Lecture10 outilier l0_svdd

Lecture10 outilier l0_svdd

• 1.
  Lecture 10: Robustoutlier detection with L0-SVDD Stéphane Canu [email protected] Sao Paulo 2014 February 28, 2014
• 2.
  Roadmap 1 Robust outlierdetection with L0-SVDD L0 SVDD 4 iterations of Adaptive L0 SVDD
• 3.
  Recall SVDD    min R,c,ξ R +C n i=1 ξi with xi − c 2 ≤ R + ξi , i = 1, . . . , n and ξi ≥ 0, i = 1, . . . , n (1) Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 3 / 11
• 4.
  SVDD + outlier C=1/16 C =1/8 C =1/4 C = 1/2 ( ) Figure: Example of SVDD solutions with different C values, m = 0 (red) and m = 5 (magenta). The circled data points represent support vectors for both m. Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 4 / 11
• 5.
  The L0 norm ξ0 ≤ t    min c∈IRp ,R∈IR,ξ∈IRn R + C ξ 0 with xi − c 2 ≤ R+ξi ξi ≥ 0 i = 1, n Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 5 / 11
• 6.
  L0 relaxations p norm exponenetial piecewiselinear log    min c∈IRp ,R∈IR,ξ∈IRn R + C n i=1 log(γ + ξi ) with xi − c 2 ≤ R+ξi ξi ≥ 0 i = 1, n . Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 6 / 11
• 7.
  DC programing log(γ +t) = f (t) − g(t) with f (t) = t and g(t) = t − log(γ + t), both functions f and g being convex. The DC framework consists in minimizing iteratively (R plus a sum of) the following convex term: f (ξ) − g (ξ)ξ = ξ − 1 − 1 γ + ξold ξ = ξ γ + ξold , where ξold i denotes the solution at the previous iteration. Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 7 / 11
• 8.
  The DC ideaapplied to our L0 SVDD approximation consists in building a sequence of solutions of the following adaptive SVDD:    min c∈IRp ,R∈IR,ξ∈IRn R + C n i=1 wi ξi with xi − c 2 ≤ R+ξi ξi ≥ 0 i = 1, n with wi = 1 γ + ξold i . Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 8 / 11
• 9.
  Stationary conditions ofthe KKT give: c = n i=1 αi xi and n i=1 αi = 1 where the αi are the Lagrange multipliers associated with the inequality constraints xi − c 2 ≤ R+ξi . The dual of this problem is min α∈IRn α XX α − α diag(XX ) with n i=1 αi = 1 0 ≤ αi ≤ Cwi i = 1, n (2) Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 9 / 11
• 10.
  Algorithm 1 L0SVDD for the linear kernel Data: X, y, C , γ Result: R , c, ξ , α wi = 1; i = 1, n while not converged do (α, λ) ← solve_QP(X, C, w) % solve problem (2) c ← X α R ← λ + c c ξi ← max(0, xi − c 2 − R) i = 1, n wi ← 1/(γ + ξi ) i = 1, n end Stéphane Canu (INSA Rouen - LITIS) February 28, 2014 10 / 11
• 11.
  Bibliography Stéphane Canu (INSARouen - LITIS) February 28, 2014 11 / 11