Translation Synchronization via Truncated Least...

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Translation Synchronization via Truncated Least Squares Xiangru Huang 1 * Zhenxiao Liang 2 * Chandrajit Bajaj 1 Qixing Huang 1 1 Department of Computer Science University of Texas at Austin 2 Tsinghua University NIPS, 2017 Xiangru Huang *, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 1 / 20
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  • 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

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 2 / 20

  • From a Simple Example

    U[µ− σ, µ+ σ]

    σ−σ

    µ

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 3 / 20

  • From a Simple Example

    U[µ− σ, µ+ σ]

    σ−σ

    µ

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 4 / 20

  • From a Simple Example

    U[µ− σ, µ+ σ]

    σ−σ

    µ

    mean

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 5 / 20

  • 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 6 / 20

  • 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

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 9 / 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 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|>ǫc12. 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|>ǫc22. 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|>ǫc32. 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 − x

    gtj + 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 + 2p�ck−1

    and eventually we’ll reach an x̂

    ‖x̂ − xgt‖∞ ≤2p + cq

    c − 4pσ

    where the RHS is independent of �.

    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 − x

    gtj + 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 + 2p�ck−1

    and eventually we’ll reach an x̂

    ‖x̂ − xgt‖∞ ≤2p + cq

    c − 4pσ

    where the RHS is independent of �.

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20

  • Randomized Case

    Biased Noise Model:

    tij =

    {xgti − x

    gtj + 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 + a2σ

    )/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 − x

    gtj + 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 + a2σ

    )/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

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 18 / 20

  • 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

    }0.00.2

    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

    }0.00.2

    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

    }0.00.2

    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

    }0.00.2

    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

    Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 20 / 20

    MotivationTheoryExact Recovery

    ExperimentsSyntheticReal