Localization from Incomplete Noisy Distance Measurements Adel - PowerPoint PPT Presentation

Localization from Incomplete Noisy Distance Measurements Adel Javanmard and Andrea Montanari Stanford University August 3, 2011 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 1 / 47

A chemistry question Which physical conformations are produced by given chemical bonds? Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 2 / 47

Other Motivations (a) Manifold Learning (b) Sensor Net. Localization (c) Indoor Positioning Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 3 / 47

General ‘geometric inference’ problem Given partial/noisy information about a cloud of points. Reconstruct the points positions. Notes Positions can be reconstructed up to rigid motions Well-posed problem only if G is connected In general, the problem (even uniqueness of reconstruction) is NP-hard [Saxe 1979] Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 4 / 47

General ‘geometric inference’ problem Given partial/noisy information about a cloud of points. Reconstruct the points positions. Notes Positions can be reconstructed up to rigid motions Well-posed problem only if G is connected In general, the problem (even uniqueness of reconstruction) is NP-hard [Saxe 1979] Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 4 / 47

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 R.G.G. G ✭ n ❀ r ✮ x 1 ❀ ✁ ✁ ✁ ❀ x n ✷ ❬ � 0 ✿ 5 ❀ 0 ✿ 5 ❪ d r ✕ ☛ ✭ log n ❂ n ✮ 1 ❂ d Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 5 / 47

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 adversarial noise R.G.G. G ✭ n ❀ r ✮ ❥ ⑦ d 2 ij � d 2 x 1 ❀ ✁ ✁ ✁ ❀ x n ✷ ❬ � 0 ✿ 5 ❀ 0 ✿ 5 ❪ d ij ❥ ✔ ✁ r ✕ ☛ ✭ log n ❂ n ✮ 1 ❂ d Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 6 / 47

Related work Triangulation Multidimensional scaling Divide and conquer (Singer 2008) Few performance guarantees, especially in presence of noise Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 7 / 47

Outline SDP relaxation and robust reconstruction 1 Lower bound 2 Rigidity theory and upper bound 3 Discussion 4 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 8 / 47

SDP relaxation and robust reconstruction Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 9 / 47

Optimization formulation n ❳ ❦ x i ❦ 2 minimize 2 i ❂ 1 ☞ ☞ ☞ ☞ 2 � ❡ ☞ ❦ x i � x j ❦ 2 d 2 subject to ☞ ✔ ✁ ij Nonconvex Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 10 / 47

Optimization formulation n ❳ minimize Q ii i ❂ 1 ☞ ☞ ☞ ☞ ☞ Q ii � 2 Q ij ✰ Q jj � ❡ d 2 subject to ☞ ✔ ✁ ij Q ij ❂ ❤ x i ❀ x j ✐ Nonconvex Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 11 / 47

Optimization formulation (better notation) minimize Tr ✭ Q ✮ ☞ ☞ ☞ ☞ ☞ ❤ M ij ❀ Q ✐ � ❡ d 2 subj ✿ to ☞ ✔ ✁ ij Q ij ❂ ❤ x i ❀ x j ✐ e ij e T ❂ ij ❀ M ij e ij ❂ ✭ 0 ❀ ✿ ✿ ✿ ❀ 0 ❀ ✰ 1 ❀ 0 ❀ ✿ ✿ ✿ ❀ 0 ❀ � 1 ❀ 0 ❀ ✿ ✿ ✿ ❀ 0 ✮ ⑤④③⑥ ⑤④③⑥ i j Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 12 / 47

Semidefinite programing relaxation minimize Tr ✭ Q ✮ ☞ ☞ ☞ ☞ ☞ ❤ M ij ❀ Q ✐ � ❡ d 2 subj ✿ to ☞ ✔ ✁ ij ✭ ✭✭✭✭✭✭ Q ✗ 0 Q ij ❂ ❤ x i ❀ x j ✐ e ij e T ❂ ij ❀ M ij e ij ❂ ✭ 0 ❀ ✿ ✿ ✿ ❀ 0 ❀ ✰ 1 ❀ 0 ❀ ✿ ✿ ✿ ❀ 0 ❀ � 1 ❀ 0 ❀ ✿ ✿ ✿ ❀ 0 ✮ ⑤④③⑥ ⑤④③⑥ i j Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 13 / 47

Semidefinite programing relaxation SDP-based Localization Input : Distance measurements ❡ d ij , ✭ i ❀ j ✮ ✷ G Output : Low-dimensional coordinates x 1 ❀ ✿ ✿ ✿ ❀ x n ✷ R d 1: Solve the following SDP problem: minimize Tr ✭ Q ✮ , ☞ ☞ ☞ ✔ ✁ , ☞ ❤ M ij ❀ Q ✐ � ❡ d 2 s.t. ✭ i ❀ j ✮ ✷ G , ij Q ✗ 0. Eigendecomposition Q ❂ U ✝ U T ; 2: Top d e-vectors X ❂ U d ✝ 1 ❂ 2 3: ; d Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 14 / 47

Robustness? Theorem (Javanmard, Montanari ’11) ♣ d ✭ log n ❂ n ✮ 1 ❂ d . Then, w.h.p., Assume r ✕ 10 X ✮ ✔ C 1 ✭ nr d ✮ 5 ✁ d ✭ X ❀ ❫ r 4 ❀ Further, there exists a set of ‘adversarial’ measurements such that ✁ d ✭ X ❀ ❫ X ✮ ✕ C 2 r 4 ✿ n ❳ X ✮ ✙ 1 d ✭ X ❀ ❫ ❦ x i � ❜ x i ❦ n i ❂ 1 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 15 / 47

Robustness? Theorem (Javanmard, Montanari ’11) ♣ d ✭ log n ❂ n ✮ 1 ❂ d . Then, w.h.p., Assume r ✕ 10 X ✮ ✔ C 1 ✭ nr d ✮ 5 ✁ d ✭ X ❀ ❫ r 4 ❀ Further, there exists a set of ‘adversarial’ measurements such that ✁ d ✭ X ❀ ❫ X ✮ ✕ C 2 r 4 ✿ n ❳ X ✮ ✙ 1 d ✭ X ❀ ❫ ❦ x i � ❜ x i ❦ n i ❂ 1 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 15 / 47

Lower bound Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 16 / 47

Proof: Lower bound Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 17 / 47

Proof: Lower bound (first attempt) 0.5
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 q ✁ Scale the coordinates by a ❂ r 2 ✰ 1 X ✮ ✕ ✁ d ✭ X ❀ ❫ r 2 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 18 / 47

Proof: Lower bound ❚ ✿ ❬ � 0 ✿ 5 ❀ 0 ✿ 5 ❪ d ✦ R d ✰ 1 R ❂ r 2 ❚ ✭ t 1 ❀ t 2 ❀ ✁ ✁ ✁ ❀ t d ✮ ❂ ✭ R sin t 1 R ❀ t 2 ❀ ✁ ✁ ✁ ❀ t d ❀ R ✭ 1 � cos t 1 R ✮✮ ❀ ♣ ✁ Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 19 / 47

Rigidity theory and upper bound Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 20 / 47

Uniqueness ✱ Global rigidity Global rigidity Assume noiseless measurements. Is the reconstruction unique? (up to rigid motions) Depends both on G and on ✭ x 1 ❀ ✿ ✿ ✿ ❀ x n ✮ Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 21 / 47

Global rigidity: Characterization Theorem (Connelly 1995; Gortler, Healy, Thurston, 2007) ✭ G ❀ ❢ x i ❣ ✮ is globally rigid in R d ✱ ✭ G ❀ ❢ x i ❣ ✮ admits a stress matrix ✡ , with rank ✭✡✮ ❂ n � d � 1 . Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 22 / 47

Stress matrix Definition ✡ ✷ R n ✂ n is a stress matrix if supp ✭✡✮ ✒ E and ✡ u ❂ ✡ x ✭ 1 ✮ ❂ ✿ ✿ ✿ ✡ x ✭ d ✮ ❂ 0 ✿ u ❂ ✭ 1 ❀ ✿ ✿ ✿ ❀ 1 ✮ ✷ R n x ✭ ❵ ✮ ✷ R n vector of positions’ ❵ -th coordinate rank ✭✡✮ ✔ n � d � 1 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 23 / 47

Stress matrix: Some intuition ✿ ✿ ✿ imagine putting springs on the edges ✿ ✿ ✿ ✦ ij Equilibrium x 1 ❀ ✿ ✿ ✿ ❀ x n : ❳ [force on i ] ❂ ✦ ij ✭ x j � x i ✮ ❂ 0 j ✷ ❅ i ✡ ij ❂ ✦ ij , ✡ ii ❂ � P j ✷ ❅ i ✦ ij Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 24 / 47

Stress matrix: Some intuition ✿ ✿ ✿ imagine putting springs on the edges ✿ ✿ ✿ ✦ ij Equilibrium x 1 ❀ ✿ ✿ ✿ ❀ x n : ❳ [force on i ] ❂ ✦ ij ✭ x j � x i ✮ ❂ 0 j ✷ ❅ i ✡ ij ❂ ✦ ij , ✡ ii ❂ � P j ✷ ❅ i ✦ ij Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 24 / 47

Infinitesimal rigidity Consider a continuos motion preserving distances instantaneously ✭ x i � x j ✮ T ✭ ❴ x i � ❴ x j ✮ ❂ 0 ❀ ✽ ✭ i ❀ j ✮ ✷ E Trivial motions A ❂ � A T ✷ R d ✂ d x i ❂ Ax i ✰ b ❀ ❴ ✡ ✣ ❏ ❪ ✡ ❏ rotation translation Definition ✭ G ❀ ❢ x i ❣ ✮ is infinitesimally rigid if rotations and translations are the only infinitesimal motions. Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 25 / 47

Infinitesimal rigidity Consider a continuos motion preserving distances instantaneously ✭ x i � x j ✮ T ✭ ❴ x i � ❴ x j ✮ ❂ 0 ❀ ✽ ✭ i ❀ j ✮ ✷ E Trivial motions A ❂ � A T ✷ R d ✂ d x i ❂ Ax i ✰ b ❀ ❴ ✡✡ ✣ ❪ ❏ ❏ rotation translation Definition ✭ G ❀ ❢ x i ❣ ✮ is infinitesimally rigid if rotations and translations are the only infinitesimal motions. Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 25 / 47

Rigidity matrix ✭ x i � x j ✮ T ✭ ❴ x i � ❴ x j ✮ ❂ 0 ❀ ✽ ✭ i ❀ j ✮ ✷ E ✷ ✸ ❴ x 1 ✻ . ✼ . R G ❀ X ✁ ✺ ❂ 0 ✹ . ❴ x n Definition R G ❀ X ✷ R ❥ E ❥✂ nd is the rigidity matrix of framework ✭ G ❀ ❢ x i ❣ ✮ . ✥ ✦ dim ✭ null ✭ R G ❀ X ✮✮ ✕ d ✭ d � 1 ✮ d ✰ 1 ✰ d ❂ ⑤④③⑥ 2 2 ⑤ ④③ ⑥ b A � d ✰ 1 ✁ . ✭ G ❀ ❢ x i ❣ ✮ is infinitesimally rigid if rank ✭ R G ❀ X ✮ ❂ nd � 2 Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 26 / 47