[PPT] - Localization from Incomplete Noisy Distance Measurements Adel PowerPoint Presentation

SLIDE 1

Localization from Incomplete Noisy Distance Measurements

Adel Javanmard and Andrea Montanari

Stanford University

August 3, 2011

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A chemistry question

Which physical conformations are produced by given chemical bonds?

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Other Motivations

(a) Manifold Learning (b) Sensor Net. Localization (c) Indoor Positioning

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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]

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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]

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This talk

0.5  0  ‐0.5  ‐0.5  0  0.5 

R.G.G. G✭n❀ r✮

x1❀ ✁ ✁ ✁ ❀ xn ✷ ❬0✿5❀ 0✿5❪d r ✕ ☛✭log n❂n✮1❂d

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This talk

0.5  0  ‐0.5  ‐0.5  0  0.5 

R.G.G. G✭n❀ r✮ x1❀ ✁ ✁ ✁ ❀ xn ✷ ❬0✿5❀ 0✿5❪d r ✕ ☛✭log n❂n✮1❂d adversarial noise ❥⑦ d2

ij d2 ij ❥ ✔ ✁ Javanmard, Montanari (Stanford) Localization Problem August 3, 2011 6 / 47

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Related work

Triangulation Multidimensional scaling Divide and conquer (Singer 2008) Few performance guarantees, especially in presence of noise

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Outline

1

SDP relaxation and robust reconstruction

2

Lower bound

3

Rigidity theory and upper bound

4

Discussion

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SDP relaxation and robust reconstruction

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Optimization formulation

minimize

n

❳

i❂1

❦xi❦2

2

subject to

☞ ☞ ☞❦xi xj ❦2

2 ❡

d2

ij

☞ ☞ ☞ ✔ ✁

Nonconvex

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Optimization formulation

minimize

n

❳

i❂1

Qii subject to

☞ ☞ ☞Qii 2Qij ✰ Qjj ❡

d2

ij

☞ ☞ ☞ ✔ ✁

Qij ❂ ❤xi❀ xj ✐ Nonconvex

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Optimization formulation (better notation)

minimize Tr✭Q✮ subj✿to

☞ ☞ ☞❤Mij ❀ Q✐ ❡

d2

ij

☞ ☞ ☞ ✔ ✁

Qij ❂ ❤xi❀ xj ✐ Mij ❂ eij eT

ij ❀

eij ❂ ✭0❀ ✿ ✿ ✿ ❀ 0❀ ✰1

⑤④③⑥

i

❀ 0❀ ✿ ✿ ✿ ❀ 0❀ 1

⑤④③⑥

j

❀ 0❀ ✿ ✿ ✿ ❀ 0✮

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Semidefinite programing relaxation

minimize Tr✭Q✮ subj✿to

☞ ☞ ☞❤Mij ❀ Q✐ ❡

d2

ij

☞ ☞ ☞ ✔ ✁ ✭✭✭✭✭✭ ✭

Qij ❂ ❤xi❀ xj ✐ Q ✗ 0 Mij ❂ eij eT

ij ❀

eij ❂ ✭0❀ ✿ ✿ ✿ ❀ 0❀ ✰1

⑤④③⑥

i

❀ 0❀ ✿ ✿ ✿ ❀ 0❀ 1

⑤④③⑥

j

❀ 0❀ ✿ ✿ ✿ ❀ 0✮

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Semidefinite programing relaxation

SDP-based Localization Input : Distance measurements ❡ dij , ✭i❀ j ✮ ✷ G Output : Low-dimensional coordinates x1❀ ✿ ✿ ✿ ❀ xn ✷ Rd 1: Solve the following SDP problem: minimize Tr✭Q✮, s.t.

☞ ☞❤Mij ❀ Q✐ ❡

d2

ij

☞ ☞ ✔ ✁,

✭i❀ j ✮ ✷ G, Q ✗ 0. 2: Eigendecomposition Q ❂ U✝U T; 3: Top d e-vectors X ❂ Ud✝1❂2

d

;

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Robustness?

Theorem (Javanmard, Montanari ’11)

Assume r ✕ 10 ♣ d✭log n❂n✮1❂d. Then, w.h.p., d✭X ❀ ❫ X ✮ ✔ C1✭nr d✮5 ✁ r 4 ❀ Further, there exists a set of ‘adversarial’ measurements such that d✭X ❀ ❫ X ✮ ✕ C2 ✁ r 4 ✿ d✭X ❀ ❫ X ✮ ✙ 1 n

n

❳

i❂1

❦xi ❜ xi❦

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Robustness?

Theorem (Javanmard, Montanari ’11)

Assume r ✕ 10 ♣ d✭log n❂n✮1❂d. Then, w.h.p., d✭X ❀ ❫ X ✮ ✔ C1✭nr d✮5 ✁ r 4 ❀ Further, there exists a set of ‘adversarial’ measurements such that d✭X ❀ ❫ X ✮ ✕ C2 ✁ r 4 ✿ d✭X ❀ ❫ X ✮ ✙ 1 n

n

❳

i❂1

❦xi ❜ xi❦

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Lower bound

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Proof: Lower bound

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Proof: Lower bound (first attempt)

0.5  0  ‐0.5  ‐0.5  0  0.5 

Scale the coordinates by a ❂

q

✁ r 2 ✰ 1

d✭X ❀ ❫ X ✮ ✕ ✁ r 2

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Proof: Lower bound

❚ ✿ ❬0✿5❀ 0✿5❪d ✦ Rd✰1 ❚ ✭t1❀ t2❀ ✁ ✁ ✁ ❀ td✮ ❂ ✭R sin t1 R ❀ t2❀ ✁ ✁ ✁ ❀ td❀ R✭1 cos t1 R ✮✮❀ R ❂ r 2 ♣ ✁

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Rigidity theory and upper bound

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Uniqueness ✱ Global rigidity

Global rigidity

Assume noiseless measurements. Is the reconstruction unique? (up to rigid motions) Depends both on G and on ✭x1❀ ✿ ✿ ✿ ❀ xn✮

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Global rigidity: Characterization

Theorem (Connelly 1995; Gortler, Healy, Thurston, 2007)

✭G❀ ❢xi❣✮ is globally rigid in Rd ✱ ✭G❀ ❢xi❣✮ admits a stress matrix ✡, with rank✭✡✮ ❂ n d 1.

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Stress matrix

Definition

✡ ✷ Rn✂n is a stress matrix if supp✭✡✮ ✒ E and ✡u ❂ ✡x ✭1✮ ❂ ✿ ✿ ✿ ✡x ✭d✮ ❂ 0 ✿ u ❂ ✭1❀ ✿ ✿ ✿ ❀ 1✮ ✷ Rn x ✭❵✮ ✷ Rn vector of positions’ ❵-th coordinate rank✭✡✮ ✔ n d 1

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Stress matrix: Some intuition

✿ ✿ ✿ imagine putting springs on the edges ✿ ✿ ✿

✦ij

Equilibrium x1❀ ✿ ✿ ✿ ❀ xn: [force on i] ❂

❳

j ✷❅i

✦ij ✭xj xi✮ ❂ 0 ✡ij ❂ ✦ij , ✡ii ❂ P

j ✷❅i ✦ij

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Stress matrix: Some intuition

✿ ✿ ✿ imagine putting springs on the edges ✿ ✿ ✿

✦ij

Equilibrium x1❀ ✿ ✿ ✿ ❀ xn: [force on i] ❂

❳

j ✷❅i

✦ij ✭xj xi✮ ❂ 0 ✡ij ❂ ✦ij , ✡ii ❂ P

j ✷❅i ✦ij

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Infinitesimal rigidity

Consider a continuos motion preserving distances instantaneously ✭xi xj ✮T✭ ❴ xi ❴ xj ✮ ❂ 0❀ ✽✭i❀ j ✮ ✷ E Trivial motions ❴ xi ❂ Axi ✰ b❀ A ❂ AT ✷ Rd✂d

✡ ✡ ✣

rotation translation

❏ ❏ ❪

Definition

✭G❀ ❢xi❣✮ is infinitesimally rigid if rotations and translations are the

nly infinitesimal motions.

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Infinitesimal rigidity

Consider a continuos motion preserving distances instantaneously ✭xi xj ✮T✭ ❴ xi ❴ xj ✮ ❂ 0❀ ✽✭i❀ j ✮ ✷ E Trivial motions ❴ xi ❂ Axi ✰ b❀ A ❂ AT ✷ Rd✂d

✡✡ ✣

rotation translation

❏ ❏ ❪

Definition

✭G❀ ❢xi❣✮ is infinitesimally rigid if rotations and translations are the

nly infinitesimal motions.

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Rigidity matrix

✭xi xj ✮T✭ ❴ xi ❴ xj ✮ ❂ 0❀ ✽✭i❀ j ✮ ✷ E RG❀X ✁

✷ ✻ ✹

❴ x1 . . . ❴ xn

✸ ✼ ✺ ❂ 0

Definition

RG❀X ✷ R❥E❥✂nd is the rigidity matrix of framework ✭G❀ ❢xi❣✮. dim✭null✭RG❀X ✮✮ ✕ d✭d 1✮ 2

⑤ ④③ ⑥

A

✰ d

⑤④③⑥

b

❂

✥

d ✰ 1 2

✦

✭G❀ ❢xi❣✮ is infinitesimally rigid if rank✭RG❀X ✮ ❂ nd

d✰1

2 ✁.

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Rigidity matrix

✭xi xj ✮T✭ ❴ xi ❴ xj ✮ ❂ 0❀ ✽✭i❀ j ✮ ✷ E RG❀X ✁

✷ ✻ ✹

❴ x1 . . . ❴ xn

✸ ✼ ✺ ❂ 0

Definition

RG❀X ✷ R❥E❥✂nd is the rigidity matrix of framework ✭G❀ ❢xi❣✮. dim✭null✭RG❀X ✮✮ ✕ d✭d 1✮ 2

⑤ ④③ ⑥

A

✰ d

⑤④③⑥

b

❂

✥

d ✰ 1 2

✦

✭G❀ ❢xi❣✮ is infinitesimally rigid if rank✭RG❀X ✮ ❂ nd

d✰1

2 ✁.

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Rigidity matrix

✭xi xj ✮T✭ ❴ xi ❴ xj ✮ ❂ 0❀ ✽✭i❀ j ✮ ✷ E RG❀X ✁

✷ ✻ ✹

❴ x1 . . . ❴ xn

✸ ✼ ✺ ❂ 0

Definition

RG❀X ✷ R❥E❥✂nd is the rigidity matrix of framework ✭G❀ ❢xi❣✮. dim✭null✭RG❀X ✮✮ ✕ d✭d 1✮ 2

⑤ ④③ ⑥

A

✰ d

⑤④③⑥

b

❂

✥

d ✰ 1 2

✦

✭G❀ ❢xi❣✮ is infinitesimally rigid if rank✭RG❀X ✮ ❂ nd

d✰1

2 ✁.

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(Idea of the) proof of the upper bound

Noise is analogous to stretching/compressing the edges Need to measure the stability of the framework Global/infinitesimal rigidity is checked by rank of Stress/Rigidity matrix. Needed: Quantitative rigidity theory (Rank ✮ bounds on singular values)

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An experiment

(d) Graph I (e) Graph II

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Quantitative rigidity theory

U✭X ✮ ❂

❳

✭i❀j ✮✷E

1 2✭❦xi xj ❦2 d2

ij ✮2

❳

i

f T

i xi

❴ X ❂ rX U xi ✰ ✍xi equilibrium positions in the presence of force ✭✡ ✡ Id ✰ RG❀X RT

G❀X ✮ ✁ ✍x ✙ f

✍x ❂ P❄

❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✰ RG❀X RT G❀X ✭P❄ ❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮✮ ✙ f

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✙ f ✮ ✕min✭✡❥❤X ❀u✐❄✮❦P❄ ❤X ❀u✐✭✍x✮❦ ✙ ❦f ❦

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Quantitative rigidity theory

U✭X ✮ ❂

❳

✭i❀j ✮✷E

1 2✭❦xi xj ❦2 d2

ij ✮2

❳

i

f T

i xi

❴ X ❂ rX U xi ✰ ✍xi equilibrium positions in the presence of force ✭✡ ✡ Id ✰ RG❀X RT

G❀X ✮ ✁ ✍x ✙ f

✍x ❂ P❄

❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✰ RG❀X RT G❀X ✭P❄ ❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮✮ ✙ f

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✙ f ✮ ✕min✭✡❥❤X ❀u✐❄✮❦P❄ ❤X ❀u✐✭✍x✮❦ ✙ ❦f ❦

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Quantitative rigidity theory

U✭X ✮ ❂

❳

✭i❀j ✮✷E

1 2✭❦xi xj ❦2 d2

ij ✮2

❳

i

f T

i xi

❴ X ❂ rX U xi ✰ ✍xi equilibrium positions in the presence of force ✭✡ ✡ Id ✰ RG❀X RT

G❀X ✮ ✁ ✍x ✙ f

✍x ❂ P❄

❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✰ RG❀X RT G❀X ✭P❄ ❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮✮ ✙ f

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✙ f ✮ ✕min✭✡❥❤X ❀u✐❄✮❦P❄ ❤X ❀u✐✭✍x✮❦ ✙ ❦f ❦

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Quantitative rigidity theory

U✭X ✮ ❂

❳

✭i❀j ✮✷E

1 2✭❦xi xj ❦2 d2

ij ✮2

❳

i

f T

i xi

❴ X ❂ rX U xi ✰ ✍xi equilibrium positions in the presence of force ✭✡ ✡ Id ✰ RG❀X RT

G❀X ✮ ✁ ✍x ✙ f

✍x ❂ P❄

❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✰ RG❀X RT G❀X ✭P❄ ❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮✮ ✙ f

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✙ f ✮ ✕min✭✡❥❤X ❀u✐❄✮❦P❄ ❤X ❀u✐✭✍x✮❦ ✙ ❦f ❦

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Quantitative rigidity theory

U✭X ✮ ❂

❳

✭i❀j ✮✷E

1 2✭❦xi xj ❦2 d2

ij ✮2

❳

i

f T

i xi

❴ X ❂ rX U xi ✰ ✍xi equilibrium positions in the presence of force ✭✡ ✡ Id ✰ RG❀X RT

G❀X ✮ ✁ ✍x ✙ f

✍x ❂ P❄

❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✰ RG❀X RT G❀X ✭P❄ ❤X ❀u✐✭✍x✮ ✰ P❤X ❀u✐✭✍x✮✮ ✙ f

✭✡ ✡ Id✮P❄

❤X ❀u✐✭✍x✮ ✙ f ✮ ✕min✭✡❥❤X ❀u✐❄✮❦P❄ ❤X ❀u✐✭✍x✮❦ ✙ ❦f ❦

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SLIDE 40

(A piece of the) proof of the upper bound

Solution of SDP ✦ Q Gram matrix ✦ Q0 (Q0 ❂ XX T, X ❂ ❬x1❥x2❥ ✁ ✁ ✁ ❥xn❪T) Q ❂ Q0 ✰ R ❂ Q0 ✰ XY T ✰ YX T ✰ R❄

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SLIDE 41

(A piece of the) proof of the upper bound

Solution of SDP ✦ Q Gram matrix ✦ Q0 (Q0 ❂ XX T, X ❂ ❬x1❥x2❥ ✁ ✁ ✁ ❥xn❪T) Q ❂ Q0 ✰ R ❂ Q0 ✰ XY T ✰ YX T ✰ R❄

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SLIDE 42

(A piece of the) proof of the upper bound

Solution of SDP ✦ Q Gram matrix ✦ Q0 (Q0 ❂ XX T, X ❂ ❬x1❥x2❥ ✁ ✁ ✁ ❥xn❪T) Q ❂ Q0 ✰ R ❂ Q0 ✰ XY T ✰ YX T

⑤ ④③ ⑥

Rigidity matrix

✰ R❄

⑤④③⑥

Stress matrix

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SLIDE 43

Steps of the proof

Lemma

For a stress matrix ✡ ✗ 0, ❦R❄❦✄ ✔ ✕max✭✡✮ ✕min✭✡❥❤u❀x✐❄✮❥E❥✁✿

Lemma

✕min✭✡❥❤u❀x✐❄✮ ✕ C1✭nr d✮2r 4❀ ✕max✭✡✮ ✔ C2✭nr d✮2✿

Lemma

❦XY T ✰ YX T❦1 ✔ C✭nr d✮5 n2 r 4 ✿

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Discussion

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SLIDE 45

Manifold learning

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SLIDE 46

Discussion: Manifold learning

Data: x1❀ ✿ ✿ ✿ xn ✷ Rm Assumption: ❢xi❣i✷❬n❪ are close to d-dimensional submanifold of Rm Objective: d-dimensional embedding z1❀ ✿ ✿ ✿ ❀ zn ✷ Rd that preserves local geometry.

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SLIDE 47

Where does the assumption come from?

You need some assumption! PCA assumes linear approximately linear manifold:

❤

x1

☞ ☞ ☞ ✁ ✁ ✁ ☞ ☞ ☞xn ✐

✙ UV ✄ ❀ U ✷ Rm✂d ❀ V ✷ Rn✂d ❀ xi ✙

d

❳

a❂1

va❀i ua Vision.

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SLIDE 48

Where does the assumption come from?

You need some assumption! PCA assumes linear approximately linear manifold:

❤

x1

☞ ☞ ☞ ✁ ✁ ✁ ☞ ☞ ☞xn ✐

✙ UV ✄ ❀ U ✷ Rm✂d ❀ V ✷ Rn✂d ❀ xi ✙

d

❳

a❂1

va❀i ua Vision.

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SLIDE 49

Vision 1

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SLIDE 50

Vision 2

Human brain effectively projects on low dimension (?).

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SLIDE 51

Vision 2

Human brain effectively projects on low dimension (?).

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SLIDE 52

Manifold learning

Manifold Learning Input : Data points x1❀ ✿ ✿ ✿ ❀ xn ✷ Rm Output : Low-dimensional coordinates z1❀ ✿ ✿ ✿ ❀ zn ✷ Rd, d ✜ m 1: Proximity graph G on ❢1❀ ✿ ✿ ✿ ❀ n❣; 2: For ✭i❀ j ✮ ✷ E set ❡ dij ❂ ❦xi xj ❦2; 3: Find positions zi ✷ Rd, such that ❦zi zj ❦2 ✙ ❡ dij

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SLIDE 53

Discussion : Manifold learning

(f) Data set (g) Proximity graph

⑦ dij ❂ ❦xi xj ❦Rm❀ dij ❂ d▼✭xi❀ xj ✮ ✁ ❂❄

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SLIDE 54

Discussion : Manifold learning

(h) Data set (i) Proximity graph

⑦ dij ❂ ❦xi xj ❦Rm❀ dij ❂ d▼✭xi❀ xj ✮ ✁ ❂❄

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SLIDE 55

Manifold learning: Reconstruction error

Lemma (M. Bernstain et. al., 2000)

Let r0 ❂ r0✭▼✮ be the radius curvature defined by 1 r0 ❂ max

✌❀t ❢❦⑧

✌✭t✮❦❣✿ Then ✭1 d2

ij

24r 2 ✮dij ✔ ⑦ dij ✔ dij ✿ ✁ ✴ r 4 r 2 d✭X ❀ ❫ X ✮ ✔ C ✭nr d✮5 r 2

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SLIDE 56

Manifold learning: Reconstruction error

Lemma (M. Bernstain et. al., 2000)

Let r0 ❂ r0✭▼✮ be the radius curvature defined by 1 r0 ❂ max

✌❀t ❢❦⑧

✌✭t✮❦❣✿ Then ✭1 d2

ij

24r 2 ✮dij ✔ ⑦ dij ✔ dij ✿ ✁ ✴ r 4 r 2 d✭X ❀ ❫ X ✮ ✔ C ✭nr d✮5 r 2

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SLIDE 57

References

R. Connelly, Generic global rigidity, Discrete & Comp. Geometry,

33:549-563,2005

S. J. Gortler, A. D. Healy, and D. P. Thurston, Characterizing

generic global rigidity, Amer. Journal of Math.,132:897-939,2010.

P. Biswas and Y. Ye, Semidefinite programming for ad hoc

wireless sensor network localization, IPSN 2004.

A. Singer, A remark on global positioning from local distances,

PNAS 2008.

A. Javanmard and Andrea Montanari, Localization from

incomplete noisy distance measurements, arXiv:1103.1417v3 Thanks!

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SLIDE 58

Maximum Variance Unfolding

[cf. Weinberger and Saul, 2006]

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SLIDE 59

Optimization formulation

maximize

n

❳

1✔i❀j ✔n

❦xi xj ❦2 subj✿to ❦xi xj ❦2 ❂ d2

ij

✽✭i❀ j ✮ ✷ E

n

❳

i❂1

xi ❂ 0 Nonconvex

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SLIDE 60

Optimization formulation (better notation)

minimize Tr✭Q✮ subj✿to ❤Mij ❀ Q✐ ❂ d2

ij

uTQu ❂ 0 Qij ❂ ❤xi❀ xj ✐ Mij ❂ eij eT

ij ❀

eij ❂ ✭0❀ ✿ ✿ ✿ ❀ 0❀ ✰1

⑤④③⑥

i

❀ 0❀ ✿ ✿ ✿ ❀ 0❀ 1

⑤④③⑥

j

❀ 0❀ ✿ ✿ ✿ ❀ 0✮

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