1. Outlier Processing via L1-Principal Subspaces
Shubham Chamadia and Dimitris A. Pados
Department of Electrical Engineering,
The State University of New York at Buffalo, NY 14260
{shubhamc,

2. Outline Outlier detection classification Limitations of L1-PCA
The proposed outlier processing scheme
Simulation results
Conclusion

3. What is an outlier?
[Hawkins, 1980] an observation that “deviates so much from other
observations as to arouse suspicions that it was generated by
a different mechanism”.
[Jarrell, 1994] a point that is far outside the norm for a variable
or population.
[Moore and McCabe, 1999] an observation that lies
outside the overall pattern of distribution.

4. Outliers detection classification
Supervised:
Semi-supervised:
Unsupervised:
Training data with labeled normal and abnormal classes.
Classification is highly unbalanced, labelling often done manually.
Availability of labeled samples from only one class.
Declare test instance not belonging to one class as belonging to other.
Most preferred, makes no assumption about availability of labeled training samples.
Assumption: normal instances are more frequent than abnormal.
we focus on unsupervised technique

5. Unsupervised detection
Principal Component Analysis (PCA) overview X2
Most valuable tool to reduce dimensionality of data.
Finds new directions over which:
Preserves most of samples’ information.
Maximizes the data variance. X1
Problem formulation
Given a real-valued data matrix
Maximizes the norm of projection of over

6. Unsupervised detectionX2
Genesis of -PCA
Outlier X1
Sensitive to extreme outliers.
Nearest preferred solution
often referred as norm PCA (robust to outliers).
Q. To what extent L1 –PCA is robust?

7. Motivation to outlier processing
Setup
Generate a 2D-data matrix
Add 4 outlier points
Limitation of and to some extent limitation of
Urgent need for outlier removal in conjunction with robust

8. Proposed outlier processing via L1-PCs
Step 1-4:
For , obtain -principal components:
Ways to calculate
If sample support is small, use optimal -PCA algorithm[1]
For medium to high sample, use iterative suboptimal algorithm[2]

9. Proposed outlier processing via L1-PCs
Step 2-4:
Evaluate reliability weight corresponding to each sample of X2
: often known as rank -reconstruction error of each sample
Outlier
: relative degree (normalized) of outliers
Ideally, corrupted sample lie far away from direction,
high reconstruction error high weighted-value Þ X1

10. Proposed outlier processing via L1-PCs
Step 3-4:
Outlier detection and removal
Prior knowledge of number of corrupted sample (say p)
Discard p-highest weighted samples.
Apriori knowledge often non-practical.
No prior knowledge of corruption
Implement (K=2)-means clustering over scalar reliability weight vector
Extract sample index corresponding to higher mean cluster (potential outlier cluster)
Discard samples contained in index set

11. Proposed outlier processing via L1-PCs
Step 4-4:
Recalculate -principal components over outlier processed data
Expected reveals a deeper insight of given data matrix than

12. Proposed outlier processing via L1-PCs
Step 1-4
Step 2-4
Scheme denoted by L1 + L1
Step 3-4
Step 4-4

13. Simulation result 1-3 Experiment 1: Data dimensionality reduction
Setup
Generate data matrix , i. i. d. Gaussian distribution
Corrupt certain percentage of samples by outlier vector , i. i. d.
Obtain P=2 principal components
Metric: Average representation error (ARE)
is the PC over corrupted data

14. Simulation result 2-3
Experiment 2: Direction-of-Arrival (DoA) estimation
Setup
Uniform linear antenna array D = 7 elements, recording N = 30 complex observation . A1 A7 A2 x
: received signal amplitude,
: Bernoulli equiprobable bits
: additive white complex Gaussian noise vector
: array response vector , with signal of interest at

15. Simulation result 2-3
Experiment 2: Direction-of-Arrival (DoA) estimation
Setup cont.
Adding jammer corruption . A1 A7 A2 x
Number of corrupted samples = 3 (out of 30)
Number of jammers (random location) corrupting = 3
Jammer SNR = 10 dB

16. Simulation result 2-3
Experiment 2: Direction-of-Arrival (DoA) estimation
Setup cont.
MUSIC-type DoA spectrum function:
: principal component over corrupted data
Jammer angle
signal angle

17. Simulation result 2-3
Experiment 2: Direction-of-Arrival (DoA) estimation
Setup cont.
MUSIC-type DoA spectrum function:
: principal component over corrupted data
Jammer angle
signal angle
Root mean square error (RMSE)
: estimated angle at realization

18. Simulation result 3-3 Experiment 3: Robust image fusion Setup
Original image
N = 10 grayscale image (256*256)
Adding independent noise
P10
Add AWGN, 𝜎 2 =50 P2
40% of 8 (out of 10) images with salt and pepper P1 P0
An extreme outlier
Append with a baboon image

19. Simulation result 3-3 Experiment 3: Robust image fusion Processing . .
Restoration block (32*32) P2 P1 P0 x x x
Total restoration blocks 256 x x 32 =64 x x . x . . .
Proposed outlier scheme over x x x

20. Simulation result 3-3 Experiment 3: Robust image fusion original noisy
Salt-pepper
Baboon

21. Outlier processing via L1-PCs
Conclusion:
With advent of big-data, requires robust outlier detection technique.
Conventional subspaces have limitations to extreme outliers.
Low complexity clustering over scalar reliability weights.
Simulation exhibits the demand of such a robust outlier removal scheme.