「発表用数式」の編集履歴（バックアップ）一覧に戻る
発表用数式 - (2010/04/27 (火) 09:04:59) のソース
<math>p(\mathbf{a},b) \sim |T(\mathbf{a},b)|</math>
<math>p(\mathbf{a},b) = \frac{\mathrm{sign}[T(\mathbf{a},b)] T(\mathbf{a},b)}{\int_{\mathbb{R}^{m+1}} |T(\mathbf{a},b)| d\mathbf{a} db}}</math>
<math>c(\mathbf{a},b)p(\mathbf{a},b) = T(\mathbf{a},b)</math>
<math></math>
<math>\mathbf{f}(x) := \begin{pmatrix} \sin \, 2 \pi x \\ \cos \, 2 \pi x \end{pmatrix}</math>
<math>f(x) := \sin \, 2 \pi x</math>
<math>f(x) := \sin \, 10 \pi x</math>
<math>\mathbb{E}\left \{ \frac{K}{J} \sum_{j=1}^J c_{kj} \phi_c( \mathbf{a}_j^\mathrm{T} \mathbf{x} - b_j ) \right \}</math>
<math>\mathbb{E}\left \{ \frac{K}{J} \sum_{j=1}^J c_{kj} \phi_c( \mathbf{a}_j^\mathrm{T} \mathbf{x} - b_j ) \right \} = f_k(\mathbf{x})</math>
<math>= K \mathbb{E}\left[c_{kj} \phi_c( \mathbf{a}_j^\mathrm{T} \mathbf{x} - b_j ) \right]</math>
<math>= K \int c_k(\mathbf{a},b) \phi_c( \mathbf{a}^\mathrm{T} \mathbf{x} - b ) \left\{ \frac{1}{K}\sum_{l=1}^K p_l(\mathbf{a}, b) \right\} d\mathbf{a} db</math>
<math>= \sum_{l=1}^K \int c_k(\mathbf{a},b) \phi_c( \mathbf{a}^\mathrm{T} \mathbf{x} - b ) p_l(\mathbf{a}, b) d\mathbf{a} db</math>
<math> \int c_k(\mathbf{a},b) \phi_c( \mathbf{a}^\mathrm{T} \mathbf{x} - b ) p_l(\mathbf{a}, b) d\mathbf{a} db = 0</math>
<math>= \int c_k(\mathbf{a},b) \phi_c( \mathbf{a}^\mathrm{T} \mathbf{x} - b ) p_k(\mathbf{a}, b) d\mathbf{a} db</math>
<math>= f_k(\mathbf{x})</math>
<math>\mathbb{E}\left \{ \frac{K}{J} \sum_{j=1}^J c_{kj} \phi_c( \mathbf{a}_j^\mathrm{T} \mathbf{x} - b_j ) \right \}</math>
<math>c_{kj}</math>
<math>p_k</math>
<math>\mathbf{a}_j^k,b_j^k</math>
<math> (\mathbf{a}_j, b_j) \sim \frac{1}{out} \sum_{k=1}^{out} p_k(\mathbf{a},b) =:p(\mathbf{a},b)</math>
<math>(\mathbf{a}_j, b_j) \sim p(\mathbf{a},b) \propto |T(\mathbf{a},b)|</math>
<math>c_j \xleftarrow{Linear Regression} (\mathbf{a}_j, b_j)</math>
<math> C(\mathbb{R}^m) \ni f(\mathbf{x}) \xrightarrow{\ \phi_c \ } T(\mathbf{a},b) \in C(\mathbb{R}^{m+1}) </math>
<math> f : [0,1]^{in} \to \mathbb{R}^{out}</math>
<math> f : \mathbb{R}^{m} \to \mathbb{R}</math>
<math>\phi_c \longleftarrow \sigma</math>
<math>(x_n, \sin x_n)</math>
<math> \sigma(z) = \frac{1}{1+e^{-z}} </math>
<math>y_k(\mathbf{x}) = \sum_{j=1}^{hidden} w^{(2)}_{kj} \sigma \left( \sum_{i=1}^{in} w^{(1)}_{ji} x_i + \theta^{(1)}_j \right) + \theta^{(2)}_k </math>
<math> \mathbf{y}(\mathbf{x}) = \mathbf{w}^{(2)\mathrm{T}}_{k} \sigma \left( \mathbf{w}^{(1)\mathrm{T}}_{j} \mathbf{x} + \theta^{(1)}_j \right) + \theta^{(2)}_k </math>
<math> \mathbf{a} \in \mathbb{R}^m, \, b \in \mathbb{R}</math>
<math> T(\mathbf{a},b) := \frac{1}{(2 \pi)^m C_{\phi_c, \phi_c}} \int_{\mathbb{R}^m} \overline{\phi _d (\mathbf{a}^\mathrm{T} \mathbf{x} - b)} f(\mathbf{x}) d \mathbf{x} </math>
<math> T(\mathbf{a},b) := \frac{1}{C} \int_{\mathbb{R}^m} \overline{\phi _d (\mathbf{a}^\mathrm{T} \mathbf{x} - b)} f(\mathbf{x}) d \mathbf{x} </math>
<math> \phi_d(t) := \begin{cases} \rho^{(m)} & m \mbox{ is even}\\ \rho^{(m+1)} & m \mbox{ is odd}\end{cases} </math>
<math> \rho(z) := \begin{cases} e^{-\frac{1}{1-|z|^2}} & |z| < 1 \\ 0 & \mbox{otherwize} \end{cases} </math>
<math> C_{\phi_c, \phi_d} := \int_\mathbb{R} \frac{ \overline{\widehat{\phi_d}(w)} \widehat{\phi_c}(w)}{|w|^m} dw </math>
<math> f : \mathbb{R}^m \to \mathbb{R} </math>
<math> \phi_c : \mathbb{R} \to \mathbb{R}</math>
<math> \phi_d : \mathbb{R} \to \mathbb{R}</math>
<math>f(\mathbf{x}) = \int_{\mathbb{R}^{m+1}} T(\mathbf{a},b) \phi_c (\mathbf{a}^\mathrm{T} \mathbf{x} - b) d\mathbf{a} db</math>
<math> = \int_{\mathbb{R}^{m+1}} c(\mathbf{a},b) \phi_c (\mathbf{a}^\mathrm{T} \mathbf{x} - b) p(\mathbf{a},b) d\mathbf{a} db </math>
<math> \approx \frac{1}{J} \sum_{j=1}^J c_j \phi_c (\mathbf{a}_j^\mathrm{T} \mathbf{x} -b_j) \qquad (\mathbf{a}_j, b_j) \sim p(\mathbf{a},b)</math>
<math> f(\mathbf{x}) \approx \frac{1}{J} \sum_{j=1}^J c_j \phi_c (\mathbf{a}_j^\mathrm{T} \mathbf{x} -b_j)</math>
<math> f(\mathbf{x}) \approx \sum_{j=1}^J c'_j \phi_j(\mathbf{x})</math>
<math> \phi_j(\mathbf{x}) := \phi_c(\mathbf{a}_j^\mathrm{T} \mathbf{x} -b_j) </math>
<math>\{ \phi_1(\mathbf{x}), \phi_2(\mathbf{x}), \cdots, \phi_J(\mathbf{x}) \}</math>
<math>\phi_j \sim |T(\mathbf{a},b)|</math>
<math>\phi_j</math>
<math> \int_{\mathbb{R}^{m+1}} | T(\mathbf{a},b) | d\mathbf{a} db < \infty</math>
<math>f \in L^1(\mathbb{R}^m) \cap L^p(\mathbb{R}^m) (1\leq p < \infty)</math>
<math>f : \mbox{bounded, uniformly continuous} </math>
<math> \phi_c \in C^\infty(\mathbb{R}^m), \phi_c(-z) = \phi_c(z) </math>
<math> \phi_d </math>
<math> \phi_c </math>
<math> \phi_c(z) := \frac{1}{H} \left \{ \sigma(z+h) - \sigma(z-h) \right \} </math>
<math> f_N(\mathbf{x}) := \frac{1}{N} \sum_{n=1}^N c_n \phi_c (\mathbf{a}_n^\mathrm{T} \mathbf{x} -b_n) </math>
<math>(\mathbf{a}_n, b_n) \sim k|T(\mathbf{a},b)| \quad \Rightarrow \quad \mathbb{E} f_N = f</math>
<math>f(\mathbf{x})</math>
<math>|T(\mathbf{a},b)|</math>
<math> f_N(\mathbf{x}) := \frac{1}{N} \sum_{n=1}^N c_n \phi_c (\mathbf{a}_n^\mathrm{T} \mathbf{x} -b_n) </math>
<math> = \sum_{j=1}^J \left \{ \frac{c_j}{J H} \sigma (\mathbf{a}_j^\mathrm{T} \mathbf{x} -b_j +h ) - \frac{c_j}{J H} \sigma (\mathbf{a}_j^\mathrm{T} \mathbf{x} -b_j -h ) \right \}</math>
<math> = \sum_{j=1}^{J/2} w_j^{(2)} \sigma ( \mathbf{w}_j^{(1)\mathrm{T}} \mathbf{x} + \theta_j^{(1)}) </math>
