Translation Synchronization via Truncated Least Squares Xiangru - PowerPoint PPT Presentation

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

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 mean U [ µ − σ, µ + σ ] − σ σ µ Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 5 / 20

Outliers 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 6 / 20

Outliers 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 7 / 20

Outliers From a Simple Example mean U [ µ − σ, µ + σ ] − σ σ µ Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 8 / 20

Outliers From a Simple Example median mean U [ µ − σ, µ + σ ] − σ σ µ Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 9 / 20

Outliers From a Simple Example median mean U [ µ − σ, µ + σ ] − σ σ µ 1. Delete Sample t j if | t j − mean | > ǫ Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 10 / 20

Outliers From a Simple Example median mean U [ µ − σ, µ + σ ] − σ σ µ 1. Delete Sample t j if | t j − mean | > ǫ Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 11 / 20

Outliers From a Simple Example median mean U [ µ − σ, µ + σ ] − σ σ µ 1. Delete Sample t j if | t j − mean | > ǫc 1 2. Recompute mean and Shrink threshold Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 12 / 20

Outliers From a Simple Example median mean U [ µ − σ, µ + σ ] − σ σ µ 1. Delete Sample t j if | t j − mean | > ǫc 2 2. Recompute mean and Shrink threshold Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 13 / 20

Outliers From a Simple Example median mean U [ µ − σ, µ + σ ] − σ σ µ 1. Delete Sample t j if | t j − mean | > ǫc 3 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 { x i } Relative measurements t ij = x i − x j + noise ∀ i , j ∈ E Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20

Translation Synchronization Ground Truth { x i } Relative measurements t ij = x i − x j + noise ∀ i , j ∈ E Algorithm: iteratively update x and E . Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20

Translation Synchronization Ground Truth { x i } Relative measurements t ij = x i − x j + 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

Exact Recovery Biased Noise Model (Unbounded Outliers): � x gt − x gt + U [ − σ, σ ] with probability p i j t ij = (1) Any real number with probability 1 − p Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20

Exact Recovery Biased Noise Model (Unbounded Outliers): � x gt − x gt + U [ − σ, σ ] with probability p i j t ij = (1) Any real number with probability 1 − p For some constants p , q only depend on graph structure, during optimization we have � x ( k ) − x gt � ∞ ≤ q σ + 2 p ǫ c k − 1 and eventually we’ll reach an ˆ x x − x gt � ∞ ≤ 2 p + cq � ˆ c − 4 p σ where the RHS is independent of ǫ . Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20

Randomized Case Biased Noise Model: � x gt − x gt + U [ − σ, σ ] with probability p i j t ij = (2) x gt − x gt + U [ − a , b ] with probability 1 − p i j Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 17 / 20

Randomized Case Biased Noise Model: � x gt − x gt + U [ − σ, σ ] with probability p i j t ij = (2) x gt − x gt + U [ − a , b ] with probability 1 − p i j Theorem � There exists a constant c so that if p > c / log( n ), then w.h.p, � x ( k ) − x gt � ∞ ≤ (1 − p / 2) k ( b − a ) , ∀ k = 0 , · · · , [ − log( b + a 2 σ ) / 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 } Dense Regular Dense Irregular Normalized Error (Min, Median, Max) Normalized Error (Min, Median, Max) ℓ 1 min ℓ 1 min TranSync TranSync 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.4, 0.01 0.4, 0.04 0.8, 0.01 0.8, 0.04 0.4, 0.01 0.4, 0.04 0.8, 0.01 0.8, 0.04 � � � � p, σ p, σ Sparse Regular Sparse Irregular Normalized Error (Min, Median, Max) Normalized Error (Min, Median, Max) ℓ 1 min ℓ 1 min TranSync TranSync 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.8, 0.01 0.8, 0.04 1.0, 0.01 1.0, 0.04 0.8, 0.01 0.8, 0.04 1.0, 0.01 1.0, 0.04 � � � � p, σ p, σ 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) ℓ 1 Minimization Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 20 / 20