搜索中的PLS和RMLS match

PLS:partial least square

• two space $X \in R^m$ and $Y \in R^n$
• training data $\{(x_i, y_i, r_i)\}_{i=1}^N, r \in \{+1,-1\}$ or $r_i \in R $

• model

  • dot product as similarity:$f(x, y)=<L_X^Tx, L_Y^Ty>=x^TL_XL_Yy$
  • $L_X, L_Y$ are two linear (and orthonormal) transformations

• objective function

  $$ argmax_{L_X, L_Y}\sum _{r_i=+1} x_i^TL_XL_Y^Ty – \sum _{r_i = -1} x_i^TL_XL_Y^Ty $$

  s.t. $L_X^TL_X=I_{K*K}, L_Y^TL_Y=I_{K*K}$

RMLS:Regularized mapping to latent space

• two space $X \in R^m$ and $Y \in R^n$
• training data $\{(x_i, y_i, r_i)\}_{i=1}^N, r \in \{+1,-1\}$ or $r_i \in R $

• model

  • dot product as similarity:$f(x, y)=<L_X^Tx, L_Y^Ty>=x^TL_XL_Yy$
  • $L_X, L_Y$ are two linear transformations with $l_1$ and $l_2$ regularizations(sparse transformations)

• objective function

  $$ argmax_{L_X, L_Y}\sum _{r_i=+1} x_i^TL_XL_Y^Ty – \sum _{r_i = -1} x_i^TL_XL_Y^Ty $$

  s.t. $|L_X| \leq |\lambda_X|,|L_Y| \leq |\lambda_Y|,||L_X|| \leq v_X,||L_Y|| \leq v_X$