transpose vectors

Author: yixuan

python/benchmark/benchmark_FairSVM/README.md

@@ -13,8 +13,8 @@ $$\sum_{i=1}^n z_{ij} = 0,$$
 
 such as gender and/or race. The constraints limit the correlation between the $d_0$-length sensitive features $\mathbf{z}_ i \in \mathbb{R}^{d_0}$ and the decision function $\mathbf{\beta}^\intercal \mathbf{x}$, and the constants $\mathbf{\rho} \in \mathbb{R}_+^{d_0}$ trade-offs predictive accuracy and fairness. Note that the FairSVM can be rewritten as a ReHLine optimization with
 ```math
-\mathbf{U} \leftarrow -C \mathbf{y}/n, \quad
-\mathbf{V} \leftarrow C \mathbf{1}_n/n, \quad
+\mathbf{U} \leftarrow -C \mathbf{y}^\intercal/n, \quad
+\mathbf{V} \leftarrow C \mathbf{1}^\intercal_n/n, \quad
 \mathbf{A} \leftarrow
 \begin{pmatrix}
   \mathbf{Z}^\intercal \mathbf{X} / n \\

python/benchmark/benchmark_SVM/README.md

@@ -6,8 +6,8 @@ SVMs solve the following optimization problem:
 ```
 where $\mathbf{x}_i \in \mathbb{R}^d$ is a feature vector, and $y_i \in \{-1, 1\}$ is a binary label. Note that the SVM can be rewritten as a ReHLine optimization with
 ```math
-\mathbf{U} \leftarrow -C \mathbf{y}/n, \quad
-\mathbf{V} \leftarrow C \mathbf{1}_n/n, \quad
+\mathbf{U} \leftarrow -C \mathbf{y}^\intercal/n, \quad
+\mathbf{V} \leftarrow C \mathbf{1}^\intercal_n/n,
 ```
 where $\mathbf{1}_n = (1, \cdots, 1)^\intercal$ is the $n$-length one vector, $\mathbf{X} \in \mathbb{R}^{n \times d}$ is the feature matrix, and $\mathbf{y} = (y_1, \cdots, y_n)^\intercal$ is the response vector.
 ### Benchmarking solvers

python/benchmark/benchmark_sSVM/README.md

@@ -6,8 +6,8 @@ Smoothed SVMs solve the following optimization problem:
 ```
 where $V(\cdot)$ is the smoothed hinge loss, $\mathbf{x}_i \in \mathbb{R}^d$ is a feature vector, and $y_i \in \{-1, 1\}$ is a binary label. Smoothed SVM can be rewritten as a ReHLine optimization with
 ```math
-\mathbf{S} \leftarrow -\sqrt{C} \mathbf{y}/n, \quad
-\mathbf{T} \leftarrow \sqrt{C} \mathbf{1}_n/n, \quad
+\mathbf{S} \leftarrow -\sqrt{C} \mathbf{y}^\intercal/n, \quad
+\mathbf{T} \leftarrow \sqrt{C} \mathbf{1}^\intercal_n/n, \quad
 \mathbf{\tau} \leftarrow \sqrt{C},
 ```
 where $\mathbf{1}_n = (1, \cdots, 1)^\intercal$ is the $n$-length one vector, $\mathbf{X} \in \mathbb{R}^{n \times d}$ is the feature matrix, and $\mathbf{y} = (y_1, \cdots, y_n)^\intercal$ is the response vector.