SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Huikang Liu
School of Information Management and Engineering
liuhuiqang@shufe.edu.cn
Xiao Li*
lixiao@cuhk.edu.cn
This work presents ReSync, a Riemannian subgradient-based algorithm for solving the robust rotation synchronization problem, which arises in various engineering applications. ReSync solves a least-unsquared minimization formulation over the rotation group, which is nonsmooth and nonconvex, and aims at recovering the underlying rotations directly. We provide strong theoretical guarantees for ReSync under the random corruption setting. Specifically, we first show that the initialization procedure of ReSync yields a proper initial point that lies in a local region around the ground-truth rotations. We next establish the weak sharpness property of the aforementioned formulation and then utilize this property to derive the local linear convergence of ReSync to the ground-truth rotations. By combining these guarantees, we conclude that ReSync converges linearly to the ground-truth rotations under appropriate conditions. Experiment results demonstrate the effectiveness of ReSync.
Rotation synchronization (RS) is a fundamental problem in many engineering applications. For instance, RS (also known as "rotation averaging") is an important subproblem of structure from motion (SfM) and simultaneous localization and mapping (SLAM) in computer vision [20, 22, 17], where the goal is to compute the absolute orientations of objects from relative rotations between pairs of objects. RS has also been applied to sensor network localization [41, 12], signal recovery from phaseless observations [2], digital communications [36], and cryo-EM imaging [35, 33].
Practical measurements of relative rotations are often incomplete and corrupted, leading to the problem of robust rotation synchronization (RRS) [28, 21, 22, 39, 9, 31]. The goal of RRS is to reconstruct a set of ground-truth rotations  $\mathbf{X}_1^{\star}, \dots, \mathbf{X}_i^{\star}, \dots, \mathbf{X}_n^{\star} \in \mathrm{SO}(d)$  from measurements of
relative rotations represented as
$$
\mathbf {Y} _ {i j} = \left\{ \begin{array}{l l} \mathbf {X} _ {i} ^ {\star} \mathbf {X} _ {j} ^ {\star^ {\top}}, & (i, j) \in \mathcal {A}, \\ \mathbf {O} _ {i j}, & (i, j) \in \mathcal {E} \backslash \mathcal {A}, \\ \mathbf {0}, & (i, j) \in \mathcal {E} ^ {c}, \end{array} \right. \quad \text {w i t h} \quad (i, j) \in \left\{ \begin{array}{l l} \mathcal {A}, & \text {w i t h r a t i o p q}, \\ \mathcal {E} \backslash \mathcal {A}, & \text {w i t h r a t i o (1 - p) q}, \\ \mathcal {E} ^ {c}, & \text {o t h e r w i s e}. \end{array} \right. \tag {1}
$$
Here,  $\mathrm{SO}(d) := \{R \in \mathbb{R}^{d \times d} : R^{\top} R = I, \det(R) = 1\}$  denotes the rotation group (also known as the special orthogonal group),  $\mathcal{E}$  represents the indices of all available observations,  $\mathcal{A}$  denotes the indices of true observations,  $\mathcal{A}^c := \mathcal{E} \setminus \mathcal{A}$  is the indices of outliers,  $O_{ij} \in \mathrm{SO}(d)$  is an outlying observation, and the missing observations are set to be  $\mathbf{0}$  by convention; see, e.g., [23, section 2.1]. We use  $q \in (0,1)$  to denote the observation ratio and  $p \in (0,1)$  to denote the ratio of true observations.
Related works. Due to the vast amount of research in this field, our overview will necessarily focus on theoretical investigations of RS. In the case where no outliers exist in the measurement model (1), i.e.,  $p = 1$ , a natural formulation is to minimize a smooth least-squares function  $\sum_{(j,j) \in \mathcal{E}} \| X_i X_j^\top - Y_{ij} \|_F^2$  over  $X_i \in \mathrm{SO}(d)$ ,  $1 \leq i \leq n$ . Spectral relaxation and semidefinite relaxation (SDR) are typical approaches for addressing this problem [34, 3, 6, 5, 4, 30], where they provide strong recovery guarantees. However, these results cannot be directly applied to the corrupted model (1) due to the existence of outliers (i.e.,  $p < 1$ ) and the sensitivity of the least-squares solution to outlying observations.
Theoretical understanding of RRS is still rather limited. One typical setting for theoretical analysis of RRS is the random corruption model (RCM); see Section 2.2. The work [39] introduces a least-unsquared formulation and applies the SDR method to tackle it. Under the RCM and in the full observation case where  $q = 1$ , it is shown that the minimizer of the SDR reformulation exactly recovers the underlying Gram matrix (hence the ground-truth rotations) under the conditions that the true observation ratio  $p \geq 0.46$  for SO(2) (and  $p \geq 0.49$  for SO(3)) and  $n \to \infty$ . In [23], the authors established the relationship between cycle-consistency and exact recovery and introduced a message-passing algorithm. Their method is tailored to find the corruption level in the graph, rather than recovering the ground-truth rotations directly. They provided linear convergence guarantees for their algorithm once the ratios satisfy  $p^8 q^2 = \Omega (\log n / n)$  under the RCM. However, it is unclear how this message-passing algorithm is related to other optimization procedures for solving the problem. Let us mention that they also provided guarantees for other compact groups and corruption settings. Following partly the framework established in [23], the work [32] presents an interesting nonconvex quadratic programming formulation of RRS. It is shown that the global minimizer of the nonconvex formulation recovers the true corruption level (still not the ground-truth rotations directly) when  $p^2 q^2 = \Omega (\log n / n)$  under the RCM. Unfortunately, the work does not provide a concrete algorithm that provably finds a global minimizer of the nonconvex formulation. In [29], the authors introduced and analyzed a depth descent algorithm for recovering the underlying rotation matrices. In the context of the RCM, they showed asymptotic convergence of their algorithm to the underlying rotations without providing a specific rate. The result is achieved under the conditions that the algorithm is initialized near  $X^{\star}$ ,  $q \geq \mathcal{O}(\log n / n)$ , and  $p \geq 1 - 1 / (d(d - 1) + 2)$ . The latter requirement translates to  $p \geq 3 / 4$  for SO(2) and  $p \geq 7 / 8$  for SO(3). It is important to note, however, that the primary focus of their research lies in the adversarial corruption setup rather than the RCM.
Main contributions. Towards tackling the RRS problem under the measurement model (1), we consider the following least-unsquared formulation, which was introduced in [39] as the initial step for applying the SDR method:
$$
\underset {\boldsymbol {X} \in \mathbb {R} ^ {n d \times d}} {\text {m i n i m i z e}} f (\boldsymbol {X}) := \sum_ {(i, j) \in \mathcal {E}} \| \boldsymbol {X} _ {i} \boldsymbol {X} _ {j} ^ {\top} - \boldsymbol {Y} _ {i j} \| _ {F} \tag {2}
$$
$$
\text {s u b j e c t} \quad X _ {i} \in \operatorname {S O} (d), 1 \leq i \leq n.
$$
Note that this problem is nonsmooth and nonconvex due to the unsquared Frobenius-norm loss and the rotation group constraint, respectively. We design a Riemannian Subgradient synchronization algorithm (ReSync) for addressing problem (2); see Algorithm 1. ReSync will first call an initialization procedure named Spectrln (see Algorithm 2), which is a spectral relaxation method. Then, it implements an iterative Riemannian subgradient procedure. ReSync targets at directly recovering the ground-truth rotations  $\pmb{X}^{\star} \in \mathrm{SO}(d)^{n}$  rather than the Gram matrix or the corruption level. Under the RCM (see Section 2.2), we provide the following strong theoretical guarantees for ReSync:
(S.1) Initialization. The first step of ReSync is to call SPECTRln for computing the initial point  $\mathbf{X}^0$ . Theoretically, we establish that  $\mathbf{X}^0$  can be relatively close to  $\mathbf{X}^\star$  depending on  $p$  and  $q$ ; see Theorem 2.
(S.2) Weak sharpness. We then establish a problem-intrinsic property of the formulation (2) called weak sharpness; see Theorem 3. This property characterizes the geometry of problem (2) and is of independent interest.
(S.3) Convergence analysis. Finally, we derive the local linear rate of convergence for ReSync based on the established weak sharpness property; see Theorem 4.
The main idea is that the weak sharpness property in (S.2) helps to show linear convergence of ReSync to  $X^{\star}$  in (S.3). However, this result only holds locally. Thus, we need the initialization guarantee in (S.1) to initialize our algorithm in this local region and then argue that it will not leave this region once initialized. We refer to Sections 3.1 to 3.3 for more technical challenges and our proof ideas. Combining the above theoretical results yields our overall guarantee: ReSync converges linearly to the ground-truth rotations  $X^{\star}$  when  $p^7 q^2 = \Omega (\log n / n)$ ; see Theorem 1.
Notation. Our notation is mostly standard. We use  $\mathbb{R}^{nd\times d}\ni X = (X_1;\ldots ;X_n)\in \mathrm{SO}(d)^n$  to represent the Cartesian product of all the variables  $X_{i}\in \mathrm{SO}(d),1\leq i\leq n$ . The same applies to the ground-truth rotations  $X^{\star} = (X_{1}^{\star};\dots ;X_{n}^{\star})$ . Let  $\mathcal{E}_i = \{j\mid (i,j)\in \mathcal{E}\}$ ,  $\mathcal{A}_i = \{j\mid (i,j)\in \mathcal{A}\}$ , and  $\mathcal{A}_i^c = \mathcal{E}_i\setminus \mathcal{A}_i$ . We also define  $\mathcal{A}_{ij} = \mathcal{A}_i\cap \mathcal{A}_j$  for simplicity. For a set  $S$ , we use  $|S|$  to denote its cardinality. For any matrix  $X,Y\in \mathbb{R}^{nd\times d}$ , we define the following distance up to a global rotation:
$$
\operatorname {d i s t} \left(\boldsymbol {X}, \boldsymbol {Y}\right) = \| \boldsymbol {X} - \boldsymbol {Y} \boldsymbol {R} ^ {\star} \| _ {F}, \text {w h e r e} \boldsymbol {R} ^ {\star} = \underset {\boldsymbol {R} \in \mathrm {S O} (d)} {\arg \min } \| \boldsymbol {X} \boldsymbol {R} - \boldsymbol {Y} \| _ {F} ^ {2} = \mathcal {P} _ {\mathrm {S O} (d)} \left(\boldsymbol {X} ^ {\top} \boldsymbol {Y}\right).
$$
Besides, we introduce the following distances up to the global rotation  $R^{\star}$  defined above:
$$
\operatorname {d i s t} _ {1} \left(\boldsymbol {X}, \boldsymbol {Y}\right) = \sum_ {i = 1} ^ {n} \| \boldsymbol {X} _ {i} - \boldsymbol {Y} _ {i} \boldsymbol {R} ^ {\star} \| _ {F}, \quad \operatorname {d i s t} _ {\infty} \left(\boldsymbol {X}, \boldsymbol {Y}\right) = \max  _ {1 \leq i \leq n} \| \boldsymbol {X} _ {i} - \boldsymbol {Y} _ {i} \boldsymbol {R} ^ {\star} \| _ {F}.
$$
2 Algorithm and Setup
2.1 ReSync: Algorithm Development
In this subsection, we present ReSync for tackling the nonsmooth nonconvex formulation (2); see Algorithm 1. Our algorithm has two main parts, i.e., initialization and an iterative Riemannian subgradient procedure.
Initialization. ReSync first calls a procedure Spectrln (see Algorithm 2) for initialization. Spectrln is a spectral relaxation-based initialization technique. Spectrln computes the first  $d$  leading unit eigenvectors of the data matrix to form  $\Phi \in \mathbb{R}^{nd\times d}$ . We multiply  $\sqrt{n}$  to those eigenvectors to ensure that its norm matches that of  $\mathrm{SO}(d)^n$ . We also construct  $\Psi$ , which reverses the sign of the last column of  $\Phi$  so that the determinants of  $\Phi$  and  $\Psi$  differ by a sign.
Then, we compute the projection of  $\Phi$  and  $\Psi$  onto  $\mathrm{SO}(d)^n$ . The projection is computed in a block-wise manner, namely
$$
\widetilde {\Phi} _ {i} = \mathcal {P} _ {\mathrm {S O} (d)} (\Phi_ {i}), \quad 1 \leq i \leq n,
$$
where  $\Phi_i, \widetilde{\Phi}_i \in \mathbb{R}^{d \times d}$  are the  $i$ -th block of  $\Phi$  and  $\widetilde{\Phi}$ , respectively. The projection can be explicitly evaluated as
$$
\widetilde {\boldsymbol {\Phi}} _ {i} = \left\{ \begin{array}{l l} \boldsymbol {P} _ {i} \boldsymbol {Q} _ {i} ^ {\top}, & \text {i f} \det  (\boldsymbol {\Phi} _ {i}) > 0, \\ \widehat {\boldsymbol {P}} _ {i} \boldsymbol {Q} _ {i} ^ {\top}, & \text {o t h e r w i s e}, \end{array} \right. \quad 1 \leq i \leq n.
$$
Here,  $P_{i}, Q_{i} \in \mathbb{R}^{d \times d}$  are the left and right singular vectors of  $\Phi_{i}$  (with descending order of singular values), respectively, and  $\widehat{P}_{i}$  is obtained by reversing the sign of the last column of  $P_{i}$ . The initial point  $X^{0}$  is chosen as  $\widetilde{\Phi}$  or  $\widetilde{\Psi}$ , depending on which is closer to  $\mathrm{SO}(d)^n$ .
Let us mention that the computation of  $\widetilde{\Psi}$  and Steps 5 - 9 in Spectrln can practically improve the approximation error  $\mathrm{dist}(\mathbf{X}^0, \mathbf{X}^\star)$ . We demonstrate such a phenomenon in Figure 1, in which "Naive Spectrln" refers to outputting  $\mathbf{X}^0 = \widetilde{\Phi}$  directly in Algorithm 2.
Figure 1: The average under 100 simulations of the initial distance  $\mathrm{dist}(\pmb{X}^0, \pmb{X}^\star)$  computed by Algorithm 2 versus naive spectral initialization (i.e., outputting  $\pmb{X}^0 = \widetilde{\pmb{\Phi}}$  directly) with  $p = 0.2$ ,  $q = 0.2$  and  $d = 3$ .
Riemannian subgradient update. ReSync then implements an iterative Riemannian subgradient procedure after obtaining the initial point  $\pmb{X}^{0}$ . The key is to compute the search direction (Riemannian subgradient)  $\widetilde{\nabla}_{\mathcal{R}}f(\pmb{X}_i^k)$  and the retraction  $\operatorname{Retr}_{\pmb{X}_i^k}(\cdot)$  onto  $\mathrm{SO}(d)$  for  $1 \leq i \leq n$ . Towards providing concrete formulas for the Riemannian subgradient update, let us impose the Euclidean inner product  $\langle \pmb{A}, \pmb{B} \rangle = \operatorname{trace}(\pmb{A}^\top \pmb{B})$  as the inherent Riemannian metric. Consequently, the tangent space to  $\mathrm{SO}(d)$  at  $\pmb{R} \in \mathrm{SO}(d)$  is given by  $\mathrm{T}_{\pmb{R}} := \{ \pmb{R}\pmb{S} : \pmb{S} \in \mathbb{R}^{d \times d}, \pmb{S} + \pmb{S}^\top = 0 \}$ . The Riemannian subgradient  $\widetilde{\nabla}_{\mathcal{R}}f(\pmb{X}_i)$  can be computed as [40, Theorem 5.1]
$$
\widetilde {\nabla} _ {\mathcal {R}} f \left(\boldsymbol {X} _ {i}\right) = \mathcal {P} _ {\mathrm {T} _ {\boldsymbol {X} _ {i}}} \left(\widetilde {\nabla} f \left(\boldsymbol {X} _ {i}\right)\right), \quad 1 \leq i \leq n, \tag {3}
$$
where the projection can be computed as  $\mathcal{P}_{\mathrm{T}\mathbf{x}_i}(\mathbf{B}) = \mathbf{X}_i\big(\mathbf{X}_i^\top \mathbf{B} - \mathbf{B}^\top \mathbf{X}_i\big) / 2$  for any  $\mathbf{B} \in \mathbb{R}^{d \times d}$  and  $\widetilde{\nabla} f(\mathbf{X}_i)$  is the Euclidean subgradient of  $f$  with respect to the  $i$ -th block variable  $\mathbf{X}_i$ . Let us define  $f_{i,j}(\mathbf{X}) \coloneqq \| \mathbf{X}_i\mathbf{X}_j^\top - \mathbf{Y}_{ij}\|_F$ . The Euclidean subdifferential  $\partial f(\mathbf{X}_i)$  with respect to the block variable  $\mathbf{X}_i$  is given by
$$
\partial f(\boldsymbol{X}_{i}) = 2\sum_{j:(i,j)\in \mathcal{E}}\partial f_{i,j}(\boldsymbol{X}_{i}),\quad \text{with}\quad \partial f_{i,j}(\boldsymbol{X}_{i}) = \left\{ \begin{array}{l}\frac{\boldsymbol{X}_{i} - \boldsymbol{Y}_{ij}\boldsymbol{X}_{j}}{\|\boldsymbol{X}_{i}\boldsymbol{X}_{j}^{\top} - \boldsymbol{Y}_{ij}\|_{F}},\text{if}\| \boldsymbol{X}_{i}\boldsymbol{X}_{j}^{\top} - \boldsymbol{Y}_{ij}\|_{F}\neq 0,\\ \boldsymbol {V}\in \mathbb{R}^{d\times d},\| \boldsymbol {V}\|_{F}\leq 1,\text{otherwise}. \end{array} \right.
$$
Algorithm 1 ReSync: Riemannian Subgradient Synchronization
Require: Initialize  $\mathbf{X}^0 = \operatorname{Spectr}\ln(\mathbf{Y})$  (Algo-rithm 2), where  $\mathbf{Y} \in \mathbb{R}^{nd \times nd}$  and its  $(i,j)$ -th block is  $\mathbf{Y}_{i,j} \in \mathbb{R}^{d \times d}$ ;
1: Set iteration count  $k = 0$
2: while stopping criterion not met do
3: Update the step size  $\mu_{k}$
4: Riemannian subgradient update:
$$
\boldsymbol {X} _ {i} ^ {k + 1} = \operatorname {R e t r} _ {\boldsymbol {X} _ {i} ^ {k}} \left(- \mu_ {k} \widetilde {\nabla} _ {\mathcal {R}} f (\boldsymbol {X} _ {i} ^ {k})\right)
$$
for  $1 < i < n$ ;
5: Update iteration count  $k = k + 1$
6: end while
Algorithm 2 Spectrln: Spectral Initialization
1: Input:  $\mathbf{Y} \in \mathbb{R}^{nd \times nd}$ ;
2: Compute the  $d$  leading unit eigenvectors of  $\mathbf{Y}$ :  $\{\pmb{u}_1, \dots, \pmb{u}_d\}$ ;
3: Set  $\Phi = \sqrt{n} [\pmb{u}_1, \pmb{u}_2, \dots, \pmb{u}_d] \in \mathbb{R}^{nd \times d}$  and  $\Psi = \sqrt{n} [\pmb{u}_1, \pmb{u}_2, \dots, \pmb{u}_{d-1}, -\pmb{u}_d]$ ;
4: Compute  $\tilde{\Phi} = \mathcal{P}_{\mathrm{SO}(d)^n}(\Phi)$  and  $\tilde{\Psi} = \mathcal{P}_{\mathrm{SO}(d)^n}(\Psi)$ ;
5: if  $\| \tilde{\Phi} -\Phi \| _F\leq \| \tilde{\Psi} -\Psi \| _F$  then
6:  $X^0 = \Phi$
7: else
8:  $X^0 = \tilde{\Psi}$
9: end if
10: Output: Initial point  $X^0$ .
Any element  $\widetilde{\nabla} f(\mathbf{X}_i) \in \partial f(\mathbf{X}_i)$  is called a Euclidean subgradient. In ReSync, one can choose an arbitrary subgradient  $\widetilde{\nabla} f(\mathbf{X}_i) \in \partial f(\mathbf{X}_i)$  at  $\mathbf{X}_i$ .
Mimicking the gradient method to update along the search direction  $\widetilde{\nabla}_{\mathcal{R}}f(\boldsymbol{X}_i)$  provides a point  $\boldsymbol{X}_i^+ = \boldsymbol{X}_i - \mu \widetilde{\nabla}_{\mathcal{R}}f(\boldsymbol{X}_i)$  on the tangent space  $\mathrm{T}_{\boldsymbol{X}_i}$  at  $\boldsymbol{X}_i$ , which may violate the manifold constraint " $X_i^+ \in \mathrm{SO}(d)$ ". One common approach in Riemannian optimization is to employ a retraction operator to address the feasibility issue. For  $\mathrm{SO}(d)$ , we can use a QR decomposition-based retraction and implement the Riemannian subgradient step as
$$
\boldsymbol {X} _ {i} ^ {+} = \operatorname {R e t r} _ {\boldsymbol {X} _ {i}} \left(- \mu \widetilde {\nabla} _ {\mathcal {R}} f (\boldsymbol {X} _ {i})\right) = \operatorname {Q r} \left(\boldsymbol {X} _ {i} - \mu \widetilde {\nabla} _ {\mathcal {R}} f (\boldsymbol {X} _ {i})\right), \quad 1 \leq i \leq n. \tag {4}
$$
Here,  $\mathrm{Qr}(B)$  returns the Q-factor in the thin QR decomposition of  $B$ , while the diagonal entries of the R-factor are restricted to be positive [7].
Finally, setting  $\mathbf{X}_i = \mathbf{X}_i^k$ ,  $\mathbf{X}_j = \mathbf{X}_j^k$  for all  $j$  such that  $(i,j) \in \mathcal{E}$ ,  $\mu = \mu_k$  in (3) and (4) yields a concrete implementation of Step 4 in ReSync and leads to  $\mathrm{SO}(d) \ni \mathbf{X}_i^{k+1} = \mathbf{X}_i^+$  for  $1 \leq i \leq n$ . This completes the description of one full iteration of ReSync. Note that the per-iteration complexity of the Riemannian subgradient procedure is  $\mathcal{O}(n^2 q)$ , and Algorithm 2 has computational cost  $\mathcal{O}(n^3)$ .
2.2 RCM Setup for Theoretical Analysis
We develop our theoretical analysis of ReSync by adopting the random corruption model (RCM). The RCM was previously used in many works to analyze the performance of various synchronization algorithms; see, e.g., [39, 19, 23, 32]. Specifically, we can represent our measurement model (1) on a graph  $\mathcal{G}(\mathcal{V},\mathcal{E})$ , where  $\mathcal{V}$  is a set of  $n$  nodes representing  $\{\pmb {X}_1^{\star},\dots ,\pmb {X}_n^{\star}\}$  and  $\mathcal{E}$  is a set of edges
containing all the available measurements  $\{\mathbf{Y}_{i,j}, (i,j) \in \mathcal{E}\}$ . We assume that the graph  $\mathcal{G}$  follows the well-known Erdős-Rényi model  $\mathrm{G}(n,q)$ , which implies that each edge  $(i,j) \in \mathcal{E}$  is observed with probability  $q$ , independently from every other edge. Each edge  $(i,j) \in \mathcal{E}$  is a true observation (i.e.,  $(i,j) \in \mathcal{A}$ ) with probability  $p$  and an outlier (i.e.,  $(i,j) \in \mathcal{A}^c$ ) with probability  $1 - p$ . Furthermore, the outliers  $\{\mathbf{O}_{i,j}\}_{(i,j) \in \mathcal{A}^c}$  are assumed to be independently and uniformly distributed on  $\mathrm{SO}(d)$ .
In this section, we present our theoretical results for ReSync. Our main results are summarized in the following theorem, which states that our proposed algorithm can converge at a linear rate to the underlying rotations  $X^{\star}$ . Our standing assumption in this section is stated below.
All our theoretical results in this section are based on the RCM; see Section 2.2.
Theorem 1 (overall). Suppose that the ratios  $p$  and  $q$  satisfy
$$
p ^ {7} q ^ {2} = \Omega \left(\frac {\log n}{n}\right).
$$
With probability at least  $1 - \mathcal{O}(1 / n)$ , ReSync with  $\mu_k = \mu_0\gamma^k$ , where  $\mu_0 = \Theta(p^2 / n)$  and  $\gamma = 1 - \frac{pq}{16}$ , converges linearly to the ground-truth rotations  $\pmb{X}^\star$  (up to a global rotation), i.e.,
$$
\operatorname {d i s t} \left(\boldsymbol {X} ^ {k}, \boldsymbol {X} ^ {\star}\right) \leq \xi_ {0} \gamma^ {k}, \quad \operatorname {d i s t} _ {\infty} \left(\boldsymbol {X} ^ {k}, \boldsymbol {X} ^ {\star}\right) \leq \delta_ {0} \gamma^ {k}, \quad \forall k \geq 0.
$$
Here,  $\xi_0 = \Theta (\sqrt{np^5q})$  and  $\delta_0 = \Theta (p^2)$
The basic idea of the proof is to establish the problem-intrinsic property of weak sharpness and then use it to derive a linear convergence result. However, the result only holds locally. Thus, we develop a procedure to initialize the algorithm in this local region and argue that ReSync will not leave this region afterwards. In the remaining parts of this section, we implement the above ideas and highlight the challenges and approaches to overcoming them.
3.1 Analysis of SpectrIn with Leave-One-Out Technique
Theorem 2 (initialization). Let  $\mathbf{X}^0$  be generated by SpectrIn (see Algorithm 2). Suppose that the ratios  $p$  and  $q$  satisfy
$$
p ^ {2} q = \Omega \left(\frac {\log n}{n}\right).
$$
Then, with probability at least  $1 - \mathcal{O}(1 / n)$ , we have
$$
\operatorname {d i s t} \left(\boldsymbol {X} ^ {0}, \boldsymbol {X} ^ {\star}\right) = \mathcal {O} \left(\frac {\sqrt {\log n}}{p \sqrt {q}}\right) \quad a n d \quad \operatorname {d i s t} _ {\infty} \left(\boldsymbol {X} ^ {0}, \boldsymbol {X} ^ {\star}\right) = \mathcal {O} \left(\frac {\sqrt {\log n}}{p \sqrt {n q}}\right). \tag {5}
$$
The works [34] and [11] show that exact reconstruction of  $X^{\star}$  is information-theoretically possible if the condition  $p^2 q = \Omega (\log n / n)$  holds for the cases  $d = 2$  and  $d = 3$ , respectively. Though Theorem 2 does not provide exact recovery, it achieves an optimal sample complexity for reconstructing an approximate solution in the infinity norm. Specifically, Theorem 2 shows that, as long as  $p^2 q \geq C \log n / n$  for some constant  $C > 0$  large enough, the  $\ell_{\infty}$ -distance  $\mathrm{dist}_{\infty}(X^{0}, X^{\star})$  (i.e.,  $\max_{1 \leq i \leq n} \mathrm{dist}(X_i, X_i^{\star})$ ) can be made relatively small. However, the  $\ell_{2}$ -distance  $\mathrm{dist}(X^{0}, X^{\star})$  is of the order  $\Omega(\sqrt{n})$  under such a sample complexity.
The work [26] considers orthogonal and permutation group synchronization and shows that spectral relaxation-based methods achieve near-optimal performance bounds. Our result differs from that of [26] in twofold: 1) Our approach follows the standard leave-one-out analysis based on the standard "Dist" (up to  $\mathrm{O}(d)$  invariance) defined above Lemma 3 in the Appendix. Nonetheless, we have to transfer the results to "dist" due to the structure of  $\mathrm{SO}(d)$  in Lemma 5, which is a nontrivial step due to the specific structure of  $\mathrm{SO}(d)$ . 2) Our result can handle incomplete observations (i.e.,  $q < 1$ ). In the case of incomplete observations, the construction in (17) in the Appendix becomes more intricate; it has the additional third column, rendering the analysis of our Lemma 2 more involved.
We prove Theorem 2 with some matrix concentration bounds and the leave-one-out technique. We provide the proof sketch below and refer to Appendix A for the full derivations.