Online Manifold Regularization by Dual Ascending Procedure

Algorithm 6

A template online manifold regularization algorithm based on two-step update.

INPUT: two positive scalars: 𝑐₁ and 𝑐₂; edge weights 𝑤_𝑖𝑗.

INITIALIZE: a coefficient vector 𝜶₀ and its associated decision boundary vector 𝝎₀.

PROCESS: For 𝑡=1,2,…,𝑇

      Receive a training point 𝐱_𝑡,

      Choose a new coefficient vector 𝜶_𝑡−(1/2) that satisfies 𝐷𝜶_𝑡−(1/2)≥𝐷𝜶_𝑡−1,

      Return a new associated boundary vector 𝝎_𝑡−(1/2) in (18).

      Receive 𝑦_𝑡,𝜎_𝑡,

      If 𝜎_𝑡=0, 𝜶_𝑡=𝜶_𝑡−(1/2), 𝝎_𝑡=𝝎_𝑡−(1/2), predict 𝑦_𝑡=sign(𝝎_𝑡,𝐱_𝑡);

      Else if 𝜎_𝑡=1, choose a new coefficient vector 𝜶_𝑡 that satisfies 𝐷𝜶_𝑡≥𝐷𝜶_𝑡−(1/2), return a new associated

      boundary vector 𝝎_𝑡 in (18).