Fast and robust sparse recoveryNew Algorithms and Applications

The Chinese University of Hong Kong

The Institute of Network Coding

ShengCai

EricChan Minghua

ChenSidharth

JaggiMohammad Jahangoshahi

VenkateshSaligrama

Mayank BakshiINC, CUHK

? n

2

Fast and robust sparse recovery

m

m<n

k

Unknown x

MeasurementMeasurement output

Reconstruct x

A. Compressive sensing

4

?

k ≤ m<n

? n

m

k

A. Robust compressive sensing

y=A(x+z)+eApproximate sparsity

Measurement noise

5

?

z

e

TomographyComputerized Axial

(CAT scan)

B. Tomography

Estimate x given y and T

y = Tx

B. Network Tomography

Measurements y:• End-to-end packet delays

Transform T:• Network connectivity matrix (known a priori)

Infer x:• Link/node congestion

Hopefully “k-sparse”

Compressive sensing?

Challenge:• Matrix T “fixed”• Can only take “some”

types of measurements

9

n-dd

1 0q

1q

For Pr(error)< ε , Lower bound:

Noisy Combinatorial OMP:

What’s known…[CCJS11]

0

C. Robust group testing

A. Robust compressive sensing

y=A(x+z)+eApproximate sparsity

Measurement noise

11

?

z

e

Apps: 1. Compression

12

W(x+z)

BW(x+z) = A(x+z)

M.A. Davenport, M.F. Duarte, Y.C. Eldar, and G. Kutyniok, "Introduction to Compressed Sensing,"in Compressed Sensing: Theory and Applications, 2012

x+z

Apps: 2. Fast(er) Fourier Transform

13

H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse fourier transform. In Proceedings of the 44th symposium on Theory of Computing (STOC '12).

Apps: 3. One-pixel camera

http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif

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y=A(x+z)+e

15

y=A(x+z)+e

16

y=A(x+z)+e

17

y=A(x+z)+e

18

y=A(x+z)+e

(Information-theoretically) order-optimal19

(Information-theoretically) order-optimal

• Support Recovery

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SHO-FA:SHO(rt)-FA(st)

O(k) measurements,O(k) time

1. Graph-Matrix

n ck

d=3

24

A

1. Graph-Matrix

25

n ck

Ad=3

26

1. Graph-Matrix

2. (Most) x-expansion

≥2|S||S|27

3. “Many” leafs

≥2|S||S|L+L’≥2|S|

3|S|≥L+2L’

L≥|S|L+L’≤3|S|

L/(L+L’) ≥1/3L/(L+L’) ≥1/2

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4. Matrix

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Encoding – Recap.

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0

1

0

1

0

Decoding – Initialization

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Decoding – Leaf Check(2-Failed-ID)

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Decoding – Leaf Check (4-Failed-VER)

33

Decoding – Leaf Check(1-Passed)

34

Decoding – Step 4 (4-Passed/STOP)

35

Decoding – Recap.

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0

0

0

0

0

?

?

?0

0

0

1

0

Decoding – Recap.

28

0

1

0

1

0

Noise/approx. sparsity

39

Meas/phase error

40

Correlated phase meas.

41

Correlated phase meas.

42

Correlated phase meas.

43

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• Goal: Infer network characteristics (edge or node delay)• Difficulties:

– Edge-by-edge (or node-by node) monitoring too slow– Inaccessible nodes

Network Tomography

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• Goal: Infer network characteristics (edge or node delay)• Difficulties:

– Edge-by-edge (or node-by node) monitoring too slow– Inaccessible nodes

• Network Tomography:– with very few end-to-end measurements– quickly– for arbitrary network topology

Network Tomography

B. Network Tomography

Measurements y:• End-to-end packet delays

Transform T:• Network connectivity matrix

(known a priori)

Infer x:• Link/node congestion

Hopefully “k-sparse”

Compressive sensing?

Idea:• “Mimic” random matrix

Challenge:• Matrix T “fixed”• Can only take “some”

types of measurements

Our algorithm: FRANTIC• Fast Reference-based Algorithm for Network

Tomography vIa Compressive sensing

SHO-FA

49

n ck

Ad=3

50

T

1. Integer valued CS [BJCC12] “SHO-FA-INT”

2. Better mimicking of desired T

Node delay estimation

1v3v4v2v

Node delay estimation

4v2v3v

1v

4v2v1v3v

Node delay estimation

Edge delay estimation

1e 5e6e 3e4e

2e

Idea 1: Cancellation

, ,

Idea 2: “Loopy” measurements

•Fewer measurements•Arbitrary packet injection/

reception•Not just 0/1 matrices (SHO-FA)

,

C. GROTESQUE: Noisy GROup TESting (QUick and Efficient)

63

n-dd

1 0q

1q

For Pr(error)< ε , Lower bound:

Noisy Combinatorial OMP:

What’s known…[CCJS11]

0

Decoding complexity

# Tests

Lower bound

Lower bound

Adaptive

Non-Adaptive

2-Stage Adaptive

This work

O(poly(D)log(N)),O(D2log(N))

O(DN),O(Dlog(N))

[NPR12]

Decoding complexity

# Tests

This work

Hammer: GROTESQUE testing

Multiplicity

?

Localization

?

Noiseless:

Noisy:

Nail: “Good” Partioning

GROTESQUE

n itemsd defectives

Adaptive Group Testing

O(n/d)

Adaptive Group Testing

O(n/d)

GROTESQUEGROTESQUE

GROTESQUE

GROTESQUE

O(dlog(n)) time, tests, constant fraction recovered

Adaptive Group Testing

•Each stage constant fraction recovered•# tests, time decaying geometrically

Adaptive Group Testing

T=O(logD)

Non-Adaptive Group Testing

Constant fraction “good”

O(Dlog(D))

Non-Adaptive Group Testing

Iterative Decoding

2-Stage Adaptive Group Testing

=D2

D. Threshold Group Testing

l u # defective items in a group

Prob

abili

ty th

at

Out

put i

s po

sitiv

e

0

1

n itemsd defectives

Each test:

Goal: find all d defectives

Our result: tests suffice; Previous best algorithms:

Summary• Fast and Robust Sparse Recovery algorithms

• Compressive sensing: Order optimal complexity, # of measurements

• Network Tomography: Nearly optimal complexity, # of measurements

• Group Testing: Optimal complexity, nearly optimal # of tests- Threshold Group Testing: Nearly optimal # of tests

THANK YOU謝謝

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