Translation Synchronizationvia Truncated Least Squares
Xiangru Huang1* Zhenxiao Liang2*Chandrajit Bajaj 1 Qixing Huang 1
1Department of Computer ScienceUniversity of Texas at Austin
2Tsinghua University
NIPS, 2017
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 1 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
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From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
µ
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From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
µ
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From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
µ
mean
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From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
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From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 7 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
mean
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 8 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
meanmedian
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From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
meanmedian
1. Delete Sample tj if |tj −mean|>ǫ
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 10 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
meanmedian
1. Delete Sample tj if |tj −mean|>ǫ
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 11 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
meanmedian
1. Delete Sample tj if |tj −mean|>ǫc1
2. Recompute mean and Shrink threshold
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 12 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
meanmedian
1. Delete Sample tj if |tj −mean|>ǫc2
2. Recompute mean and Shrink threshold
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 13 / 20
From a Simple Example
U[µ− σ, µ+ σ]
σ−σ
Outliers
µ
meanmedian
1. Delete Sample tj if |tj −mean|>ǫc3
2. Recompute mean and Shrink threshold
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 14 / 20
Translation Synchronization
Ground Truth {xi}Relative measurements tij = xi − xj + noise ∀i , j ∈ E
Algorithm: iteratively update x and E .Can be applied to Pairwise Ranking, Joint Alignment of point clouds,and etc.
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20
Translation Synchronization
Ground Truth {xi}Relative measurements tij = xi − xj + noise ∀i , j ∈ EAlgorithm: iteratively update x and E .
Can be applied to Pairwise Ranking, Joint Alignment of point clouds,and etc.
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20
Translation Synchronization
Ground Truth {xi}Relative measurements tij = xi − xj + noise ∀i , j ∈ EAlgorithm: iteratively update x and E .Can be applied to Pairwise Ranking, Joint Alignment of point clouds,and etc.
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20
Exact Recovery
Biased Noise Model (Unbounded Outliers):
tij =
{xgti − xgtj + U[−σ, σ] with probability p
Any real number with probability 1− p(1)
For some constants p, q only depend on graph structure, duringoptimization we have
‖x (k) − xgt‖∞ ≤ qσ + 2pεck−1
and eventually we’ll reach an x̂
‖x̂ − xgt‖∞ ≤2p + cq
c − 4pσ
where the RHS is independent of ε.
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20
Exact Recovery
Biased Noise Model (Unbounded Outliers):
tij =
{xgti − xgtj + U[−σ, σ] with probability p
Any real number with probability 1− p(1)
For some constants p, q only depend on graph structure, duringoptimization we have
‖x (k) − xgt‖∞ ≤ qσ + 2pεck−1
and eventually we’ll reach an x̂
‖x̂ − xgt‖∞ ≤2p + cq
c − 4pσ
where the RHS is independent of ε.
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20
Randomized Case
Biased Noise Model:
tij =
{xgti − xgtj + U[−σ, σ] with probability p
xgti − xgtj + U[−a, b] with probability 1− p(2)
TheoremThere exists a constant c so that if p > c/
√log(n), then w.h.p,
‖x (k) − xgt‖∞ ≤ (1− p/2)k(b − a),
∀ k = 0, · · · , [− log(b + a
2σ)/log(1− p/2)].
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 17 / 20
Randomized Case
Biased Noise Model:
tij =
{xgti − xgtj + U[−σ, σ] with probability p
xgti − xgtj + U[−a, b] with probability 1− p(2)
TheoremThere exists a constant c so that if p > c/
√log(n), then w.h.p,
‖x (k) − xgt‖∞ ≤ (1− p/2)k(b − a),
∀ k = 0, · · · , [− log(b + a
2σ)/log(1− p/2)].
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 17 / 20
Experiments on Synthetic Graphs{Dense, Sparse } × { Regular, Irregular }
{Dense, Sparse} × {Regular, Irregular}
(a) Regular (b) Irregular
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Experiments on Synthetic Graphs{Dense, Sparse } × { Regular, Irregular }
0.4, 0.01 0.4, 0.04 0.8, 0.01 0.8, 0.04{p, σ
}0.0
0.2
0.4
0.6
0.8
1.0
Norm
alized Error (M
in, Median, Max) Dense Regular
ℓ1 min
TranSync
0.4, 0.01 0.4, 0.04 0.8, 0.01 0.8, 0.04{p, σ
}0.0
0.2
0.4
0.6
0.8
1.0
Norm
alized Error (M
in, Median, Max) Dense Irregular
ℓ1 min
TranSync
0.8, 0.01 0.8, 0.04 1.0, 0.01 1.0, 0.04{p, σ
}0.0
0.2
0.4
0.6
0.8
1.0
Norm
alized Error (M
in, Median, Max) Sparse Regular
ℓ1 min
TranSync
0.8, 0.01 0.8, 0.04 1.0, 0.01 1.0, 0.04{p, σ
}0.0
0.2
0.4
0.6
0.8
1.0
Norm
alized Error (M
in, Median, Max) Sparse Irregular
ℓ1 min
TranSync
Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 19 / 20
Joint Alignment of 6K Lidar Scans
(c) Ground Truth (d) Our Method (e) `1 Minimization
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