Git patch 1 (#6)

* doc: Update the mathematical formula of MF in API

- correct the dimension and formula of MF constraint
- modified a display error in plq_Ridge_Regressor.predict()

* doc: fixed a notation error in MF example page

Author: DataboyUsen

doc/source/examples/MF.ipynb

@@ -25,7 +25,7 @@
     "      \\pmb{\\beta} \\in \\mathbb{R}^m\n",
     "  }} \n",
     "  \\left[\n",
-    "      \\sum_{(u,i)\\in \\Omega} C \\cdot \\text{PLQ}(r_{ui}, \\ \\mathbf{p}_u^T \\mathbf{q}_i + \\alpha_u + \\beta_i) \n",
+    "      \\sum_{(u,i)\\in \\Omega} C \\cdot \\phi(r_{ui}, \\ \\mathbf{p}_u^T \\mathbf{q}_i + \\alpha_u + \\beta_i) \n",
     "  \\right]  \n",
     "  + \n",
     "  \\left[ \n",
@@ -36,9 +36,9 @@
     "\n",
     "$$\n",
     "  \\ \\text{ s.t. } \\ \n",
-    "  \\mathbf{A} \\begin{pmatrix} \\alpha_u \\\\ \\mathbf{p}_u \\end{pmatrix} + \\mathbf{b} \\geq \\mathbf{0},\\ \\forall u \\in [n]\n",
+    "  \\mathbf{A} \\begin{pmatrix} \\alpha_u \\\\ \\mathbf{p}_u \\end{pmatrix} + \\mathbf{b} \\geq \\mathbf{0},\\ u = 1,\\dots,n\n",
     "  \\quad \\text{and} \\quad\n",
-    "  \\mathbf{A} \\begin{pmatrix} \\beta_i \\\\ \\mathbf{q}_i \\end{pmatrix} + \\mathbf{b} \\geq \\mathbf{0},\\ \\forall i \\in [m]\n",
+    "  \\mathbf{A} \\begin{pmatrix} \\beta_i \\\\ \\mathbf{q}_i \\end{pmatrix} + \\mathbf{b} \\geq \\mathbf{0},\\ i = 1,\\dots,m\n",
     "$$\n",
     "\n",
     "where\n",

doc/source/tutorials/ReHLine_MF.rst

@@ -26,9 +26,9 @@ Considering a User-Item-Rating triplet dataset :math:`(u, i, r_{ui})` derived fr
 
 .. math::
         \ \text{ s.t. } \ 
-        \mathbf{A} \begin{pmatrix} \alpha_u \\ \mathbf{p}_u \end{pmatrix} + \mathbf{b} \geq \mathbf{0},\ \forall u \in [n]
+        \mathbf{A} \begin{pmatrix} \alpha_u \\ \mathbf{p}_u \end{pmatrix} + \mathbf{b} \geq \mathbf{0},\ u = 1,\dots,n
         \quad \text{and} \quad
-        \mathbf{A} \begin{pmatrix} \beta_i \\ \mathbf{q}_i \end{pmatrix} + \mathbf{b} \geq \mathbf{0},\ \forall i \in [m]
+        \mathbf{A} \begin{pmatrix} \beta_i \\ \mathbf{q}_i \end{pmatrix} + \mathbf{b} \geq \mathbf{0},\ i = 1,\dots,m
 
 where
 

rehline/_mf_class.py

@@ -16,9 +16,9 @@ class plqMF_Ridge(_BaseReHLine, BaseEstimator):
 
     .. math::
         \min_{\substack{
-            \mathbf{P} \in \mathbb{R}^{n \times r}\ 
+            \mathbf{P} \in \mathbb{R}^{n \times k}\ 
             \pmb{\alpha} \in \mathbb{R}^n \\
-            \mathbf{Q} \in \mathbb{R}^{m \times r}\ 
+            \mathbf{Q} \in \mathbb{R}^{m \times k}\ 
             \pmb{\beta} \in \mathbb{R}^m
         }} 
         \left[
@@ -32,15 +32,9 @@ class plqMF_Ridge(_BaseReHLine, BaseEstimator):
 
     .. math::
         \ \text{ s.t. } \ 
-        \mathbf{A} \begin{bmatrix}
-                        \pmb{\alpha} & \mathbf{P}
-                    \end{bmatrix}^T + 
-                    \mathbf{b}\mathbf{1}_{n}^T \geq \mathbf{0}
-        \ \text{ and } \ 
-        \mathbf{A} \begin{bmatrix}
-                        \pmb{\beta} & \mathbf{Q}
-                    \end{bmatrix}^T + 
-                    \mathbf{b}\mathbf{1}_{m}^T \geq \mathbf{0}
+        \mathbf{A} \begin{pmatrix} \alpha_u \\ \mathbf{p}_u \end{pmatrix} + \mathbf{b} \geq \mathbf{0},\ u = 1,\dots,n
+        \quad \text{and} \quad
+        \mathbf{A} \begin{pmatrix} \beta_i \\ \mathbf{q}_i \end{pmatrix} + \mathbf{b} \geq \mathbf{0},\ i = 1,\dots,m
         
     The function supports various loss functions, including:
         - 'hinge', 'svm' or 'SVM'

rehline/_sklearn_mixin.py

@@ -442,6 +442,7 @@ def decision_function(self, X):
     def predict(self, X):
         """
         Predict targets as the linear decision function.
+        
         Parameters
         ----------
         X : ndarray of shape (n_samples, n_features)