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Transcript of Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal...
![Page 1: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/1.jpg)
Low-rankLow-rank
By: Yanglet
Date: 2012/12/2
![Page 2: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/2.jpg)
Included Works.
Yin Zhang, Lili Qiu
― Spatio-Temporal Compressive Sensing and Internet Traffic Matrices,
SIGCOMM 2009.
― Exploiting Temporal Stability and Low-rank Structure for Localization in
Mobile Networks, MobiCom 2011.
Zhi Li
― Compressive Sensing Approach to Urban Traffic Sensing, ICDCS 2011.
Linghe Kong
― Environment Reconstruction in Sensor Networks with Massive Data Loss, INFOCOM 2013.
Hongjian Wang
― Compressive Sensing based Monitoring with Vehicular Networks,
INFOCOM 2013.
We do not discuss Dina’s work here.
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Efficient and Reliable Low-Power Backscatter Networks
SIGCOMM 2012
Jue Wang, Haitham Hassanieh, Dina Katabi, Piotr Indyk
Networks@MIT
Presented by: Yanglet
Date: 2012/10/12
![Page 4: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/4.jpg)
Faster GPS via the Sparse Fourier Transform
MobiCom 2012
Haitham Hassanieh, Fadel Adib, Dina Katabi, Piotr Indyk
Networks@MIT
Presented by: Yanglet
Date: 2012/10/29
![Page 5: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/5.jpg)
Outline
Low-rank and Sparsity
Yin Zhang’s SIGCOMM 2009 Paper
The Rest Papers
Rethinks
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Sparsity & Low-rank
“Sparsity”
e.g. , for vector X
“Low-rank”
The singular value vector is sparse!!6
1
0
X
X with
NR
K K N
,
1
1
A
A=VRU
=V
0
0
V: ; ;
M NC
U
M M R M N U N N
![Page 7: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/7.jpg)
Compressive Sensing
Compressive Sensing Approach― Y is the random linear encoding results of K-sparse vector X
7
~
1
1
X arg min X
. . XM M N Ns t Y A
Results
We need only to recovery Xlog( / )M CK N K N
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Spatio-Temporal Compressive Sensing and Internet Traffic Matrices
SIGCOMM 2009
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10
Q: How to fill in missing values in a matrix?― Traffic matrix
― Delay matrix
― Social proximity matrix
Matrix Completion
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11
Internet Traffic Matrices
Traffic Matrix (TM)
― Gives traffic volumes between origins and destinations
Essential for many networking tasks
― what-if analysis, traffic engineering, anomaly detection
• Lots of prior research– Measurement, e.g.
[FGLR+01, VE03]
– Inference, e.g. [MTSB+02, ZRDG03, ZRLD03, ZRLD05, SLTP+06, ZGWX06]
– Anomaly detection, e.g.
[LCD04, ZGRG05, RSRD07]
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12
Missing Values: Why Bother?
Missing values are common in TM
measurements― Direct measurement is infeasible/expensive
― Measurement and data collection are unreliable
― Anomalies/outliers hide non-anomaly-related traffic
― Future traffic has not yet appeared
The need for missing value interpolation― Many networking tasks are sensitive to missing values
― Need non-anomaly-related traffic for diagnosis
― Need predicted TMs in what-if analysis, traffic engineering,
capacity planning, etc.
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The Problem
1
3
2router
route 1
route 3
route 2 link 2
link 1
link 3
6,3
6,2
6,1
5,3
5,2
5,1
4,13,32,3
4,13,22,2
4,13,12,1
1,3
1,2
1,1
x
x
x
x
x
x
xxx
xxx
xxx
x
x
x
X
xr,t : traffic volume on route r at time
t
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14
,t,t,t xxy 321 indirect: only measure at links
The Problem
1
3
2router
route 1
route 3
route 2 link 2
link 1
link 3
6,3
6,2
6,1
5,3
5,2
5,1
4,13,32,3
4,13,22,2
4,13,12,1
1,3
1,2
1,1
x
x
x
x
x
x
xxx
xxx
xxx
x
x
x
X
Interpolation: fill in missing values from incomplete and/or indirect measurements
futureanomalymissing
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15
The Problem
E.g., link loads only: AX=Y• A: routing matrix;
Y: link load matrix
E.g., direct measurements only:
M.*X=M.*D• M(r,t)=1 X(r,t) exists;
D: direct measurements
1
3
2router
route 1
route 3
route 2 link 2
link 1
link 3
A(X)=BChallenge: In real networks, the problem is
massively underconstrained!
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16
Spatio-Temporal Compressive Sensing
Idea 1: Exploit low-rank nature of TMs― Observation: TMs are low-rank [LPCD+04, LCD04]:
Xnxm Lnxr * RmxrT (r
« n,m)
Idea 2: Exploit spatio-temporal properties― Observation: TM rows or columns close to each other (in
some sense) are often close in value
Idea 3: Exploit local structures in TMs― Observation: TMs have both global & local structures
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Spatio-Temporal Compressive Sensing
Idea 1: Exploit low-rank nature of TMs― Technique: Compressive Sensing
Idea 2: Exploit spatio-temporal properties― Technique: Sparsity Regularized Matrix Factorization (SRMF)
Idea 3: Exploit local structures in TMs― Technique: Combine global and local interpolation
![Page 18: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/18.jpg)
18
Compressive Sensing
Basic approach: find X=LRT s.t. A(LRT)=B― (m+n)*r unknowns (instead of m*n)
Challenges― A(LRT)=B may have many solutions which to pick?
― A(LRT)=B may have zero solution, e.g. when X is approximately
low-rank, or there is noise
Solution: Sparsity Regularized SVD (SRSVD)
― minimize |A(LRT) – B|2 // fitting error
+ (|L|2+|R|2) // regularization
― Similar to SVD but can handle missing values and indirect
measurements
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19
Sparsity Regularized Matrix Factorization
Motivation
― The theoretical conditions for compressive sensing
to perform well may not hold on real-world TMs
Sparsity Regularized Matrix Factorization― minimize |A(LRT) – B|2 // fitting error
+ (|L|2+|R|2) // regularization
+ |S(LRT)|2 // spatial constraint
+ |(LRT)TT|2 // temporal
constraint
― S and T capture spatio-temporal properties of TMs
― Can be solved efficiently via alternating least-
squares
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20
Alternating Least Squares
Goal: minimize |A(LRT) – B|2 + (|L|2+|R|2)
Step 1: fix L and optimize R
― A standard least-squares problem
Step 2: fix R and optimize L
― A standard least-squares problem
Step 3: goto Step 1 unless MaxIter is reached
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21
Spatio-Temporal Constraints
Temporal constraint matrix T
― Captures temporal smoothness
― Simple choices suffice, e.g.:
Spatial constraint matrix S
― Captures which rows of X are close to each other
― Challenge: TM rows are ordered arbitrarily
― Our solution: use a initial estimate of X to
approximate similarity between rows of X
100
110
011
T
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22
Combining Global and Local Methods
Local correlation among individual elements
may be stronger than among TM
rows/columns
― S and T in SRMF are chosen to capture global
correlation among entire TM rows or columns
SRMF+KNN: combine SRMF with local
interpolation
― Switch to K-Nearest-Neighbors if a missing
element is temporally close to observed
elements
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23
Generalizing Previous Methods
Tomo-SRMF: find a solution that is close to LRT yet satisfies A(X)=B
solution subspace A(X)=B
Tomo-SRMF solution
SRMF solution: LRT
Tomo-SRMF generalizes the tomo-gravity method for inferring TM from link loads
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24
Applications
Inference (a.k.a. tomography)
― Can combine both direct and indirect measurements for
TM inference
Prediction
― What-if analysis, traffic engineering, capacity planning
all require predicted traffic matrix
Anomaly Detection
― Project TM onto a low-dimensional, spatially &
temporally smooth subspace (LRT) normal trafficSpatio-temporal compressive sensing provides a
unified approach for many applications
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25
Evaluation Methodology
Data sets
Metrics― Normalized Mean Absolute Error for missing values
― Other metrics yield qualitatively similar results.
0),(:,
0),(:,est
|),(|
|),(),(|
jiMji
jiMji
jiX
jiXjiX
NMAE
Network Date Duration
Resolution
Size
Abilene 03/2003
1 week 10 min. 121x1008
Commercial ISP
10/2006
3 weeks
1 hour 400x504
GEANT 04/2005
1 week 15 min. 529x672
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26
Algorithms Compared
Algorithm Description
Baseline Baseline estimate via rank-2 approximation
SRSVD Sparsity Regularized SVD
SRSVD-base SRSVD with baseline removal
NMF Nonnegative Matrix Factorization
KNN K-Nearest-Neighbors
SRSVD-base+KNN
Hybrid of SRSVD-base and KNN
SRMF Sparsity Regularized Matrix Factorization
SRMF+KNN Hybrid of SRMF and KNN
Tomo-SRMF Generalization of tomo-gravity
![Page 27: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/27.jpg)
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Interpolation: Random Loss
Our method isalways the best
Only ~20% error even with 98% loss
Dataset: Abilene
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Interpolation: Structured Loss
Our method is always the best; sometimes dramatically better
Only ~20% error even with 98% loss
Dataset: Abilene
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Tomography Performance
Dataset: Commercial ISP
Can halve the error of Tomo-Gravity
by measuring only 2% elements!
![Page 30: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/30.jpg)
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Other Results
Prediction
― Taking periodicity into account helps prediction
― Our method consistently outperforms other methods• Smooth, low-rank approximation improves prediction
Anomaly detection
― Generalizes many previous methods• E.g., PCA, anomography, time domain methods
― Yet offers more• Can handle missing values, indirect measurements
• Less sensitive to contamination in normal subspace
• No need to specify exact # of dimensions for normal subspace
― Preliminary results also show better accuracy
![Page 31: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/31.jpg)
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Conclusion
Spatio-temporal compressive sensing― Advances ideas from compressive sensing― Uses the first truly spatio-temporal model of TMs― Exploits both global and local structures of TMs
General and flexible― Generalizes previous methods yet can do much
more― Provides a unified approach to TM estimation,
prediction, anomaly detection, etc.
Highly effective― Accurate: works even with 90+% values missing― Robust: copes easily with highly structured loss― Fast: a few seconds on TMs we tested
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Lots of Future Work
Other types of network matrices― Delay matrices, social proximity matrices
Better choices of S and T― Tailor to both applications and datasets
Extension to higher dimensions― E.g., 3D: source, destination, time
Theoretical foundation― When and why our approach works so well?
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To be con’t!
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Exploiting Temporal Stability and Low-rank Structure for Localization in Mobile Networks,
MobiCom 2011
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To be con’t!
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Compressive Sensing Approach to Urban Traffic Sensing, ICDCS 2011
Zhi Li
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To be con’t!
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Compressive Sensing based Monitoring with Vehicular Networks, INFOCOM 2013.
Hongjian Wang
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To be con’t!
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Environment Reconstruction in Sensor Networks with Massive Data Loss, INFOCOM 2013.
Linghe Kong
![Page 49: Low-rank By: Yanglet Date: 2012/12/2. Included Works. Yin Zhang, Lili Qiu ―Spatio-Temporal Compressive Sensing and Internet Traffic Matrices, SIGCOMM.](https://reader036.fdocuments.net/reader036/viewer/2022062719/56649ed35503460f94be3385/html5/thumbnails/49.jpg)
Thank you!