SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof outline of Theorem 2. According to (1) and the fact  $\mathbb{E}(\pmb{O}_{ij}) = \mathbf{0}$  since outliers are assumed to be independently and uniformly distributed on  $\mathrm{SO}(d)$  in the RCM (see Appendix A), we know that  $\mathbb{E}(\pmb{Y}_{ij}) = pq\pmb{X}_i^\star \pmb{X}_j^{\star \top}$  for all  $(i,j)\in [n]\times [n]$ . This motivates us to introduce the noise matrix  $\pmb{W}_{ij} = \pmb{Y}_{ij} - pq\pmb{X}_i^\star \pmb{X}_j^{\star \top}$ , i.e.,
$$
\boldsymbol {Y} = p q \boldsymbol {X} ^ {\star} \boldsymbol {X} ^ {\star^ {\top}} + \boldsymbol {W}. \tag {6}
$$
The condition  $p^2 q = \Omega (\log n / n)$  in Theorem 2 ensures that the expectation  $pq\pmb{X}^{\star}\pmb{X}^{\star \top}$  will dominate the noise matrix  $\pmb{W}$  in the decomposition (6).
We first discuss how to bound  $\mathrm{dist}(\pmb{X}^0, \pmb{X}^\star)$ . Notice that  $\pmb{X}^0$  and  $\pmb{X}^\star$  are the  $d$  leading eigenvectors of  $\pmb{Y}$  (after projection onto  $\mathrm{SO}(d)^n$ ) and  $pq\pmb{X}^\star \pmb{X}^{\star \top}$ , respectively. We can then use the matrix perturbation theory (see Lemma 3) to bound  $\mathrm{dist}(\pmb{X}^0, \pmb{X}^\star)$ . Towards this end, we need to estimate the operator norm  $\| \dot{\pmb{W}} \|_2$ , which could be done by applying the standard matrix Bernstein concentration inequality [38] since the blocks  $\{\pmb{W}_{ij}\}$  are i.i.d. white noise with bounded operator norms and variances; see Lemma 2.
We next turn to bound the initialization error in the infinity norm, i.e.,  $\mathrm{dist}_{\infty}(\pmb{X}^0,\pmb{X}^\star)$ . Let us use  $(\pmb{W}\pmb{X}^0)_m\in \mathbb{R}^{d\times d}$  to denote the  $m$ -th block of  $\pmb{W}\pmb{X}^0\in \mathbb{R}^{nd\times d}$  for  $1\leq m\leq n$ . The main technical challenge lies in deriving a sharp bound for the term  $\max_{1\leq m\leq n}\| (\pmb{W}\pmb{X}^0)_m\| _F$ , as it involves two dependent random quantities, i.e., the noise matrix  $\pmb{W}$  and the initial  $\pmb{X}^0$  that is obtained by projecting the first  $d$  leading eigenvectors of  $\pmb{Y}$  onto  $\mathrm{SO}(d)^n$ . To overcome such a statistical dependence, we utilize the leave-one-out technique. This technique was utilized in [42] to analyze the phase synchronization problem and was later applied to many other synchronization problems [1, 10, 15, 18, 26]. Let us define
$$
\boldsymbol {Y} ^ {(m)} = p q \boldsymbol {X} ^ {\star} \boldsymbol {X} ^ {\star^ {\top}} + \boldsymbol {W} ^ {(m)} \quad \text {w i t h} \quad \boldsymbol {W} _ {k l} ^ {(m)} = \boldsymbol {W} _ {k l} \cdot \mathbf {1} _ {\{k \neq m \}} \cdot \mathbf {1} _ {\{l \neq m \}}. \tag {7}
$$
That is, we construct  $\pmb{W}^{(m)} \in \mathbb{R}^{nd \times nd}$  by setting the  $m$ -th block-wise row and column of  $\pmb{W}$  to be 0. Then, it is easy to see that  $\pmb{Y}^{(m)}$  is statistically independent of  $\pmb{W}_m^\top \in \mathbb{R}^{d \times nd}$ , where the latter denotes the  $m$ -th block-wise row of  $\pmb{W}$ . Let  $\pmb{X}^{(m)}$  be the  $d$  leading eigenvectors of  $\pmb{Y}^{(m)}$ . Consequently,  $\pmb{X}^{(m)}$  is also independent of  $\pmb{W}_m^\top$ . Based on the above discussions, we can bound each  $\| (\pmb{W}\pmb{X}^0)_m\|_F$  in the following way:
$$
\left\| \left(\boldsymbol {W} \boldsymbol {X} ^ {0}\right) _ {m} \right\| _ {F} = \left\| \boldsymbol {W} _ {m} ^ {\top} \boldsymbol {X} ^ {0} \right\| _ {F} \leq \left\| \boldsymbol {W} _ {m} ^ {\top} \boldsymbol {X} ^ {(m)} \right\| _ {F} + \left\| \boldsymbol {W} _ {m} ^ {\top} \left(\boldsymbol {X} ^ {0} - \boldsymbol {X} ^ {(m)}\right) \right\| _ {F}. \tag {8}
$$
The first term  $\| \pmb{W}_m^\top \pmb{X}^{(m)}\| _F$  can be bounded using an appropriate concentration inequality due to the statistical independence between  $\pmb{W}_m^\top$  and  $\pmb{X}^{(m)}$ . The second term can be bounded as
$$
\left\| \boldsymbol {W} _ {m} ^ {\top} \left(\boldsymbol {X} ^ {0} - \boldsymbol {X} ^ {(m)}\right) \right\| _ {F} \leq \left\| \boldsymbol {W} _ {m} \right\| _ {2} \cdot \left\| \boldsymbol {X} ^ {0} - \boldsymbol {X} ^ {(m)} \right\| _ {F},
$$
in which  $\| W_m\| _2$  can be further bounded by matrix concentration inequality (see Lemma 2) and  $\| X^0 -X^{(m)}\| _F$  can be bounded using standard matrix perturbation theory (see Lemma 4).
3.2 Weak Sharpness and Exact Recovery
We next present a property that is intrinsic to problem (2) in the following theorem.
Theorem 3 (weak sharpness). Suppose that the ratios  $p$  and  $q$  satisfy
$$
p ^ {2} q ^ {2} = \Omega \left(\frac {\log n}{n}\right).
$$
Then, with probability at least  $1 - \mathcal{O}(1 / n)$ , for any  $\pmb {X}\in \mathrm{SO}(d)^n$  satisfying  $\mathrm{dist}_{\infty}(\pmb {X},\pmb{X}^{\star}) = \mathcal{O}(p)$  we have
$$
f (\boldsymbol {X}) - f \left(\boldsymbol {X} ^ {\star}\right) \geq \frac {n p q}{8} \operatorname {d i s t} _ {1} \left(\boldsymbol {X}, \boldsymbol {X} ^ {\star}\right).
$$
Some remarks on Theorem 3 are in order. This theorem shows that problem (2) possesses the weak sharpness property [8], which is intrinsic to the problem and independent of the algorithm used to solve it. It is known that with this property, various subgradient-type methods can achieve linear convergence [14, 25]. We will establish a similar linear convergence result for ReSync in the next subsection based on Theorem 3.
The weak sharpness property shown in Theorem 3 is of independent interest, as it could be helpful when analyzing other optimization algorithms (not just ReSync) for solving problem (2). Currently,
only a few applications are known to produce sharp optimization problems, such as robust low-rank matrix recovery [25], robust phase retrieval [16], and robust subspace recovery [24]. Furthermore, sharp instances of manifold optimization problems are especially scarce. Hence, Theorem 3 extends the list of optimization problems that possess the weak sharpness property and contributes to the growing literature on the geometry of structured nonsmooth nonconvex optimization problems.
It is worth noting that Theorem 3 also establishes the exact recovery property of the formulation (2). Specifically, up to a global rotation, the ground-truth  $\pmb{X}^{\star}$  is guaranteed to be the unique global minimizer of  $f$  over the region  $\mathrm{SO}(d)^n \cap \{ \pmb{X} : \mathrm{dist}_{\infty}(\pmb{X}, \pmb{X}^{\star}) = \mathcal{O}(p) \}$ . Consequently, recovering the underlying  $\pmb{X}^{\star}$  reduces to finding the global minimizer of  $f$  over the aforementioned region. As we will show in the next subsection, ReSync will converge linearly to the global minimizer  $\pmb{X}^{\star}$  when initialized in this region. However, the initialization requirement is subject to the stronger condition  $p^4 q = \Omega (\log n / n)$  on the ratios  $p$  and  $q$ , which is ensured by Theorem 2.
We list our main ideas for proving Theorem 3 below. The full proof can be found in Appendix B.
Proof outline of Theorem 3. Note that the objective function  $f$  can be decomposed into two parts:
$$
f (\boldsymbol {X}) = \overbrace {\sum_ {(i , j) \in \mathcal {A}} \| \boldsymbol {X} _ {i} ^ {\top} \boldsymbol {X} _ {j} - \boldsymbol {X} _ {i} ^ {\star \top} \boldsymbol {X} _ {j} ^ {\star} \| _ {F}} ^ {g (\boldsymbol {X})} + \overbrace {\sum_ {(i , j) \in \mathcal {A} ^ {c}} \| \boldsymbol {X} _ {i} ^ {\top} \boldsymbol {X} _ {j} - \boldsymbol {O} _ {i j} \| _ {F}} ^ {h (\boldsymbol {X})}. \tag {9}
$$
It is easy to see that  $g(\pmb{X}^{\star}) = 0$  and  $g(\pmb{X}) \geq 0$ . Based on the fact that the true observation is uniformly distributed in all the indices, we have  $\mathbb{E}\left(g(\pmb{X})\right) = pq\sum_{1 \leq i,j \leq n}\| \pmb{X}_i^\top \pmb{X}_j - \pmb{X}_i^{\star \top}\pmb{X}_j^{\star}\| _F \geq \frac{npq}{2}\mathrm{dist}_1(\pmb{X},\pmb{X}^{\star})$ ; see Appendix B.2 for the last inequality. A traditional way to lower bounding  $g(\pmb{X})$  using  $\mathbb{E}(g(\pmb{X}))$  for all  $\pmb{X} \in \mathrm{SO}(d)$  is to apply concentration inequality and an epsilon-net covering argument. Unfortunately, the sample complexity condition  $p^2 q^2 = \Omega (\log n / n)$  does not lead to a high probability result in this way. Instead, our approach is to apply the concentration theory on the cardinalities of index sets rather than on  $\pmb{X}$  directly; see the following lemma.
Lemma 1 (concentration of cardinalities of index sets). Given any  $\epsilon = \Omega\left(\frac{\sqrt{\log n}}{\sqrt{n}pq}\right)$ , with probability at least  $1 - \mathcal{O}(1/n)$ , we have
$$
(1 - \epsilon) n q \leq | \mathcal {E} _ {i} | \leq (1 + \epsilon) n q,
$$
$$
(1 - \epsilon) n p q \leq | \mathcal {A} _ {i} | \leq (1 + \epsilon) n p q,
$$
$$
(1 - \epsilon) n p q ^ {2} \leq | \mathcal {E} _ {i} \cap \mathcal {A} _ {j} | \leq (1 + \epsilon) n p q ^ {2},
$$
$$
(1 - \epsilon) n p ^ {2} q ^ {2} \leq | \mathcal {A} _ {i j} | \leq (1 + \epsilon) n p ^ {2} q ^ {2}
$$
for any  $1 \leq i, j \leq n$ . See Section 1 for the notation.
We then provide a sharp lower bound on  $g(\mathbf{X})$  based on Lemma 1.
Proposition 1. Under the conditions of Theorem 3, with probability at least  $1 - \mathcal{O}(1 / n)$ , we have
$$
g (\boldsymbol {X}) \geq \frac {3 n p q}{1 6} \operatorname {d i s t} _ {1} \left(\boldsymbol {X}, \boldsymbol {X} ^ {\star}\right), \quad \forall \boldsymbol {X} \in \mathrm {S O} (d) ^ {n}. \tag {10}
$$
Next, to lower bound  $h(\mathbf{X}) - h(\mathbf{X}^{\star}) = \sum_{(i,j)\in \mathcal{A}^{c}}\left(\| \mathbf{X}_{i}^{\top}\mathbf{X}_{j} - \mathbf{O}_{ij}\|_{F} - \| \mathbf{X}_{i}^{\star \top}\mathbf{X}_{j}^{\star} - \mathbf{O}_{ij}\|_{F}\right)$  we first bound
$$
h (\boldsymbol {X}) - h (\boldsymbol {X} ^ {\star}) \geq \sum_ {(i, j) \in \mathcal {A} ^ {c}} \left\langle \frac {\boldsymbol {X} _ {i} ^ {\star \top} \boldsymbol {X} _ {j} ^ {\star} - \boldsymbol {O} _ {i j}}{\| \boldsymbol {X} _ {i} ^ {\star \top} \boldsymbol {X} _ {j} ^ {\star} - \boldsymbol {O} _ {i j} \| _ {F}}, \boldsymbol {X} _ {i} ^ {\top} \boldsymbol {X} _ {j} - \boldsymbol {X} _ {i} ^ {\star \top} \boldsymbol {X} _ {j} ^ {\star} \right\rangle ,
$$
where the inequality comes from the convexity of the norm function  $U \mapsto \|U\|_F$  whenever  $X_i^{\star \top}X_j^{\star} - O_{ij} \neq 0$ . Then, using the orthogonality of each block of  $X^{\star}$ , we further have
$$
h (\boldsymbol {X}) - h \left(\boldsymbol {X} ^ {\star}\right) \geq \sum_ {(i, j) \in \mathcal {A} ^ {c}} \left\langle \frac {\boldsymbol {I} - \boldsymbol {X} _ {i} ^ {\star} \boldsymbol {O} _ {i j} \boldsymbol {X} _ {j} ^ {\star^ {\top}}}{\| \boldsymbol {I} - \boldsymbol {X} _ {i} ^ {\star} \boldsymbol {O} _ {i j} \boldsymbol {X} _ {j} ^ {\star^ {\top}} \| _ {F}}, \boldsymbol {X} _ {i} ^ {\star} \boldsymbol {X} _ {i} ^ {\top} \boldsymbol {X} _ {j} \boldsymbol {X} _ {j} ^ {\star^ {\top}} - \boldsymbol {I} \right\rangle . \tag {11}
$$
Recall that since the outliers  $\{O_{i,j}\}_{(i,j)\in \mathcal{A}^c}$  are independently and uniformly distributed on  $\mathrm{SO}(d)$ , so are  $\{X_i^\star O_{ij}X_j^{\star \top}\}_{(i,j)\in \mathcal{A}^c}$ . This observation indicates that  $\left\{I - X_i^\star O_{ij}X_j^{\star \top} / \| I - X_i^\star O_{ij}X_j^{\star \top}\| _F\right\}_{(i,j)\in \mathcal{A}^c}$  are i.i.d. random matrices. Hence, by invoking concentration results that utilize the randomness of the outliers  $\{O_{i,j}\}_{(i,j)\in \mathcal{A}^c}$  and the cardinalities  $(i,j)\in \mathcal{A}^c$ , we obtain the following result.
Proposition 2. Under the conditions of Theorem 3, with probability at least  $1 - \mathcal{O}(1 / n)$ , we have
$$
h (\boldsymbol {X}) - h \left(\boldsymbol {X} ^ {\star}\right) \geq - \frac {n p q}{1 6} \operatorname {d i s t} _ {1} \left(\boldsymbol {X}, \boldsymbol {X} ^ {\star}\right) \tag {12}
$$
for all  $X\in \mathrm{SO}(d)^n$  satisfying  $\mathrm{dist}_{\infty}(\pmb {X},\pmb{X}^{\star}) = \mathcal{O}(p)$
Combining Proposition 1 and Proposition 2 gives Theorem 3.
3.3 Convergence Analysis and Proof of Theorem 1
Let us now turn to utilize the weak sharpness property shown in Theorem 3 to establish the local linear convergence of ReSync. As a quick corollary of Theorem 3, we have the following result.
Corollary 1. Under the conditions of Theorem 3, with probability at least  $1 - \mathcal{O}(1 / n)$ , for any  $\pmb{X} \in \mathrm{SO}(d)^n$  satisfying  $\mathrm{dist}_{\infty}(\pmb{X}, \pmb{X}^{\star}) = \mathcal{O}(p)$ , we have
$$
\left\langle \widetilde {\nabla} _ {\mathcal {R}} f (\boldsymbol {X}), \boldsymbol {X} ^ {\star} - \boldsymbol {X} \right\rangle \leq - \frac {n p q}{1 6} \operatorname {d i s t} _ {1} (\boldsymbol {X}, \boldsymbol {X} ^ {\star}), \quad \forall \widetilde {\nabla} _ {\mathcal {R}} f (\boldsymbol {X}) \in \partial_ {\mathcal {R}} f (\boldsymbol {X}). \tag {13}
$$
This condition indicates that any Riemannian subgradient  $\widetilde{\nabla}_{\mathcal{R}}f(\pmb {X})$  provides a descent direction pointing towards  $X^{\star}$ . However, it only holds for  $\pmb {X}\in \mathrm{SO}(d)^n$  satisfying  $\mathrm{dist}_{\infty}(\pmb {X},\pmb{X}^{\star}) = \mathcal{O}(p)$ . Our key idea for establishing local convergence is to show that the Riemannian subgradient update in ReSync is a contraction operator in both the Euclidean and infinity norm-induced distances using Corollary 1, i.e., if  $\pmb{X}^{k}$  lies in the local region, then  $\pmb{X}^{k + 1}$  also lies in the region. This idea motivates us to define two sequences of neighborhoods as follows:
$$
\mathcal {N} _ {F} ^ {k} = \left\{\boldsymbol {X} \mid \operatorname {d i s t} \left(\boldsymbol {X}, \boldsymbol {X} ^ {\star}\right) \leq \xi_ {k} \right\} \quad \text {a n d} \quad \mathcal {N} _ {\infty} ^ {k} = \left\{\boldsymbol {X} \mid \operatorname {d i s t} _ {\infty} \left(\boldsymbol {X}, \boldsymbol {X} ^ {\star}\right) \leq \delta_ {k} \right\}. \tag {14}
$$
Here,  $\xi_{k} = \xi_{0}\gamma^{k},\delta_{k} = \delta_{0}\gamma^{k}$ , where  $\xi_0,\delta_0$  and  $\gamma \in (0,1)$  will be specified later. Thus, these two sequences of sets  $\{\mathcal{N}_F^k\}$  and  $\{\mathcal{N}_{\infty}^{k}\}$  will linearly shrink to the ground-truth. It remains to show that if  $X^{k}\in \mathcal{N}_{F}^{k}\cap \mathcal{N}_{\infty}^{k}$ , then  $X^{k + 1}\in \mathcal{N}_F^{k + 1}\cap \mathcal{N}_\infty^{k + 1}$ , which is summarized in the following theorem.
Theorem 4 (convergence analysis). Suppose that  $\delta_0 = \mathcal{O}(p^2)$  and  $\xi_0 = \mathcal{O}(\sqrt{npq}\delta_0)$ . Set  $\gamma = 1 - \frac{pq}{16}$  and  $\mu_k = \delta_k / n$  in ReSync. If  $X^k\in \mathcal{N}_F^k\cap \mathcal{N}_\infty^k$  for any  $k\geq 0$ , then with probability at least  $1 - \mathcal{O}(1 / n)$ , we have
$$
\boldsymbol {X} ^ {k + 1} \in \mathcal {N} _ {F} ^ {k + 1} \cap \mathcal {N} _ {\infty} ^ {k + 1}.
$$
Proof outline of Theorem 4. The proof consisted of two parts. On the one hand, we need to show that  $\pmb{X}^{k + 1}\in \mathcal{N}_F^{k + 1}$ , which can be achieved by applying Corollary 1. On the other hand, in order to show that  $\pmb{X}^{k + 1}\in \mathcal{N}_{\infty}^{k + 1}$ , we need a good estimate of each block of  $\widetilde{\nabla}_{\mathcal{R}}f(\pmb {X})$ . See Appendix C.
Having developed the necessary tools, we are now ready to prove Theorem 1.
Proof of Theorem 1. Based on Theorem 2, we know that  $\pmb{X}^0 \in \mathcal{N}_F^0 \cap \mathcal{N}_\infty^0$  if  $\xi_0$  and  $\delta_0$  satisfy
$$
\xi_ {0} = \mathcal {O} \left(\frac {\sqrt {\log n}}{p \sqrt {q}}\right) \quad \text {a n d} \quad \delta_ {0} = \mathcal {O} \left(\frac {\sqrt {\log n}}{p \sqrt {n q}}\right). \tag {15}
$$
According to Theorem 4, by choosing  $\delta_0 = \Theta(p^2)$  and  $\xi_0 = \Theta(\sqrt{np^5q})$ , condition (15) holds when  $p^7 q^2 = \Omega$  ( $\log n/n$ ). This completes the proof of Theorem 1.
In this section, we conduct experiments on ReSync for solving the RRS problem on both synthetic and real data, providing empirical support for our theoretical findings. Our experiments are conducted on a personal computer with a 2.90GHz 8-core CPU and 32GB memory. All our experiment results are averaged over 20 independent trials. Our code is available at https://github.com/Huikang2019/ReSync.
We consider the rotation group  $\mathrm{SO}(3)$  in all our experiments. We generate  $X_1^\star, \ldots, X_n^\star$  by first generating matrices of the same dimension with i.i.d. standard Gaussian entries and then projecting
(a)  $n = 400$
(b)  $n = 600$
(c)  $n = 800$
Figure 2: Convergence of ReSync with  $p = q = (\log n / n)^{1 / 3}$ .
(d)  $n = 1000$
(a)  $q = 0.2,\sigma = 0.0$
(b)  $q = 0.2, \sigma = 1.0$
(c)  $p = 0.2, \sigma = 0.0$
Figure 3: Comparison with state-of-the-art synchronization algorithms.
(d)  $p = 0.2, \sigma = 1.0$
each of them onto  $\mathrm{SO}(3)$ . The underlying graph, outliers, and relative rotations in the measurement model (1) are generated according to the RCM as described in Section 2.2. In our experiments, we also consider the case where the true observations are contaminated by additive noise, namely,  $\{\mathbf{Y}_{i,j}\}_{(i,j)\in \mathcal{A}}$  in (1) is generated using the formula
$$
\mathbf {Y} _ {i, j} = \mathcal {P} _ {\mathrm {S O} (3)} \left(\mathbf {X} _ {i} ^ {\star \top} \mathbf {X} _ {j} ^ {\star} + \sigma \mathbf {G} _ {i, j}\right) \quad \text {f o r} \quad (i, j) \in \mathcal {A}, \tag {16}
$$
where  $G_{i,j}$  consists of i.i.d. entries following the standard Gaussian distribution and  $\sigma \geq 0$  controls the variance level of the noise.
Convergence verification of ReSync. We evaluate the convergence performance of ReSync with the noise level  $\sigma = 0$  in (16). We set  $p = q = (\log n / n)^{1 / 3}$  in the measurement model (1), which satisfies  $p^2 q = \log n / n$ . We use the initial step size  $\mu_0 = 1 / npq$  and the decaying factor  $\gamma \in \{0.7, 0.8, 0.85, 0.90, 0.95, 0.98\}$  in ReSync. We test the performance for various  $n$  selected from  $\{400, 600, 800, 1000\}$ . Figure 2 displays the experiment results. It can be observed that (i) ReSync converges linearly to ground-truth rotations for a wide range of  $\gamma$  and (ii) a smaller  $\gamma$  often leads to faster convergence speed. These corroborate our theoretical findings. However, it is worth noting that excessively small  $\gamma$  values may result in an early stopping phenomenon (e.g.,  $\gamma \leq 0.8$  when  $n = 400$ ). In addition, ReSync performs better with a larger  $n$ , as it allows for a smaller  $\gamma$  (e.g.,  $\gamma = 0.7$  when  $n = 1000$ ) and hence converges to the ground-truth rotations faster.
Comparison with the state-of-the-arts. We next compare ReSync with state-of-the-art synchronization algorithms, including IRLS_L12 [9], MPLS [31], CEMP_GCW [23, 32], DESC [32], and LUD [39]. We obtain the implementation of the first four algorithms from https: //github.com/ColeWyeth/DESC, while LUD's implementation is obtained through private communication with its authors. In our comparisons, we use their default parameter settings. For ReSync, we set the initial step size to  $\mu_0 = 1 / npq$  and the decaying factor to  $\gamma = 0.95$ , as suggested by the previous experiment. We fix  $n = 200$  and vary the true observation ratio  $p$  (or the observation ratio  $q$ ) while keeping  $q = 0.2$  (or  $p = 0.2$ ) fixed. We display the experiment results for  $\sigma = 0$  and  $\sigma = 1$  in Figures 3a and 3b, respectively, where  $p$  is selected from  $\{0.2, 0.3, 0.4, \ldots, 1\}$ . When  $\sigma = 0$ , ReSync achieves competitive performance compared to other robust synchronization algorithms. When the additive noise level is  $\sigma = 1$ , ReSync outperforms other algorithms. In Figures 3c and 3d, we present the results with varying  $q$  chosen from  $\{0.2, 0.3, 0.4, \ldots, 1\}$  for noise-free ( $\sigma = 0$ ) and noisy ( $\sigma = 1$ ) cases, respectively. In the noise-free case, DESC performs best when  $q < 0.5$ , while ReSync slightly outperforms others when  $q \geq 0.5$ . In the noisy case, it is clear that ReSync achieves the best performance for a large range of  $q$ .
4.2 Real Data
We consider the global alignment problem of three-dimensional scans from the Lucy dataset, which is a down-sampled version of the dataset containing 368 scans with a total number of 3.5 million triangles. We refer to [39] for more details about the experiment setting. We apply three algorithms LUD [39], DESC [32] and our ReSync on this dataset since they have the best performance on noisy synthetic data. As Figure 4 shows, ReSync outperforms the other two methods.
Figure 4: Histogram of the unsquared residuals of LUD, DESC, and ReSync for the Lucy dataset.
5 Conclusion and Discussions on Limitations
In this work, we introduced ReSync, a Riemannian subgradient-based algorithm with spectral initialization for solving RRS. We established strong theoretical results for ReSync under the RCM. In particular, we first presented an initialization guarantee for SpectrIn, which is a procedure embedded in ReSync for initialization. Then, we established a problem-intrinsic property called weak sharpness for our nonsmooth nonconvex formulation, which is of independent interest. Based on the established weak sharpness property, we derived linear convergence of ReSync to the underlying rotations once it is initialized in a local region. Combining these theoretical results demonstrates that ReSync converges linearly to the ground-truth rotations under the RCM.
Limitations. Our overall guarantee in Theorem 1 requires the sample complexity of  $p^7 q^2 = \Omega (\log n / n)$ , which does not match the currently best known lower bound  $p^2 q = \Omega (\log n / n)$  for exact recovery [34, 11]. We showed in Theorem 2 that approximate recovery with an optimal sample complexity is possible. Moreover, we showed in Theorem 3 that exact recovery with  $p^2 q^2 = \Omega (\log n / n)$  is possible if we have a global minimizer of the objective function of problem (2) within a certain local region. However, due to the nonconvexity of problem (2), it is non-trivial to obtain the said minimizer. We circumvented this difficulty by establishing the linear convergence of ReSync to a desired minimizer in Theorem 4. Nevertheless, a strong requirement on initialization is needed, which translates to the weaker final complexity result of  $p^7 q^2 = \Omega (\log n / n)$ .
Although our theory allows for  $p \to 0$  as  $n \to \infty$ , our argument relies heavily on the randomness of the outliers  $\{O_{i,j}\}$  and the absence of additive noise. In practice, adversarial outliers that arbitrarily corrupt a measurement and additive noise contamination are prevalent. It remains unknown how well ReSync performs in such scenarios.
The above challenges are significant areas for future research and improvements.
The authors thank Dr. Zengde Deng (Cainiao Network) and Dr. Shixiang Chen (University of Science and Technology of China) for providing helpful advice. They also thank the reviewers for their insightful comments, which have helped greatly to improve the quality and presentation of the manuscript.
Huikang Liu is supported in part by the National Natural Science Foundation of China (NSFC) Grant 72192832. Xiao Li is supported in part by the National Natural Science Foundation of China (NSFC) under grants 12201534 and 72150002, and in part by the Shenzhen Science and Technology Program under grants RCBS20210609103708017 and RCYX20221008093033010. Anthony Man-Cho So is supported in part by the Hong Kong Research Grants Council (RGC) General Research Fund (GRF) project CUHK 14205421.