Datasets:

zai-org
/

RPC-Bench

@@ -257,7 +257,7 @@ $\mathbf { A } = \mathbf { C } ^ { T } \cdot \mathbf { C } .$ , A is a semi-posi
 Let $\big \{ \alpha _ { \mathrm { i } } \big \}$ be a set of eigenvectors w.r.t $\mathbf { A } ( \alpha _ { \mathrm { i } } ^ { T } \cdot \alpha _ { \mathrm { i } } = 1 )$ ), then $\{ \mathbf { H } \cdot \mathbf { \boldsymbol { \alpha } } _ { \mathrm { i } } \}$ is a set of eigenvectors w.r.t $\mathbf { H } \cdot \dot { \mathbf { A } } \cdot \dot { \mathbf { H } } ^ { - 1 }$ , $\{ \mathbf { H } ^ { - 1 } \cdot \mathbf { \bar { \alpha } } _ { \mathrm { i } } \}$ is a set of eigenvectors w.r.t $\mathbf { H } ^ { - 1 } \cdot \mathbf { A } \cdot \mathbf { H }$ , and $\{ \lambda _ { \mathrm { i } } \}$ is a set of eigenvectors shared by them.
-Let $\beta$ be an eigenvector $\begin{array} { r } { \mathbf { \operatorname { p f } } \mathbf { H } { \cdot } \mathbf { A } { \cdot } \mathbf { H } ^ { - 1 } + \mathbf { H } ^ { - 1 } { \cdot } \mathbf { A } { \cdot } \mathbf { H } , \beta = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } \times \mathbf { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \times \mathbf { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \end{array}$  $\begin{array} { r l } & { \qquad \mathrm { E } : = \dots \qquad \mathrm { A } : \mathrm { H } ^ { - 1 } + \mathrm { H } ^ { - 1 } \cdot \mathrm { A } \cdot \mathrm { H } \cdot \mathrm { \Lambda } \mathrm { H } \cdot \mathrm { \Lambda } \beta = \mathrm { \Lambda } \lambda \beta , \quad \mathrm { ~ a } : = 1 \times \mathrm { H } \cdot \mathrm { \Lambda } \alpha _ { \mathrm { i } } + \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda _ { \mathrm { i } } \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } = } \\ & { \underset { \mathrm { - i } = 1 } { \sum } a _ { \mathrm { i } } \lambda \times \mathrm { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \\ & { \underset { \mathrm { - i } = 1 } { \sum } \left( \lambda _ { \mathrm { i } } - \lambda \right) a _ { \mathrm { i } } b _ { \mathrm { i } } = - \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \alpha _ { \mathrm { i } } ^ { T } \cdot \mathrm { H } ^ { - 1 } \cdot \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } \le 0 } \end{array}$
 A is a real symmetric matrix, so we have many sets of $\big \{ \alpha _ { \mathrm { i } } \big \}$ that is orthogonal to each other. $\beta ^ { T } \cdot \beta =$ $\textstyle \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } b _ { \mathrm { i } } \geq 0$ , For every eigenvalue, the dimension of the subspace will be very high in the case of neural architecture search, so we assume that we can choose a set of $\big \{ \alpha _ { \mathrm { i } } \big \}$ orthogonal to each other from the subspace satisfying Pni+1j=ni $\sum _ { \mathrm { j = n _ { i } } } ^ { \mathrm { n _ { i + 1 } - 1 } } a _ { \mathrm { j } } b _ { \mathrm { j } } \geq 0$ in most cases $\mathrm { { \acute { n } _ { i } } }$ is the id of the first eigenvector w.r.t $\lambda _ { \mathrm { i } }$ ).

 Let $\big \{ \alpha _ { \mathrm { i } } \big \}$ be a set of eigenvectors w.r.t $\mathbf { A } ( \alpha _ { \mathrm { i } } ^ { T } \cdot \alpha _ { \mathrm { i } } = 1 )$ ), then $\{ \mathbf { H } \cdot \mathbf { \boldsymbol { \alpha } } _ { \mathrm { i } } \}$ is a set of eigenvectors w.r.t $\mathbf { H } \cdot \dot { \mathbf { A } } \cdot \dot { \mathbf { H } } ^ { - 1 }$ , $\{ \mathbf { H } ^ { - 1 } \cdot \mathbf { \bar { \alpha } } _ { \mathrm { i } } \}$ is a set of eigenvectors w.r.t $\mathbf { H } ^ { - 1 } \cdot \mathbf { A } \cdot \mathbf { H }$ , and $\{ \lambda _ { \mathrm { i } } \}$ is a set of eigenvectors shared by them.
+Let $\beta$ be an eigenvector $\begin{array} { r } { \mathbf { \operatorname { p f } } \mathbf { H } { \cdot } \mathbf { A } { \cdot } \mathbf { H } ^ { - 1 } + \mathbf { H } ^ { - 1 } { \cdot } \mathbf { A } { \cdot } \mathbf { H } , \beta = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } \times \mathbf { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \times \mathbf { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \end{array}$  $\begin{array} { r l } & { \qquad \mathrm { E } : = \dots \qquad \mathrm { A } : \mathrm { H } ^ { - 1 } + \mathrm { H } ^ { - 1 } \cdot \mathrm { A } \cdot \mathrm { H } \cdot \mathrm { \Lambda } \mathrm { H } \cdot \mathrm { \Lambda } \beta = \mathrm { \Lambda } \lambda \beta , \quad \mathrm { ~ a } : = 1 \times \mathrm { H } \cdot \mathrm { \Lambda } \alpha _ { \mathrm { i } } + \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda _ { \mathrm { i } } \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } = } \\ & { \underset { \mathrm { - i } = 1 } { \sum } a _ { \mathrm { i } } \lambda \times \mathrm { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \\ & { \underset { \mathrm { - i } = 1 } { \sum } \left( \lambda _ { \mathrm { i } } - \lambda \right) a _ { \mathrm { i } } b _ { \mathrm { i } } = - \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \alpha _ { \mathrm { i } } ^ { T } \cdot \mathrm { H } ^ { - 1 } \cdot \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } \le 0 } \end{array}$
 A is a real symmetric matrix, so we have many sets of $\big \{ \alpha _ { \mathrm { i } } \big \}$ that is orthogonal to each other. $\beta ^ { T } \cdot \beta =$ $\textstyle \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } b _ { \mathrm { i } } \geq 0$ , For every eigenvalue, the dimension of the subspace will be very high in the case of neural architecture search, so we assume that we can choose a set of $\big \{ \alpha _ { \mathrm { i } } \big \}$ orthogonal to each other from the subspace satisfying Pni+1j=ni $\sum _ { \mathrm { j = n _ { i } } } ^ { \mathrm { n _ { i + 1 } - 1 } } a _ { \mathrm { j } } b _ { \mathrm { j } } \geq 0$ in most cases $\mathrm { { \acute { n } _ { i } } }$ is the id of the first eigenvector w.r.t $\lambda _ { \mathrm { i } }$ ).