Robust Network Compressive Sensing Lili Qiu UT Austin NSF Workshop Nov. 12, 2014.
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Transcript of Robust Network Compressive Sensing Lili Qiu UT Austin NSF Workshop Nov. 12, 2014.
Network Matrices and Applications
• Network matrices– Traffic matrix– Loss matrix– Delay matrix– Channel State Information (CSI) matrix– RSS matrix
2
1
3
2router
flow 1
flow 3
flow 2
link 2link 1
link 3
flow 1
flow 2
flow 3
time 1 time 2 …
• Applications need complete network matrices– Traffic engineering– Spectrum sensing– Channel estimation– Localization– Multi-access channel design– Network coding, wireless video coding– Anomaly detection– Data aggregation– …
Missing Values: Why Bother?4
subcarrier 1
subcarrier 2
subcarrier 3
time 1 time 2 …
Vacant
freq ,loc1
freq 2, loc1
freq 3, loc1
time 1 time 2 …
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The Problem
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, erroneous, and/or indirect measurements
Anomaly FutureMissing
x1,3
State of the Art
• Existing works exploit low-rank nature of network matrices
• Many factors contribute to network matrices– Anomalies, measurement errors, and noise– These factors may destroy low-rank
structure and spatio-temporal locality– Limit the effectiveness of existing works
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Network Matrices
Network Date Duration Size (flows/links x #timeslot)
3G traffic 11/2010 1 day 472 x 144
WiFi traffic 1/2013 1 day 50 x 118
Abilene traffic 4/2003 1 week 121 x 1008
GEANT traffic 4/2005 1 week 529 x 672
1 channel CSI 2/2009 15 min. 90 x 9000
Multi. Channel CSI
2/2014 15 min. 270 x 5000
Cister RSSI 11/2010 4 hours 16 x 10000
CU RSSI 8/2007 500 frames 895 x 500
Umich RSS 4/2006 30 min. 182 x 3127
UCSB Meshnet 4/2006 3 days 425 x 1527
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LENS Decomposition: Basic Formulation
9
= +
y1,30…
…
…y3,n
0 0 0000 00000 0
000 0 0
+
[Input] D:Original matrix
[Output] X:A low rank matrix (r «
m,n)
[Output] Y:A sparse anomaly
matrix
[Output] Z:A small noise
matrix
d1,3d1,2
…
…
…
d2,n
dm,n
d3,n
d1,4
d2,1 d2,2 d2,3
d3,1 d3,4
dm,2 dm,4
…
xm,r
xr,n
…
x1,1
x2,1
x3,1
xm,1
x3,r
x2,r
x1,r
x1,1
xr,1 xr,2
x1,2
xr,3
x1,3 x1,n… …
…
…
…
LENS Decomposition: Basic Formulation
• Formulate it as a convex opt. problem:
10
min:
subject to:
= + +d1,3
d1,2 d1,4
[Input] D:Original matrix
x1,2 x1,4
[Output] X:A low rank
matrix
0 0 y1,3 0
0 0 0 0
0 0 0 0
[Output] Y:A sparse anomaly matrix
[Output] Z:A small noise
matrix
α βσ
LENS Decomposition: Support Indirect Measurement
• The matrix of interest may not be directly observable (e.g., traffic matrices)– AX + BY + CZ + W = D
• A: routing matrix• B: an over-complete anomaly profile matrix• C: noise profile matrix
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,t,t,t xxy 321 1
3
2router
flow 1
flow 3
flow 2
link 2link 1
link 3
LENS Decomposition: Account for Domain Knowledge
• Domain Knowledge– Temporal stability– Spatial locality– Initial solution
12
min:
subject to:
Optimization Algorithm
• One of many challenges in optimization:– X and Y appear in multiple places in the objective
and constraints– Coupling makes optimization hard
• Reformulation for optimization by introducing auxiliary variables
•
13
min:
subject to:
Optimization Algorithm• Alternating Direction Method (ADM)
– For each iteration, alternate among the optimization of the augmented Lagrangian function by varying each one of X, Xk, Y, Y0, Z, W, M, Mk, N while fixing the other variables
– Improve efficiency through approximate SVD
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Setting Parameters
• • •
where (mX,nX) is the size of X, (mY,nY) is the size of Y, η(D) is the fraction of entriesneither missing or erroneous, θ is a control parameter that limitscontamination of dense measurement noise
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min:α βσ σ
Setting Parameters (Cont.)
• ϒ reflects the importance of domain knowledge– e.g. temporal-stability varies across traces
• Self-tuning algorithm– Drop additional entries in the matrix– Quantify the error of the entries that were
present in the matrix but dropped intentionally during the search
– Pick ϒ that gives lowest error
16
min:σ
γ
17
Algorithms Compared
Algorithm Description
Baseline Baseline estimate via rank-2 approximation
SVD-base SRSVD with baseline removal
SVD-base +KNN Apply KNN after SVD-base
SRMF [SIGCOMM09] Sparsity Regularized Matrix Factorization
SRMF+KNN [SIGCOMM09]
Hybrid of SRMF and KNN
LENS Robust network compressive sensing
Self Learned ϒ18
Best ϒ = 0 Best ϒ = 1 Best ϒ = 10
No single ϒ works for all traces.Self tuning allows us to automatically select the best ϒ.
Conclusion
• Main contributions– Important impact of anomalies in matrix
interpolation– Decompose a matrix into
• a low-rank matrix, • a sparse anomaly matrix, • a dense but small noise matrix
– An efficient optimization algorithm– A self-learning algorithm to automatically tune the
parameters• Future work
– Applying it to spectrum sensing, channel estimation, localization, etc.
21
Evaluation Methodology
• Metric– Normalized Mean Absolute Error for missing values
• Report the average of 10 random runs• Anomaly generation
– Inject anomalies to a varying fraction of entries with varying sizes
• Different dropping models
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