{"title": "Translation Synchronization via Truncated Least Squares", "book": "Advances in Neural Information Processing Systems", "page_first": 1459, "page_last": 1468, "abstract": "In this paper, we introduce a robust algorithm, \\textsl{TranSync}, for the 1D translation synchronization problem, in which the aim is to recover the global coordinates of a set of nodes from noisy measurements of relative coordinates along an observation graph. The basic idea of TranSync is to apply truncated least squares, where the solution at each step is used to gradually prune out noisy measurements. We analyze TranSync under both deterministic and randomized noisy models, demonstrating its robustness and stability. Experimental results on synthetic and real datasets show that TranSync is superior to state-of-the-art convex formulations in terms of both efficiency and accuracy.", "full_text": "Translation Synchronization via Truncated Least\n\nSquares\n\nXiangru Huang(cid:63)\n\nThe University of Texas at Austin\n2317 Speedway, Austin, 78712\nxrhuang@cs.utexas.edu\n\nZhenxiao Liang(cid:63)\nTsinghua University\nBeijing, China, 100084\n\nliangzx14@mails.tsinghua.edu.cn\n\nChandrajit Bajaj\n\nThe University of Texas at Austin\n2317 Speedway, Austin, 78712\nbajaj@cs.utexas.edu\n\nQixing Huang\n\nThe University of Texas at Austin\n2317 Speedway, Austin, 78712\nhuangqx@cs.utexas.edu\n\nAbstract\n\nIn this paper, we introduce a robust algorithm, TranSync, for the 1D translation\nsynchronization problem, in which the aim is to recover the global coordinates of a\nset of nodes from noisy measurements of relative coordinates along an observation\ngraph. The basic idea of TranSync is to apply truncated least squares, where the\nsolution at each step is used to gradually prune out noisy measurements. We analyze\nTranSync under both deterministic and randomized noisy models, demonstrating\nits robustness and stability. Experimental results on synthetic and real datasets\nshow that TranSync is superior to state-of-the-art convex formulations in terms of\nboth ef\ufb01ciency and accuracy.\n\n1\n\nIntroduction\n\nIn this paper, we are interested in solving the 1D translation synchronization problem, where the\ninput is encoded as an observation graph G = (V,E) with n nodes (i.e. V = {1,\u00b7\u00b7\u00b7 , n}). Each node\ni \u2208 R, 1 \u2264 i \u2264 n, and each edge (i, j) \u2208 E is associated with\nis associated with a latent coordinate x(cid:63)\nj + N (\u0001ij) of the coordinate difference xi \u2212 xj under some noise\na noisy measurement tij = x(cid:63)\nmodel N (\u0001ij). The goal of translation synchronization is to recover the latent coordinates (up to a\nglobal shift) from these noisy pairwise measurements. Translation synchronization is a fundamental\nproblem that arises in many application domains, including joint alignment of point clouds [7] and\nranking from relative comparisons [8, 16].\nA standard approach to translation synchronization is to solve the following linear program:\n\ni \u2212 x(cid:63)\n\n|tij \u2212 (xi \u2212 xj)|,\n\nsubject to\n\nxi = 0,\n\n(1)\n\nminimize (cid:88)\n\n(i,j)\u2208E\n\nn(cid:88)\n\ni=1\n\nWhere the constraint ensures that the solution is unique. The major drawback of the linear pro-\ngramming formulation is that it can only tolerate up to 50% of measurements coming from biased\nnoise models (e.g., uniform samples with non-zero mean). Moreover, it is challenging to solve (1)\nef\ufb01ciently at scale. Solving (1) using interior point method becomes impractical for large-scale\ndatasets, while more scalable methods such as coordinate descent usually exhibit slow convergence.\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fIn this paper, we introduce a robust and scalable algorithm, TranSync, for translation synchronization.\nThe algorithm is rather simple, we solve a truncated least squares at each iteration k:\n\n{x(k)\n\ni } = argmin\n{xi}\n\n(cid:88)\n\n(i,j)\u2208E\n\nwij|tij \u2212 (xi \u2212 xj)|2,\n\nsubject to\n\ndixi = 0,\n\ndi :=\n\nwij.\n\nn(cid:88)\n\n(cid:112)\n\ni=1\n\n(cid:88)\n\nj\u2208N (i)\n\ni\n\nj\n\n\u2212 x(k\u22121)\n\n(2)\nwhere the weights wij = Id(|tij \u2212 (x(k\u22121)\n)| < \u03b4k) are obtained from the solution at\nthe previous iteration using a geometrically decaying truncation parameter \u03b4k. Although TranSync\nrequires solving a linear system at each step, these linear systems are fairly similar to each other,\nmeaning that the solution at the previous iteration provides an excellent warm-start for solving the\nlinear system at the current iteration. As a result, the computational ef\ufb01ciency of TranSync is superior\nto state-of-the-art methods for solving the linear programming formulation. We analyze TranSync\nunder both deterministic and randomized noise models, demonstrating its robustness and stability. In\nparticular, we show that TranSync is able to handle biased noisy measurements.\nWe have evaluated TranSync on both synthetic datasets and real datasets used in the applications of\njoint alignment of point clouds and ranking from pair-wise measurements. Experimental results show\nthat TranSync is superior to state-of-the-art solvers for the linear programming formulation in terms\nof both computational ef\ufb01ciency and accuracy.\n\n1.1 Related Work\n\nTranslation synchronization falls into the general problem of map synchronization, which takes maps\ncomputed between pairs of objects as input, and outputs consistent maps across all the objects. Map\nsynchronization appears as a crucial step in many scienti\ufb01c problems, including fusing partially\noverlapping range scans [15], assembling fractured surfaces [14], solving jigsaw puzzles [5, 11],\nmulti-view structure from motion [25], data-driven shape analysis and processing [17], and structure\nfrom motion [27].\nEarly methods for map synchronization focused on applying greedy algorithms [14, 15, 18] or\ncombinatorial optimization [20, 23, 27]. Although these methods exhibit certain empirical success,\nthey lack theoretical understanding (e.g. we do not know under what conditions can the underlying\nground-truth maps be exactly recovered).\nRecent methods for map synchronization apply modern optimization techniques such as convex\noptimization and non-convex optimization. In [13], Huang and Guibas introduce a semide\ufb01nite\nprogramming formulation for permutation synchronization and its variants. Chen et al. [4] generalize\nthe method to partial maps. In [26], Wang and Singer introduce a method for rotation synchronization.\nAlthough these methods provide tight, exact recovery conditions, the computational cost of the convex\noptimizations provide an obstruction for applying these methods to large-scale data sets.\nIn contrast to convex optimization, very recent map synchronization methods leverage non-convex\noptimization approaches such as spectral techniques and gradient-based optimization. In [21, 22],\nPachauri et al. study map synchronization from the perspective of spectral decomposition. Recently,\nShen et al. [24] provide an analysis of spectral techniques for permutation synchronization. Beyond\nspectral techniques, Zhou et al. [28] apply alternating minimization for permutation synchronization.\nFinally, Chen and Candes [3] introduce a method for the generalized permutation synchronization\nproblem using the projected power method. To the best of our knowledge, we are the \ufb01rst to\ndevelop and analyze continuous map synchronizations (e.g., translations or rotations) beyond convex\noptimization.\nOur approach can be considered as a special case of reweighted least squares (or RLS) [9, 12], which\nis a powerful method for solving convex and non-convex optimizations. The general RLS framework\nhas been applied for map synchronization (e.g. see [1, 2]). Despite the empirical success of these\napproaches, the theoretical understanding of RLS remains rather limited. The analysis in this paper\nprovides a \ufb01rst step towards the understanding of RLS for map synchronization.\n\n1.2 Notation\n\nBefore proceeding to the technical part of this paper, we introduce some notation that will be used\nlater. The unnormalized graph Laplacian of a graph G is denoted as LG. If it is obvious from the\n\n2\n\n\fAlgorithm 1 TranSync(c, kmax)\n\n1. x(\u22121) \u2190 0. \u03b4\u22121 \u2190 \u221e.\nfor k = 0, 1, 2, kmax do\n\n2. Obtain the truncated graph G(k) using x(k\u22121) and \u03b4k\u22121.\n3. Break if G(k) is disconnected\n4. Solve (2) using (4) to obtain x(k).\n|tij \u2212 (x(0)\n\n5. \u03b4k = min(cid:0) max\n\nj )|, c\u03b4k\u22121\n\ni \u2212 x(0)\n\n(cid:1).\n\n(i,j)\u2208E\n\nend for\nOutput: x(k).\n\ncontext, we will always shorten LG as L to make the notation uncluttered. Similarly, we will use\nD = diag(d1,\u00b7\u00b7\u00b7 , dn) to collect the vertex degrees and denote the vertex adjacency and vertex-edge\nadjacency matrices as A and B respectively. The peusdo-inverse of a matrix X is given by X +. In\naddition, we always sort the eigenvalues of a symmetric matrix X \u2208 Rn\u00d7n in increasing order (i.e.\n\u03bb1(X) \u2264 \u03bb2(X) \u2264 \u00b7\u00b7\u00b7 \u2264 \u03bbn(X)). Moreover, we will consider several matrix norms (cid:107) \u00b7 (cid:107), (cid:107) \u00b7 (cid:107)1,\u221e\nand (cid:107) \u00b7 (cid:107)F , which are de\ufb01ned as follows:\n\nn(cid:88)\n\n(cid:107)X(cid:107)F =(cid:0)(cid:88)\n\n(cid:1) 1\n\nx2\nij\n\n2 .\n\nj=1\n\ni,j\n\n(cid:107)X(cid:107) = \u03c3max(X),\n\n(cid:107)X(cid:107)1,\u221e = max\n1\u2264i\u2264n\n\n|xij|,\n\nNote that (cid:107)X(cid:107)1,\u221e is consistent with the L\u221e-norm of vectors.\n\n2 Algorithm\n\ni \u2212 x(0)\n\n|tij \u2212 (x(0)\n\nIn this section, we provide the algorithmic details of TranSync. The iterative scheme (1) requires an\ninitial solution x(0), an initial truncation parameter \u03b40, and a stopping condition. The initial solution\ncan be determined by solving for x(0) from (2) w.r.t. wij = 1. We set the initial truncation parameter\nj )|, so that the edge with the biggest residual is removed. We stop\n\u03b40 = max\n(i,j)\u2208E\nTranSync either after the maximum number of iterations is reached, or the truncated graph becomes\ndisconnected. Algorithm 1 provides the pseudo code of TranSync.\nClearly, the performance of TranSync is driven by the ef\ufb01ciency of solving (2) at each iteration.\nTranSync takes an iterative approach, in which we utilize a warm-start x(k\u22121) provided by the\nsolution obtained at the previous iteration. When the truncated graph is non-bipartite, we \ufb01nd\na simple weighted average scheme delivers satisfactory computational ef\ufb01ciency. Speci\ufb01cally, it\ngenerates a series of vectors xk,0 = x(k\u22121), xk,1,\u00b7\u00b7\u00b7 , xk,nmax via the following recursion:\n\nxk,l+1\ni\n\n= (1 \u2212 \u0001)\n\nwij(xk,l\n\nj + tij)/\n\nwij + \u0001xk,l\ni\n\n(cid:88)\n\nj\u2208N (i)\n\n(cid:88)\nn(cid:88)\n\ni(cid:48)=1\n\nj\u2208N (i)\n\n(cid:112)\n\n1(cid:80)n\n\ni(cid:48)=1\n\n\u221a\n\ndi\n\nxk,l+1\ni\n\n= xk,l+1\n\ni\n\n\u2212\n\ndi(cid:48)xk,l+1\n\ni(cid:48)\n\n,\n\nwhich may be written in the following matrix form:\n\nxk,l+1 = (In \u2212 1\nn\n\nD\u2212 1\n\n2 11T D\n\n2 )[(1 \u2212 \u0001)D\u22121(cid:0)Axk,l + Bt(k)(cid:1) + \u0001xk,l],\n\n1\n\nHere we add the parameter \u0001 to create a small perturbation to avoid the special case of bipartite graphs.\nFor non-bipartite graphs, \u0001 can be set to zero.\n\nRemark 2.1 The corresponding normalization constraint in (4), i.e.,(cid:80)\n\ndixi = 0, only changes\nthe solution to (2) by a constant factor. We utilize this modi\ufb01cation for the purpose of obtaining a\nconcise convergence property of the iterative scheme detailed below.\n\n\u221a\n\ni\n\nThe following proposition states that (4) admits a geometric convergence rate:\n\n3\n\n(3)\n\n(4)\n\n(5)\n\n\fProposition 2.1 xk,l geometrically converges to x(k+1). Speci\ufb01cally, \u2200l \u2265 0,\n\n2(cid:0)xk,l \u2212 x(k)\n\n1\n\nshift\n\n(cid:1)(cid:107) \u2264 (1 \u2212 (1 \u2212 \u0001)\u03c1)l(cid:107)D\n\n2(cid:0)xk,0 \u2212 x(k)\n\n1\n\n(cid:1)(cid:107),\n\nshift\n\nshift = x(k) \u2212\nx(k)\n\n(cid:107)D\n\ndi\nwhere \u03c1 < 1 is the spectral gap of the normalized Graph Laplacian of the truncated graph.\n\ni\n\n(cid:80)\ni(cid:80)\n\n\u221a\n\ndix(k)\n\u221a\n\ni\n\n1.\n\nProof. See Appendix A.\nSince the intermediate solutions are mainly used to prune outlier observations, it is clear that\nO(log(n)) iterations of (5), which induce a O(1/n) error for solving (2), are suf\ufb01cient. The com-\nplexity of checking if the graph is non-bapriatite is O(|E|). The total running time for solving (2) is\n\nthus O(cid:0)|E| log(n)(cid:1). This means the total running time of TranSync is O(|E| log(n)kmax), making it\n\nscalable to large-scale datasets.\n\n3 Analysis of TranSync\n\nIn this section, we provide exact recovery conditions of TranSync. We begin with describing an exact\nrecovery condition under a deterministic noise model in Section 3.1. We then study an exact recovery\ncondition to demonstrate that TranSync can handle biased noisy samples in Section 3.2.\n\n3.1 Deterministic Exact Recovery Condition\n\nWe consider the following deterministic noisy model: We are given the ground-truth location xgt.\nThen, for each correct measurement tij, (i, j) \u2208 G, |tij \u2212 (xgt\nj )| \u2264 \u03c3 for a threshold \u03c3. In\ni \u2212 xgt\ncontrast, each incorrect measurement tij, (i, j) \u2208 G could take any real number. The following\ntheorem provides an exact recovery condition under this noisy model.\n\nTheorem 3.1 Let dbad be the maximum number of incorrect measurements per node. De\ufb01ne\n\n\u03b1 = max\n\nk\n\n\u2020\nG,kk + max\nL\ni(cid:54)=j\n\n\u2020\nG,ij +\nL\n\nn\n2\n\nmax\ni,j,k\n\npairwisely different\n\n\u2020\n\u2020\n|L\nG,ki \u2212 L\nG,kj|,\n\nand\n\nh = \u03b1dbad,\n\np =\n\ndbad\u03b1\n1 \u2212 2h\n\n,\n\n(n \u2212 dbad)\u03b1\n\n1 \u2212 2h\n\n.\n\nq =\n\n6 (or p < 1\n\n4 ), then starting from any initial solution x(0), and for any large enough\nSuppose h < 1\ninitial truncation threshold \u0001 \u2265 2(cid:107)x(0)(cid:107)\u221e + \u03c3 and iterative step size c satisfying 4p < c < 1, we\nhave\n\nwhere\n\nk \u2264 \u2212 log\n\n(cid:107)x(k) \u2212 xgt(cid:107)\u221e \u2264 q\u03c3 + 2p\u0001ck\u22121,\n\n(cid:18) \u0001(c \u2212 4p)\n\n(cid:19)\n\n(1 + 2q) \u03c3\n\n/ log c + 1.\n\nMoreover, we can eventually reach an x(k) such that\n\n(cid:107)x(k)(cid:107)\u221e \u2264 2p + cq\nc \u2212 4p\n\n\u03c3\n\nwhich is independent of the initial solution x(0), initial truncation threshold \u0001, and values of all wrong\nmeasurements tG\\Ggood.\n\n(cid:3)\nProof: See Appendix B.\nTheorem 3.1 essentially says that TransSync can tolerate a constant fraction of arbitrary noise. To\nunderstand how strong this condition is, we consider the case where G = Kn is given by a clique.\nMoreover, we assume the nodes are divided into two clusters of equal size, where all the measurements\nwithin each cluster are correct. For measurements between different clusters, half of them are correct\nand the other half are wrong. In this case, 25% of all measurements are wrong. However, we cannot\nrecover the original xgt in this case. In fact, we can set the wrong measurements in a consistent\n\n4\n\n\fi \u2212 xgt\n\nj + b for a constant b (cid:54)= 0, leading to two competing clusters (one correct\nmanner, i.e tij = xgt\nand the other one incorrect) with equal strength. Hence, in the worst case, any algorithm can only\ntolerate at most 25% of measurements being wrong.\nWe now try to use Theorem 3.1 to analyze the case where the observation graph is a clique. In\nthis case, it is clear that \u03b1 = 1\nn , i.e the fraction of wrong measurements out of all\nmeasurements. Hence, in the clique case, we have shown that TranSync converges to a neighborhood\nof the ground truth from any initial solution if the fraction of wrong measurements is less that 1\n6 (i.e.,\n2/3 of the upper bound).\n\nn, and p = dbad\n\n3.2 Biased Random Noisy Model\n\nWe proceed to provide an exact recovery condition of TranSync under a biased random noisy model.\nTo simplify the discussion, we assume the observation graph G = Kn is a clique. However, our\nanalysis framework can be extended to handle arbitrary graphs.\nAssume \u03c3 << a + b. We consider the following noise model, where the noisy measurements are\nindependent, and they follow\n\ntij =\n\ni \u2212 xgt\ni \u2212 xgt\nxgt\n\nj + U [\u2212\u03c3, \u03c3]\nwith probability p\nj + U [\u2212a, b] with probability 1 \u2212 p\n\n(6)\n\n(cid:26) xgt\n\nb\n\na+b (1 \u2212 p) > 1\n\nIt is easy to check that the linear programming formulation is unable to recover the ground-truth\nsolution if\n2. The following theorem shows that TranSync achieves a sub-constant\nrecovery rate instead.\n\nTheorem 3.2 There exists a constant c so that if p > c/(cid:112)log(n), then w.h.p,\n\n(cid:107)x(k) \u2212 xgt(cid:107)\u221e \u2264 (1 \u2212 p/2)k(b \u2212 a),\n\n\u2200 k = 0,\u00b7\u00b7\u00b7 , [\u2212 log(\n\nb + a\n\n2\u03c3\n\n)/log(1 \u2212 p/2)].\n\nThe major dif\ufb01culty of proving Theorem 3.2 is that x(k) is dependent on tk, making it hard to\ncontrol x(k) using existing concentration bounds. We address this issue by showing that the solutions\nx(k), k = 0,\u00b7\u00b7\u00b7 , stay close to the segment between xgt and xgt + (1 \u2212 p) a+b\n2 1. Speci\ufb01cally, for\npoints on this segment, we can leverage the independence of tij to derive the following concentration\nbound for one step of TranSync:\nLemma 3.1 Consider a \ufb01xed observation graph G. Let r =\n(a+b)p+2(1\u2212p)\u03b4 and dmin be the\nminimum degree of G. Suppose dmin = \u2126(log2(n)), and p + r(1 \u2212 p) = \u2126(log2(n)/dmin) .\nConsider an initial point x(0) (independent from tij) and a threshold parameter \u03b4 such that \u2212a + \u03b4 \u2264\nmini x(0)\n\ni \u2264 maxi x(0)\ni \u2264 b \u2212 \u03b4. Then w.h.p., one step of TranSync outputs x(1) which satis\ufb01es\n(cid:32)(cid:115)\n\n(cid:107)x(1) \u2212 (1 \u2212 r)x(0) + rxgt )(cid:107)\u221e\n\n(cid:33)\n\n(a+b)p\n\n(cid:114)\n\n= O\n\nlog(n)\n\n(p + r(1 \u2212 p))dmin\u03bb2(LG)\n\n)\n\n\u00b7\n\nmax((cid:107)x(0)(cid:107)2\n\nd,\u221e, r2) + O\n\n\u03c32\n\n,\n\n(cid:17)\n\n(cid:16) p\n\nr\n\nwhere (cid:107)x(0)(cid:107)d,\u221e = max\n1\u2264i,j\u2264n\n\n|x(0)\ni \u2212 x(0)\n\nj\n\n|, and LG is the normalized graph Laplacian of G.\n\n(cid:3)\nProof: See Appendix C.1.\nRemark 3.1 Note that when G is a clique or a graph sampled from the standard Erd\u02ddos-R\u00e9nyi model\nG(n, q), then O(\n\n(cid:113) log(n)\n\n(cid:113)\n\n\u03c1 log(n)\n\n(p+r(1\u2212p))\u03bb2(LG ) ) = O(\n\n(p+r(1\u2212p))n ).\n\n4 (n)), the L\u221e distance between x(k) to the\nTo prove Theorem 3.2, we show that when k = O(log\nline segment between xgt and xgt + (1 \u2212 p) a+b\n2 1 only grows geometrically, and this distance is in\nthe order of o(p). On the other hand, (1 \u2212 p/2)k = o(p). So when k \u2265 k, that distance decays with\na geometrical rate that is small than c. The details are deferred to Appendix C.2.\n\n3\n\n5\n\n\fImproving recovery rate via sample splitting. Note that Lemma 3.1 enables us to apply standard\nsampling tricks to improve the recovery rate. To simplify the discussion, we will assume \u03c3 is\nsuf\ufb01ciently small. First of all, it is clear that if re-sampling is allowed at each iteration, then TranSync\nadmits a recovery rate of O( log(n)\n). When re-sampling is not allowed, we can improve the recovery\ndmin\nrate by dividing the observations into O( log(n)\u221a\nn ) independent sets, and apply one set of observations\nat each iteration. In this case, the recovery rate is O( log2(n)\u221a\nn ). These recovery rates suggest that the\nrecovery rate in Theorem 3.2 could potentially be improved. Nevertheless, Theorem 3.2 still shows\nthat TranSync can tolerate a sub-constant recovery rate, which is superior to the linear programming\nformulation.\n\n\u221a\n\n4 Experimental Results\n\nIn this section, we provide a detailed experimental evaluation of the proposed translation synchro-\nnization (TranSync) method. We begin with describing the experimental setup in Section 4.1. We\nthen perform evaluations on synthetic and real datasets in Section 4.2 and Section 4.3 respectively.\n\n4.1 Experimental Setup\n\nDatasets. We employ both synthetic datasets and real datasets for evaluation. The synthetic data is\ngenerated following the noisy model described in (6). In the following, we encode the noisy model\nas M(G, p, \u03c3), where G is the observation graph, p is the fraction of correct measurements, and \u03c3\ndescribes the interval of correct measurements. Besides the synthetic data, we also consider two real\ndatasets coming from the applications of joint alignment of point clouds and global ranking from\nrelative rankings.\nBaseline comparison. We choose coordinate descent for solving (1) as the baseline algorithm.\ni } are given by the\nSpeci\ufb01cally, denote the solution of xi, 1 \u2264 i \u2264 n at iteration k as x(k)\nfollowing recursion:\n\n. Then {x(k)\n\ni\n\n|xi \u2212 (x(k\u22121)\n\nj\n\n\u2212 tij)|\n\n\u2212 tij},\n\n1 \u2264 i \u2264 n,\n\nk = 1, 2,\u00b7\u00b7\u00b7 ,\n\n(7)\n\n(cid:88)\n\nj\u2208N (i)\n{x(k\u22121)\n\nj\n\nx(k)\ni = arg min\n\nxi\n\n= median\nj\u2208N (i)\n\nWe use the same initial starting point as TranSync. We also tested interior point methods, and all the\ndatasets used in our experiments are beyond their reach.\nEvaluation protocol. We report the min, median, and max of the coordinate-wise difference between\nthe solution of each algorithm and the underlying ground-truth. We also report the total running time\nof each algorithm on each dataset (See Table 1).\n\n4.2 Experimental Evaluation on Synthetic Datasets\n\nWe generate the synthetic datasets by sampling from four kinds of observation graphs and two values\nof \u03c3, i.e. \u03c3 \u2208 {0.01, 0.04}. The graphs are generated according to two modes: 1) dense graphs\nversus sparse graphs, and 2) regular graphs versus irregular graphs. To illustrate the strength of\nTranSync, we choose p \u2208 {0.4, 0.8} for dense graphs and p \u2208 {0.8, 1.0} for sparse graphs. Below is\na detailed descriptions for all kinds of observation graphs generated.\n\n\u2022 Gdr (dense, regular): The \ufb01rst graph contains n = 2000 nodes. Independently, we connect\nan edge between a pair of vertices vi, vj with a \ufb01xed probability p = 0.1. The expected\ndegree of each vertex is 200.\n\n\u2022 Gdi (dense, irregular): The second graph contains n = 2000 nodes. Independently, we\nconnect an edge between a pair of vertices vi, vj with probability p = 0.4sisj, where\nn\u22121 , 1 \u2264 i \u2264 n are scalar values associated the vertices. The expected degree\nsi = 0.2 + 0.6 i\u22121\nof each vertex is about 200.\n\n6\n\n\fTranSync\n\nG\nGdr\nGdr\nGdr\nGdr\nGdi\nGdi\nGdi\nGdi\nGsr\nGsr\nGsr\nGsr\nGsi\nGsi\nGsi\nGsi\n\np\n0.4\n0.4\n0.8\n0.8\n0.4\n0.4\n0.8\n0.8\n0.8\n0.8\n1.0\n1.0\n0.8\n0.8\n1.0\n1.0\n\n\u03c3\n0.01\n0.04\n0.01\n0.04\n0.01\n0.04\n0.01\n0.04\n0.01\n0.04\n0.01\n0.04\n0.01\n0.04\n0.01\n0.04\n\nmin\n\n0.95e-2\n3.87e-2\n0.30e-2\n1.19e-2\n2.17e-2\n5.46e-2\n0.34e-2\n1.39e-2\n0.58e-2\n2.35e-2\n0.45e-2\n1.84e-2\n0.72e-2\n2.88e-2\n0.53e-2\n2.24e-2\n\nmax\n\nCoordinate Descent\nmedian\n1.28e-2\n4.73e-2\n0.34e-2\n1.35e-2\n17.59e-2\n19.40e-2\n0.42e-2\n1.66e-2\n0.65e-2\n2.62e-2\n0.50e-2\n1.99e-2\n0.85e-2\n3.38e-2\n0.62e-2\n2.52e-2\n\n11.40e-2\n18.59e-2\n0.41e-2\n1.78e-2\n50.51e-2\n53.88e-2\n0.58e-2\n2.30e-2\n0.79e-2\n3.54e-2\n0.58e-2\n2.36e-2\n75.85e-2\n11.48e-2\n0.77e-2\n3.12e-2\n\ntime\n0.939s\n1.325s\n0.781s\n1.006s\n0.865s\n1.043s\n0.766s\n0.972s\n10.062s\n12.375s\n9.798s\n11.626s\n10.236s\n12.350s\n9.388s\n12.200s\n\nmin\n\n0.30e-2\n1.04e-2\n0.16e-2\n0.57e-2\n0.39e-2\n1.25e-2\n0.17e-2\n0.68e-2\n0.38e-2\n1.35e-2\n0.28e-2\n1.14e-2\n0.52e-2\n1.79e-2\n0.37e-2\n1.44e-2\n\nmedian\n0.37e-2\n1.22e-2\n0.18e-2\n0.70e-2\n0.52e-2\n1.55e-2\n0.24e-2\n0.86e-2\n0.45e-2\n1.55e-2\n0.32e-2\n1.29e-2\n0.64e-2\n2.16e-2\n0.43e-2\n1.72e-2\n\nmax\n\n0.60e-2\n1.59e-2\n0.28e-2\n0.87e-2\n0.93e-2\n2.42e-2\n0.33e-2\n1.16e-2\n0.61e-2\n2.05e-2\n0.39e-2\n1.60e-2\n1.10e-2\n3.59e-2\n0.57e-2\n2.47e-2\n\ntime\n0.178s\n0.155s\n0.149s\n0.133s\n0.179s\n0.169s\n0.159s\n0.141s\n1.852s\n1.577s\n0.188s\n0.179s\n1.835s\n1.610s\n0.180s\n0.187s\n\nTable 1: Experimental results comparing TranSync and Coordinate Descent (CD) under different\nsettings. All statistics (min, median, max) and mean running time are computed among 100 indepen-\ndent experiments with the same setting. As observed, TranSync outperforms Coordinate Descent in\nall experiments.\n\n\u2022 Gsr (sparse, regular): The third graph is generated in a similar fashion as the \ufb01rst graph,\nexcept that the number of nodes n = 20K, and the connecting probability is set to p = 0.003.\nThe expected degree of each vertex is 60.\n\u2022 Gsi (sparse, irregular): The fourth graph is generated in a similar fashion as the second\ngraph, except that the number of nodes n = 20K, and the connecting probability between a\nn\u22121 , 1 \u2264 i \u2264 n are scalar values\npair of vertices is p = 0.1sisj, where si = 0.07 + 0.21 i\u22121\nassociated the vertices. The expected degree of each vertex is about 60.\n\nFor this experiment, instead of using kmax as stopping condition as in Algorithm 1, we stop when we\nobserve \u03b4k < \u03b4min. Here \u03b4min does not need to be close to \u03c3. In fact, we choose \u03b4min = 0.05, 0.1 for\n\u03c3 = 0.01, 0.04, respectively. We also claim that if a small validation set (with size signi\ufb01cantly less\nthan n) of correct observations is available, our performance could be further improved.\nAs illustrated in Table 1, TranSync dominates coordinate descent in terms of both accuracy and\nprediction. In particular, TranSync is signi\ufb01cantly better than coordinate descent on dense graphs in\nterms of accuracy. In particular, on dense but irregular graphs, coordinate descent did not converge\nat all when p = 0.8. The main advantage of TranSync on sparse graphs is the computational cost,\nalthough the accuracy is still considerably better than coordinate descent.\n\n4.3 Experimental Evaluation on Real Datasets\n\nTranslation synchronization for joint alignment of point clouds.\nIn the \ufb01rst application, we\nconsider the problem of joint alignment of point clouds from pair-wise alignment [10]. To this end,\nwe utilize the Patriot Circle Lidar dataset1. We uniformly subsampled the dataset to 6K scans. We\napplied Super4PCS [19] to match each scan to 300 randomly selected scans, where each match\nreturns a pair-wise rigid transformation and a score. We then pick the top-30 matches for each scan,\nthis results in a graph with 140K edges. To create the input data for translation synchronization, we\nrun the state-of-the-art rotation synchronization algorithm described in [2] to estimate a global pose\ni tlocal\nRi for each scan. The pair-wise measurement tij from node i to node j is then given by RT\n,\nwhere tlocal\nis the translation vector obtained in pair-wise matching. The average outlier ratio of the\npair-wise matches per node is 35%, which is relatively high since the observation graph is fairly\nsparse. Since tij is a 3D vector, we run TranSync three times, one for each coordinate. As illustrated\nin Figure 1, TranSync is able to recover the the global shape of the underlying scanning trajectory. In\ncontrast, coordinate descent completely fails on this dataset.\n\nij\n\nij\n\n1http://masc.cs.gmu.edu/wiki/MapGMU\n\n7\n\n\fFigure 1: The application of TranSync in joint alignment of 6K Lidar scans around a city block. (a)\nSnapshot of the underlying scanning trajectory. (b) Reconstruction using TranSync (c) Reconstruction\nusing Coordinate Descent.\n\nMovie\n\nShakespeare in Love\n\nWitness\n\nOctober Sky\nThe Waterboy\n\nInterview with the Vampire\n\nDune\n\n1(0.247)\n2(0.217)\n3(0.213)\n6(-0.464)\n4(-0.031)\n5(-0.183)\n\n2(0.078)\n1(0.088)\n3(0.078)\n6(-0.162)\n4(-0.012)\n5(-0.069)\n\nGlobal ranking (score)\n\nMRQE Hodge-Diff. Hodge-Ratio Hodge-Binary\n1(85)\n2(77)\n3(76)\n4(66)\n5(65)\n6(44)\n\n1 (0.138)\n3(0.107)\n2(0.111)\n6(-0.252)\n4(-0.120)\n5(-0.092)\n\nTS-Init\n1(0.135)\n3(0.076)\n2(0.092)\n5(-0.134)\n4 (-0.098)\n6(-0.216)\n\nTS-Final\n1(0.219)\n2(0.095)\n3(0.0714)\n4(-0.112)\n5(-0.140)\n6(-0.281)\n\nTable 2: Global ranking of selected six movies via different methods: MRQE, HodgeRank[16] with\n1) arithmetic mean score difference, 2) geometric mean score ratio and 3) and binary comparisons,\nand the initial and \ufb01nal predictions of TranSync. TranSync results in the most consistent result with\nMRQE.\n\nRanking from relative comparisons. In the second application, we apply TranSync to predict\nglobal rankings of Net\ufb02ix movies from their relative comparisons provided by users. The Net\ufb02ix\ndataset contains 17070 movies that were rated between October, 1998 and December, 2005. We adapt\nthe procedure described in [16] to generate the input data. Speci\ufb01cally, for each pair of movies, we\naverage the relative ratings from the same users within the same month. We only consider a relative\nmeasurement if we collect more than 10 such relative ratings. We then apply TranSync to predict\nthe global rankings of all the movies. We report the initial prediction obtained by the \ufb01rst step of\nTranSync (i.e., all the relative comparisons are used) and the \ufb01nal prediction suggested by TranSync\n(i.e., after removing inconsistent relative comparisons).\nTable 2 compares TranSync with HodgeRank [16] on six representative movies that are studied\nin [16]. The experimental results show that both predictions appear to be more consistent with\nMRQE2 (the largest online directory of movie reviews on the internet) than HodgeRank [16] and its\nvariants, which were only applied on these six movies in isolation. Moreover, the \ufb01nal prediction is\nsuperior to the initial prediction. These observations indicate two key advantages of TranSync, i.e.,\nscalability on large-scale datasets and robustness to noisy relative comparisons.\n\n5 Conclusions and Future Work\n\nIn this paper, we have introduced an iterative algorithm for solving the translation synchronization\nproblem, which estimates the global locations of objects from noisy measurements of relative\nlocations. We have justi\ufb01ed the performance of our approach both experimentally and theoretically\nunder both deterministic and randomized conditions. Our approach is more scalable and accurate\nthan the standard linear programming formulation. In particular, when the pair-wise measurement\n\n2http://www.mrqe.com\n\n8\n\n\fis biased, our approach can still achieve sub-constant recovery rate, while the linear programming\napproach can tolerate no more than 50% of the measurements being biased.\nIn the future, we plan to extend this iterative scheme to other synchronization problems, such as\nsynchronizing rotations and point-based maps. Moreover, it would also be interesting to study\nvariants of the iterative scheme such as re-weighted least squares. We would also like to close the\ngap between the current recovery rate and the lower bound, which exhibits a poly-log factor. This\nrequires developing new tools for analyzing the iterative algorithm.\nAcknowledgement. Qixing Huang would like to acknowledge support this research from NSF DMS-\n1700234. Chandrajit Bajaj would like to acknowledge support for this research from the National\nInstitute of Health grants #R41 GM116300 and #R01 GM117594.\n\nReferences\n[1] F. 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In 2015\nIEEE International Conference on Computer Vision, ICCV 2015, Santiago, Chile, December 7-13, 2015,\npages 4032\u20134040, 2015.\n\n10\n\n\f", "award": [], "sourceid": 931, "authors": [{"given_name": "Xiangru", "family_name": "Huang", "institution": "University of Texas at Austin"}, {"given_name": "Zhenxiao", "family_name": "Liang", "institution": "Tsinghua University"}, {"given_name": "Chandrajit", "family_name": "Bajaj", "institution": "The University of Texas at Austin"}, {"given_name": "Qixing", "family_name": "Huang", "institution": "The University of Texas at Austin"}]}