1. Approximating Points by A Piecewise Linear Function: II
Approximating Points by A Piecewise Linear Function: II. Dealing with Outliers
Danny Z. Chen and Haitao Wang
Computer Science and Engineering
University of Notre Dame
Indiana, USA

2. Problem Definitions Input: A point set P in 2-D
Output: An approximation function f
Error: Vertical distance
e(P,f)=max{error of each point}
error

3. Two Problem Versions min-#: min-ε: Given: An error toleranceε≥ 0
Goal: f of minimized size, with e(p,f) ≤ε
min-ε:
Given: k>0
Goal: f of minimized error e(p,f), with size ≤k

4. Problem Variations
Step function (SF)
error

5. Problem Variations (cont.)
Piecewise-linear function (PF)
error

6. Problem Variations (cont.)
Weighted versions for both SF and PF
Every point has a weight ui
Error of each point: ui×vertical distance
WSF and WPF
there is a weight ui

7. Problem Variations (cont.)
Violation versions:
Allow a given number g of violation points
e(P,f)=max{error of every non-violation point}
error
violation

8. Problem Variations (cont.)
Violation versions of SF, WSF
VSF and VWSF
error
violation

9. Problem Variations (cont.)
Violation versions of PF, WPF
VPF and VWPF
error
violation

10. Problem Summary
VSF, VPF,
Weighted version: VWSF, VWPF
min-# and min-ε

11. Motivations Applications where some noisy data must be removed
Find the outliers optimally

12. Previous Work min-# min-ε VSF O(ng2) [Fournier, Vigneron,08]
O(ng2logn) [Fournier, Vigneron,08]
VWSF
VPF
VWPF

13. Our Results min-# min-ε VSF O(ng2logn) [Fournier,Vigneron,08]
O(ng3k log(log*n))
VWSF
O(ng2)
O(n2+ng2logn)
VPF
O(ng4log2n)
O(nkg2(nlogg+g3nδ))
O(nglogn(nlogg+g3nδ))
VWPF

14. Technical Contributions
One min-# framework
One min-εframework
A min-εalgorithm for VWSF
A min-εalgorithm for VWPF
Data structures which are of independent interest

15. min-# Framework Problem: Dynamic programming Givenε
Goal: A function of minimized size
Constraint: The function error is ≤ε
Dynamic programming
Sub-problems N(i,t): The minimum number of segments for Pin with t violations, for any 1≤i≤n, 0≤t≤g
Goal: N(1,g)
N(i,t)=1+min{N(riq+1,t-q)} for any 0≤q≤t

16. min-# Framework (cont.)
riq: The rightmost point such that all points from i to riq can be approximated by one segment with q violations and the error is ≤ε
q violations
point riq pi

17. min-# Framework (cont.)
N(i,t)=1+min{N(riq+1,t-q)} for any 0≤q≤t
Use one segment with q violations to approximate points from i to riq and use N(riq+1,t-q) segments with t-q violations to approximate points from riq+1 to n
N(i,t)
riq
riq+1 i n
N(riq+1,t-q)

18. min-# Framework (cont.)
Time of the framework: O(ng2+T) (O(T) is for computing all ng riq’s)
Major components: Compute riq for all 1≤i≤n, 0≤q≤g

19. min-# Framework (cont.)
A scheme for computing all riq’s
Key: A fully dynamic data structure to maintain a point set for
Point insertion: O(I) time,
Point deletion: O(D) time,
Feasibility test: Whether the current set can be approximated by one segment with q violations, O(F) time
Time of the scheme: O(ng(I+D+F+g))

20. VSF and VWSF Data structure: Four arrays, and the range-minima
Time: O(g) per operation (insertion, deletion, feasibility test)
Computing all riq’s: O(ng2) time
min-# algorithm: O(ng2) time

21. VPF and VWPF
Data structure: The fully dynamic convex hull data structure [Overmars,81] (not the one in [Brodal, 02])
Insertion and deletion: O(log2n)
Feasibility test: Algorithm for 2D general LP with q violations, O(q3log2n)
Computing all riq’s: O(ng4log2n) time

22. min-εFramework Problem: Dynamic programming Given k
Goal: A function of minimized error
Constraint: The function size is k
Dynamic programming
Sub-problems E(j,l,t): The minimum error for P1j using l segments and t violations, for any 1≤i≤n, 0≤l≤k, 0≤t≤g
Goal: E(n,k,g)
E(j,l,t)=min{max{E(i-1,l-1,t-q), wijq}} for 1<i≤j, 0≤q≤t

23. min-εFramework (Cont.)
wijq: The minimum error to approximate Pij with q violations by one segment
q violations
wijq pi pj

24. min-εFramework (Cont.)
E(j,l,t)=min{max{E(i-1,l-1,t-q), wijq}} for 1<i≤j, 0≤q≤t
Use one segment with q violations to approximate Pij and use l-1 segment with t-q violations to approximate P1,i-1
E(j,l,t)
i-1 i j 1
q outliers
E(i-1,l-1,t-q)

25. min-εFramework (Cont.)
A straightforward way: O(n2g2k W) where O(W) is the time for computing each wijq
Not efficient

26. Improvement Transform 3-D to 2-D A totally monotone 2-D matrix
Use row-minima algorithm [Aggarwal etc, 87]
Time: O(ng2kW) (instead of O(n2g2kW))

27. Major Components
Compute wijq: The minimum error to approximate Pij with q violations
q violations
wijq pi pj

28. VSF and q-range-minima
Given: An array A[1…n],
Query (i,j): The q smallest elements in A[i…j] in sorted order
An extension of range-minima

29. VSF and q-range-minima (cont)
A simple solution: (n,qlogq), by range-minima
Our solution: (nlog2q, qlog(log*n))
VSF solution: O(nkg3log(log*n))

30. VPF and VWPF To compute wijq: Total time: O(ngk(nlogq+q3nδ))
3-D feasible LP with q violations
A O(nlogq+q3nδ) time solution in [Matousek,94]
Total time: O(ngk(nlogq+q3nδ))

31. min-εAlgorithm for VWSF
Observation:ε* is determined by two points in P
Idea: Consider all pairs of points, by using the min-# algorithm as a decision procedure
Time: O(n2+ng2logn)

32. min-εAlgorithm for VWPF
Observation:ε* is equal to some Wijq
A naïve solution: Compute all Wijq’s and then findε*
Not efficient: There are O(n2g) Wijq’s
Improved solution: Binary search on sorted arrays
Only O(nglog n) Wijq’s need to be computed
min-# algorithm as a decision procedure

33. Thank You
Questions?