507 lines
7.8 KiB
TeX
507 lines
7.8 KiB
TeX
\documentclass{article}
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\usepackage{epsfig}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{mathrsfs}
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\pagestyle{empty}
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\begin{document}
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\[ f(x|\theta) = \frac{1}{2 \theta}\exp\left(-\frac{\|x - \mu\|}{\theta}\right) \]
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\pagebreak
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$\theta$
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\pagebreak
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$\mu$
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\pagebreak
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\[ d(a, b) = \frac{a^T b}{|| a || || b ||} \]
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\pagebreak
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\[ K(x, y) = \max \{0, 1 - || x - y ||^2_2 / b^2 \} \]
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\pagebreak
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$ b $
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\pagebreak
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$ K(x, y) $
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\pagebreak
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$ x $
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\pagebreak
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$ y $
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\pagebreak
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\[ \int \int K(x, y) g(x) g(y) dx dy \ge 0 \]
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\pagebreak
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$ g(x) $
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\pagebreak
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$ \mu $
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\pagebreak
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\[ K(x, y) = \exp(-\frac{|| x - y ||^2}{2 \mu^2}). \]
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\pagebreak
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$ \gamma = -\frac{1}{2 \mu^2} $
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\pagebreak
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$ s $
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\pagebreak
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$ t $
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\pagebreak
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\[ K(x, y) = \tanh(s <x, y> + t) \]
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\pagebreak
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\[ K(x, y) = \exp(-\frac{|| x - y ||}{\mu}). \]
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\pagebreak
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\[ K(x, y) = x^T y \]
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\pagebreak
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$ degree $
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\pagebreak
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$ offset $
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\pagebreak
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\[ K(x, y) = (x^T * y + offset) ^ {degree}. \]
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\pagebreak
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\[ K(x, y) = \max \{ 0, 1 - \frac{|| x - y ||_2}{b} \} \]
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\pagebreak
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$ W = x (x^T x)^{-0.5} $
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\pagebreak
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\[ d(x, y) = \sqrt{ K(x, x) + K(y, y) - 2K(x, y) }. \]
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\pagebreak
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$ n $
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\pagebreak
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\[ d(x, y) = \left( \sum_{i = 1}^{n} | x_i - y_i |^p \right)^{\frac{1}{p}}. \]
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\pagebreak
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$ d(x, y) $
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\pagebreak
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\[ d(x, y) = \sum_{i = 1}^{n} | x_i - y_i |^p \]
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\pagebreak
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$ Q $
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\pagebreak
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$ d $
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\pagebreak
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\[ d(x, y) = \sqrt{(x - y)^T Q (x - y)} \]
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\pagebreak
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\[ d(x, y) = (x - y)^T Q (x - y) \]
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\pagebreak
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$n$
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\pagebreak
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\[ A_{j + 1} = A_j + \alpha \nabla F(A) \]
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\pagebreak
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$ \alpha $
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\pagebreak
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$ F $
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\pagebreak
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$ j $
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\pagebreak
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$ \epsilon $
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\pagebreak
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\[ | F(A_{j + 1}) - F(A_j) | < \epsilon. \]
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\pagebreak
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$\epsilon$
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\pagebreak
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\[ f(A) = \sum_{i = 0}^{n} f_i(A) \]
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\pagebreak
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$ A $
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\pagebreak
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$ \{ f_{i0}(A), f_{i1}(A), \ldots, f_{i(m - 1)}(A) $
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\pagebreak
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$ m $
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\pagebreak
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\[ A_{j + 1} = A_j + \alpha \left(\sum_{k = 0}^{m - 1} \nabla f_{ik}(A) \right) \]
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\pagebreak
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$ \gamma $
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\pagebreak
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\begin{eqnarray*} r_t &=& (1 - \gamma) f'(\Delta_t)^2 + \gamma r_{t - 1} \\ v_{t + 1} &=& \frac{\alpha}{\sqrt{r_t}}f'(\Delta_t) \\ \Delta_{t + 1} &=& \Delta_t - v_{t + 1} \end{eqnarray*}
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\pagebreak
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\[ T_{n+1} = (1-\lambda) T_{n} \]
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\pagebreak
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$ 0<\lambda<1 $
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\pagebreak
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$ \lambda $
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\pagebreak
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$ \alpha = (-1 \lambda) $
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\pagebreak
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$ f_i(A) $
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\pagebreak
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$ f_i(A)$
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\pagebreak
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$ j$
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\pagebreak
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\[ | f(A_{j + n}) - f(A_j) | < \epsilon. \]
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\pagebreak
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\[ A_{j + 1} = A_j + \alpha \nabla f_i(A) \]
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\pagebreak
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$ i $
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\pagebreak
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$ V \approx WH $
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\pagebreak
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$ \sqrt{\sum_i \sum_j(V-WH)^2} $
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\pagebreak
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\[ W^T = \frac{H V^T}{H H^T} \]
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\pagebreak
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\[ H = \frac{W^T V}{W^T W} \]
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\pagebreak
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\[ W_{ia} \leftarrow W_{ia} \frac{(VH^T)_{ia}}{(WHH^T)_{ia}} \]
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\pagebreak
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\[ H_{a\mu} \leftarrow H_{a\mu} \frac{(W^T V)_{a\mu}}{(W^T WH)_{a\mu}} \]
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\pagebreak
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\[ \sum_i \sum_j (V_{ij} \log\frac{V_{ij}}{(W H)_{ij}} - V_{ij} + (W H)_{ij}) \]
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\pagebreak
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\[ W_{ia} \leftarrow W_{ia} \frac{\sum_{\mu} H_{a\mu} V_{i\mu} / (W H)_{i\mu}} {\sum_{\nu} H_{a\nu}} \]
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\pagebreak
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\[ H_{a\mu} \leftarrow H_{a\mu} \frac{\sum_{i} W_{ia} V_{i\mu}/(WH)_{i\mu}} {\sum_{k} H_{ka}} \]
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\pagebreak
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\begin{eqnarray*} f(x) &=& x \\ f'(x) &=& 1 \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \frac{1}{1 + e^{-x}} \\ f'(x) &=& f(x) * (1 - f(x)) \\ f^{-1}(y) &=& ln(\frac{y}{1-y}) \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \max(0, x) \\ f'(x) &=& \left\{ \begin{array}{lr} 1 & : x > 0 \\ 0 & : x \le 0 \end{array} \right. \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \ln(1 + e^{x}) \\ f'(x) &=& \frac{1}{1 + e^{-x}} \\ f^{-1}(y) &=& \ln(e^{y} - 1) \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \frac{x}{1 + |x|} \\ f'(x) &=& (1 - |x|)^2 \\ f(x) &=& \left\{ \begin{array}{lr} -\frac{y}{y-1} & : x > 0 \\ \frac{x}{1 + x} & : x \le 0 \end{array} \right. \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \frac{e^x - e^{-x}}{e^x + e^{-x}} \\ f'(x) &=& 1 - \tanh^2(x) \\ f^{-1}(x) &=& \arctan(x) \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} \overline{s} &=& f^{-1}(\overline{t}) \\ \Theta^{1}_{p} &\le& \overline{s} \sqrt{\frac{3}{I \sum_{i = 1}^{I} (x_{ip}^2)}} \\ \Theta^1 &=& min(\Theta_{p}^{1}); p=1,2,..,P \\ -\Theta^{1} \le w_{i}^{1} &\le& \Theta^{1} \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} \gamma &\le& w_i \le \gamma \\ \beta &=& 0.7H^{\frac{1}{I}} \\ n &=& \sqrt{\sum_{i=0}{I}w_{i}^{2}} \\ w_i &=& \frac{\beta w_i}{n} \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} b &=& |F^{-1}(1 - \epsilon) - f^{-1}(\epsilon)| \\ \hat{w} &=& \frac{b}{k \cdot n} \\ \gamma &\le& a_i \le \gamma \\ w_i &=& \hat{w} \cdot \sqrt{a_i + 1} \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \left\{ \begin{array}{lr} x & : x > 0 \\ alpha(e^x - 1) & : x \le 0 \end{array} \right f'(x) &=& \left\{ \begin{array}{lr} 1 & : x > 0 \\ y + alpha & : x \le 0 \end{array} \right \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \left\{ \begin{array}{lr} max & : x > maxValue \\ min & : x \le minValue \\ x & : otherwise \end{array} \right. \\ f'(x) &=& \left\{ \begin{array}{lr} 0 & : x > maxValue \\ 0 & : x \le minValue \\ 1 & : otherwise \end{array} \right. \end{eqnarray*}
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\pagebreak
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\begin{eqnarray*} f(x) &=& \max(x, alpha*x) \\ f'(x) &=& \left\{ \begin{array}{lr} 1 & : x > 0 \\ alpha & : x \le 0 \end{array} \right. \end{eqnarray*}
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\pagebreak
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\[ \min_{\beta} 0.5 || X \beta - y ||_2^2 + 0.5 \lambda_2 || \beta ||_2^2 \]
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\pagebreak
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$ ||\beta||_1 <= \tau $
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\pagebreak
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$ X $
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\pagebreak
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\[ \min_{\beta} 0.5 || X \beta - y ||_2^2 + \lambda_1 || \beta ||_1 + 0.5 \lambda_2 || \beta ||_2^2 \]
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\pagebreak
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$ \beta $
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\pagebreak
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$ \lambda_1 > 0 $
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\pagebreak
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$ \lambda_2 = 0 $
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\pagebreak
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$ \lambda_2 > 0 $
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\pagebreak
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$ \lambda_1 = 0 $
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\pagebreak
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$ \lambda_1 $
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\pagebreak
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\[ (1 / n) * \| y - X B \|^2_2 \]
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$ B $
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\pagebreak
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\[ \min_{D,Z} 0.5 ||X - D Z||_{F}^2\ + \lambda_1 \sum_{i=1}^m ||Z_i||_1 + 0.5 \lambda_2 \sum_{i=1}^m ||Z_i||_2^2 \]
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\pagebreak
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$ ||D_j||_2 <= 1 $
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\pagebreak
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$ 1 <= j <= k $
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\pagebreak
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$ lambda_1 > 0 $
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\pagebreak
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$ lambda_2 = 0 $
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\pagebreak
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$(0, 1)$
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$(3, 1)$
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$(5, -5)$
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\pagebreak
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$ C = X^T X $
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$ C = X X^T $
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$R$
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$Q$
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$k$
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\pagebreak
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\[ \operatorname{k-argmax}_{p_r \in R} d(p_q, p_r). \]
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$m$
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$V \in \Re^{n \times m}$
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$V_{ij}$
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$i$
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$j$
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$V$
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$V \approx WH$
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$W$
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$H$
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$\alpha$
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$S$
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$\mathbf{R}^d$
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$kd$
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$n-1 \times 3 $
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$\log N$
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$N$
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$O(N^2)$
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\[ \arg\max_{p_r \in R} K(p_q, p_r) \]
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$p_q \in Q$
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$K(\cdot, \cdot)$
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$ x_i $
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$ c_j $
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$ j \le k $
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$ k $
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\[ \sum_{j = 1}^{k} \sum_{x_i \in c_j} \| x_i - \mu_j \|^2 \]
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\pagebreak
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$\mu_j$
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$c_j$
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$x_i$
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$X$
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\pagebreak
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$O(kN)$
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\pagebreak
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$\mathbf{x_i}, 0 \le i < n$
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$d$
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$y_i, 0 \le i < n$
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\pagebreak
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$\beta_i, 0 \le i \le d$
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\[ y_i = \beta_0 + \displaystyle\sum_{j = 1}^{d} \beta_j x_{ij} \]
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\pagebreak
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$\mathbf{x_i}$
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$y_i$
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$\mathbf{X}$
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$\mathbf{y}$
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\[ \mathbf{y} = \mathbf{X} \mathbf{\beta} + \beta_0 \]
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$\mathbf{\beta}$
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$\beta_0$
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$\beta_1$
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$\beta_2$
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$y$
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\pagebreak
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$f(y)=0+1x_1$
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\pagebreak
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$x_1$
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\pagebreak
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\[ \mathbf{X}' \mathbf{X} \]
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\pagebreak
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\[ \mathbf{X}' \mathbf{X} + \lambda \mathbf{I} \]
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$\mathbf{I}$
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\pagebreak
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$\lambda$
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\pagebreak
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$\mathcal{K}(\cdot, \cdot)$
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\pagebreak
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$\mathcal{K}(a, b)$
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\pagebreak
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$ K(x, x) = 1 \; \forall x $
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\pagebreak
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$d(\cdot, \cdot)$
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\pagebreak
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$d(a, b)$
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\pagebreak
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$ S \in \mathcal{R}^{N \times d} $
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\pagebreak
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$\mathcal{R}^d$
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$c$
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\pagebreak
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$p$
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\pagebreak
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$d(p, c) - \lambda$
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\pagebreak
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\end{document}
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