• Multi-label classification
  • 1.Formal definitions
  • 2.Evaluation Metrics
  • 3.learning algotithms

Multi-label classification

1.Formal definitions

1. Learning framework

   multi-label indicators:

   • label cardinality
   $$LDcard(D)=\frac{1}{m}\sum_{i=1}^{m}|Y_i|$$

   • label density $LDen(D)=\frac{1}{|D|}LDcard(D)$

   • label diversity $LDiv(D)=|\{Y|\exists x:(x,Y)\in D\}|$

   • normalized label diversity $P LDiv(D) = \frac{1}{|D|} · LDiv(D)$

   Real value function f: $f：X\times Y\rightarrow \mathbb{R}$

    where f(x, y) can be regarded as the confidence of y ∈ Y being the proper label of x. Specifically, given a multi-label example (x, Y ), f(·, ·) should yield larger output on the relevant label $y ′ ∈ Y$ and smaller output on the irrelevant label $y^{''}\notin   Y $
   

   multi-label classifier h(·)： $h(x) = \{y | f(x, y) > t(x), y ∈ Y\}$

   where t : X → R acts as a thresholding function which dichotomizes the label space into relevant and irrelevant label sets

2. key challenge:label correlations

   • First-order strategy
   • Second-order strategy
   • High-order strategy

3. threshold calibration

   in order to decide the proper label set for unseen instance x (i.e. h(x)), the real-valued output f(x, y) on each label should be calibrated against the thresholding function output t(x)

   • constant function or inducing t(·) from the training examples
   • a linear model for t(·)

2.Evaluation Metrics

3.learning algotithms