fast and robust sparse recovery new algorithms and applications the chinese university of hong kong...
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![Page 1: Fast and robust sparse recovery New Algorithms and Applications The Chinese University of Hong Kong The Institute of Network Coding Sheng Cai Eric Chan](https://reader035.vdocuments.mx/reader035/viewer/2022062519/56649ebd5503460f94bc6335/html5/thumbnails/1.jpg)
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
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? n
2
Fast and robust sparse recovery
m
m<n
k
Unknown x
MeasurementMeasurement output
Reconstruct x
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A. Compressive sensing
4
?
k ≤ m<n
? n
m
k
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A. Robust compressive sensing
y=A(x+z)+eApproximate sparsity
Measurement noise
5
?
z
e
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TomographyComputerized Axial
(CAT scan)
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B. Tomography
Estimate x given y and T
y = Tx
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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
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9
n-dd
1 0q
1q
For Pr(error)< ε , Lower bound:
Noisy Combinatorial OMP:
What’s known…[CCJS11]
0
C. Robust group testing
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A. Robust compressive sensing
y=A(x+z)+eApproximate sparsity
Measurement noise
11
?
z
e
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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
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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).
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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
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y=A(x+z)+e
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y=A(x+z)+e
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y=A(x+z)+e
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y=A(x+z)+e
(Information-theoretically) order-optimal19
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(Information-theoretically) order-optimal
• Support Recovery
20
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SHO-FA:SHO(rt)-FA(st)
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O(k) measurements,O(k) time
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1. Graph-Matrix
n ck
d=3
24
A
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1. Graph-Matrix
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n ck
Ad=3
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1. Graph-Matrix
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2. (Most) x-expansion
≥2|S||S|27
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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
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Decoding – Initialization
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Decoding – Leaf Check(2-Failed-ID)
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Decoding – Leaf Check (4-Failed-VER)
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Decoding – Leaf Check(1-Passed)
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Decoding – Step 4 (4-Passed/STOP)
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Decoding – Recap.
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0
0
0
0
0
?
?
?0
0
0
1
0
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Decoding – Recap.
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0
1
0
1
0
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Noise/approx. sparsity
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Meas/phase error
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Correlated phase meas.
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Correlated phase meas.
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Correlated phase meas.
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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
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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
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SHO-FA
49
n ck
Ad=3
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50
T
1. Integer valued CS [BJCC12] “SHO-FA-INT”
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2. Better mimicking of desired T
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Node delay estimation
1v3v4v2v
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Node delay estimation
4v2v3v
1v
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4v2v1v3v
Node delay estimation
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Edge delay estimation
1e 5e6e 3e4e
2e
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Idea 1: Cancellation
, ,
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Idea 2: “Loopy” measurements
•Fewer measurements•Arbitrary packet injection/
reception•Not just 0/1 matrices (SHO-FA)
,
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C. GROTESQUE: Noisy GROup TESting (QUick and Efficient)
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63
n-dd
1 0q
1q
For Pr(error)< ε , Lower bound:
Noisy Combinatorial OMP:
What’s known…[CCJS11]
0
![Page 56: Fast and robust sparse recovery New Algorithms and Applications The Chinese University of Hong Kong The Institute of Network Coding Sheng Cai Eric Chan](https://reader035.vdocuments.mx/reader035/viewer/2022062519/56649ebd5503460f94bc6335/html5/thumbnails/56.jpg)
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]
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Decoding complexity
# Tests
This work
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Hammer: GROTESQUE testing
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Multiplicity
?
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Localization
?
Noiseless:
Noisy:
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Nail: “Good” Partioning
GROTESQUE
n itemsd defectives
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Adaptive Group Testing
O(n/d)
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Adaptive Group Testing
O(n/d)
GROTESQUEGROTESQUE
GROTESQUE
GROTESQUE
O(dlog(n)) time, tests, constant fraction recovered
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Adaptive Group Testing
•Each stage constant fraction recovered•# tests, time decaying geometrically
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Adaptive Group Testing
T=O(logD)
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Non-Adaptive Group Testing
Constant fraction “good”
O(Dlog(D))
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Non-Adaptive Group Testing
Iterative Decoding
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2-Stage Adaptive Group Testing
=D2
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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:
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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
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THANK YOU謝謝
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