parameterization methods for 3D rotation estimators. The method is based on operation in the complex number domain, and 3 methods are proposed including 2-vector, QMAlg and QMSVD. The proposed method is benchmarked on Wahba's problem, point cloud pose estimation, inverse Kinematics and camera pose estimation.", "questions": "- Do the proposed approaches solve the discontinuity and double cover problems of 3D rotation parameterization?\n- Do the baseline methods (e.g., GS) also solve these two problems? The GS baseline seems to work well for point cloud data and camera pose data, I am curious whether the importance of solving discontinuity or/and double cover issues are shown in the results.\n- What are the numbers of output elements need for the proposed methods to parametrize a rotation?\n- Figure 1 provides some intuitive illustration of the method. For QuadMobius, is there an interpretation for the fact that projecting to intermediate points can improve the results?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The parameterization of rotation estimator is a challenging and practically important topic in machine learning.\n- The paper introduce new theoretical results and their proofs.\n- The experimental results are extensive and the results are mostly better than the baselines.", "weaknesses": "- The paper is math heavy and may not be easy for general reader to get a gist of the approach.", "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": "Wp7RO7EX9G", "reviewer": "Reviewer_STif", "summary": "The paper develops three new representations of rotations based on unitary matrices.\n\nSection 1 introduces Wahba's problem (aka estimate a rotation matrix from vector observations) and provides a brief overview on recent works. \n\nSection 2 first discusses how to measure distances in complex projective space and uses the derived distance metric to formulate Wahba's problem in terms of unitary rotations. This allows the problem to be solved in terms of a set of linear constraints on unitary matrix parameters / quaternions (Section 2.1). Section 2.2 approximates the solution to Wahba's problem by estimating an optimal Möbius transform (a special case of unitary matrix). Section 2.3 derives another solution in terms of linear constraints on quaternions parameters.\n\nSection 3 discusses how the aforementioned solution to Wahba's problem extend to related rotation estimation tasks.\n\nSection 4 proposes to novel formulation of rotations: 2-vec (6D vector that is mapped to an optimal rotation using unitary matrices) and QuadMobius (16D vector is mapped to a Möbius transform and subsequently rotation matrix). QuadMobius is computed either via the SVD or through an algebraic method.\n\nSection 5 compares the proposed rotation representations to other representations (Euler, Quaternions, SVD) on various rotation estimation experiments.", "questions": "- Line 426 - Is this really an  \"Unsupervised learning task\"? Seems quite supervised to me as joint locations are given.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "Learning with 3D rotations affects numerous applications of machine learning, such that **providing three potentially state-of-the-art rotation representations** is a significant achievement for the ICLR community. The mathematical proofs are elegant and the experiments provide sufficient evidence on the utility of the derived representations. The code appears sound and will be (hopefully) made open-source.", "weaknesses": "The writing style of the paper is excellent. However, I personally found the paper's structure to be confusing. \n\nIn particular, important prerequisite for understanding the paper are placed in the appendix, while other seemingly unrelated topics are discussed in the main paper:\n1) The mathematical tools for this paper are given in Appendix A. Here, I would have liked a 2D illustration of the projective mapping. Also the notation in Equation 30 is a bit confusing. Perhaps, given the projection in (28) and (30) different symbols. Also an overview on the transformations between spaces (SO(3), S^2, CP1, S^3/Quaternion space, Möbius transform space) and the constraints in each space could be insightful.\n2) In Section 1, the reader is introduced to Wahba's problem and possible routes to solving it using unitary rotations. Some proofs are given in the section while others are moved into Appendix B.\n3) Section 3 talks about solving weighted and unweighted version of Wahba's problem. However, the derived equations are not used in the rest of the paper / experiments, such that I wondered what the purpose of this section was. It seemed this section could have been a paper of its own.\n4) Section 4 is somewhat comprehensive, but unfortunately the authors only very briefly mention (around line 350) that the derived representations relate to prior work on learning with rotations. I would have preferred if the authors would have moved derivations to the appendix and shorten/remove Section 3 to instead talk in depth about connections to related work.", "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": "nkj1wBcody", "reviewer": "Reviewer_gJ8i", "summary": "The paper, building on the Wahba problem, proposes two novel representations for learning rotations in neural networks. The first, 2vec, closely resembles the widely used GSO implementation, which has become a standard in neural network training. This representation retains all the desirable properties of the GSO approach, including smoothness and continuity. The second, QuadMobius, introduces a 16-dimensional parameterization for rotations, offering an alternative approach for representing rotational transformations in neural networks.", "questions": "no questions", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Solid paper - good visualization. \nwell motivated and good structure to follow along. \ncoherent notation throughout the paper", "weaknesses": "Experimental results show, as expected, no drastic benefit. \nDiscussion of timing results in the main paper can be emphasized more.", "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)