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md/train/BJlgt2EYwr/BJlgt2EYwr.md

@@ -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 $\begin{array} { r } { \mathbf { \operatorname { p f } } \mathbf { H } { \cdot } \mathbf { A } { \cdot } \mathbf { H } ^ { - 1 } + \mathbf { H } ^ { - 1 } { \cdot } \mathbf { A } { \cdot } \mathbf { H } , \beta = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } \times \mathbf { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \times \mathbf { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \end{array}$  $\begin{array} { r l } & { \qquad \mathrm { E } : = \dots \qquad \mathrm { A } : \mathrm { H } ^ { - 1 } + \mathrm { H } ^ { - 1 } \cdot \mathrm { A } \cdot \mathrm { H } \cdot \mathrm { \Lambda } \mathrm { H } \cdot \mathrm { \Lambda } \beta = \mathrm { \Lambda } \lambda \beta , \quad \mathrm { ~ a } : = 1 \times \mathrm { H } \cdot \mathrm { \Lambda } \alpha _ { \mathrm { i } } + \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda _ { \mathrm { i } } \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } = } \\ & { \underset { \mathrm { - i } = 1 } { \sum } a _ { \mathrm { i } } \lambda \times \mathrm { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \\ & { \underset { \mathrm { - i } = 1 } { \sum } \left( \lambda _ { \mathrm { i } } - \lambda \right) a _ { \mathrm { i } } b _ { \mathrm { i } } = - \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \alpha _ { \mathrm { i } } ^ { T } \cdot \mathrm { H } ^ { - 1 } \cdot \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } \le 0 } \end{array}$
+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 $\lambda _ { \mathrm { i } }$ ).
 

md/train/enoVQWLsfyL/enoVQWLsfyL.md

@@ -65,7 +65,7 @@ We generate views by perturbing examples with a viewmaker network $V$ , trained
 
 Figure $\bigstar$ summarizes our method. The encoder and viewmaker are optimized in alternating steps to minimize and maximize $\mathcal { L }$ , respectively. We use an image-to-image neural network as our viewmaker network, with an architecture adapted from work on style transfer $\left( \mathrm { J o h n s o n e t a l . } \right) \left[ \mathrm { 2 0 1 6 } \right)$ See the Appendix for more details. This network ingests the input image and outputs a perturbation that is constrained to an $\ell _ { 1 }$ sphere. The sphere’s radius is determined by the volume of the input tensor times a hyperparameter $\epsilon$ , the distortion budget, which determines the strength of the applied perturbation. This perturbation is added to the input image and optionally clamped in the case of images to ensure all pixels are in $[ 0 , 1 ]$ . Algorithm 1 describes this process precisely.
 
-Input: Viewmaker network $V$ , $C \times W \times H$ image X, $\ell _ { 1 }$ distortion budget ✏, noise    
+Input: Viewmaker network $V$ , $C \times W \times H$ image X, $\ell _ { 1 }$ distortion budget ✏, noise    
 Output: Perturbed $C \times W \times H$ image $X$   
 $P  V ( X , \delta ) \ / ,$ / generate perturbation   
 $\begin{array} { r } { P \gets \frac { \epsilon C W H } { | P | _ { 1 } } P \gets | / \langle } \end{array}$ project to $\ell _ { 1 }$ sphere