Simultaneous Learning of Trees and Representations for Extreme Classification and Density Estimation Simultaneous Learning of Trees and Representations for Extreme Classification

70 100 100 100 100 h Discrete: p (n) 1 = 6 12 = 0.5 p (n) 1|1 = 3 3 = 1, p (n) 1|2 = 3 3 = 1, p (n) 1|3 = 0 3 = 0, p (n) 1|4 = 0 3 = 0 Continuous: p (n) 1 = 1 12 (σ(100) + σ(70) + . . .+ σ(−70) + σ(−100)) ≈ 0.5 p (n) 1|1 = 1 3 (σ(100) + σ(70) + σ(100)) ≈ 1 p (n) 1|2 = 1 3 (σ(100) + σ(70) + σ(100)) ≈ 1 p (n) 1|3 = 1 3 (σ(−100) + σ(−70) + σ(−100)) ≈ 0 p (n) 1|4 = 1 3 (σ(−100) + σ(−70) + σ(−100)) ≈ 0 Figure 3. The comparison of discrete and continuous definitions of probabilities p j and p (n) j|i on a simple example with K = 4 classes and binary tree (M = 2). n is an exemplary node, e.g. root. σ denotes sigmoid function. Color circles denote data points.