LaTeX vs MathPix image — 1605.05775

f(\mathbf{x})=W \cdot \Phi(\mathbf{x}) .

f(\mathbf{x}) = W \cdot \Phi(\mathbf{x}) \ .

\Phi^{s_{1} s_{2} \cdots s_{N}}(\mathbf{x})=\phi^{s_{1}}\left(x_{1}\right) \otimes \phi^{s_{2}}\left(x_{2}\right) \otimes \cdots \phi^{s_{N}}\left(x_{N}\right) .

\Phi^{s_1 s_2 \cdots s_N}(\mathbf{x}) = \phi^{s_1}(x_1) \otimes \phi^{s_2}(x_2) \otimes \cdots \phi^{s_N}(x_N) \ .

\phi^{s_{j}}\left(x_{j}\right)=\left[\cos \left(\frac{\pi}{2} x_{j}\right), \sin \left(\frac{\pi}{2} x_{j}\right)\right]

\phi^{s_j}(x_j) = \left[ \cos{\left(\frac{\pi}{2} x_j\right)},\ \sin{\left(\frac{\pi}{2} x_j\right)} \right]

f^{\ell}(\mathbf{x})=W^{\ell} \cdot \Phi(\mathbf{x})

f^\ell(\mathbf{x}) = W^\ell \cdot \Phi(\mathbf{x})

W_{s_{1} s_{2} \cdots s_{N}}^{\ell}=\sum_{\{\alpha\}} A_{s_{1}}^{\alpha_{1}} A_{s_{2}}^{\alpha_{1} \alpha_{2}} \cdots A_{s_{j}}^{\ell ; \alpha_{j} \alpha_{j+1}} \cdots A_{s_{N}}^{\alpha_{N-1}}

W^\ell_{s_1 s_2 \cdots s_N} = \sum_{\{\alpha\}} A^{\alpha_1}_{s_1} A^{\alpha_1 \alpha_2}_{s_2} \cdots A^{\ell; \alpha_j \alpha_{j+1}}_{s_j} \cdots A^{\alpha_{N-1}}_{s_N}

C=\frac{1}{2} \sum_{n=1}^{N_{T}} \sum_{\ell}\left(f^{\ell}\left(\mathbf{x}_{n}\right)-\delta_{L_{n}}^{\ell}\right)^{2}

C = \frac{1}{2} \sum_{n=1}^{N_T} \sum_{\ell} (f^\ell(\mathbf{x}_n) - \delta^\ell_{L_n})^2

f^{\ell}\left(\mathbf{x}_{n}\right)=\sum_{\alpha_{j-1} \alpha_{j+1}} \sum_{s_{j} s_{j+1}} B_{s_{j} s_{j+1}}^{\alpha_{j-1} \ell \alpha_{j+1}}\left(\tilde{\Phi}_{n}\right)_{\alpha_{j-1} \ell \alpha_{j+1}}^{s_{j} s_{j+1}}

f^\ell(\mathbf{x}_n) = \sum_{\alpha_{j-1} \alpha_{j+1}}\sum_{s_j s_{j+1}} B^{\alpha_{j-1} \ell \alpha_{j+1}}_{s_j s_{j+1}} (\tilde{\Phi}_n)_{\alpha_{j-1} \ell \alpha_{j+1}}^{s_j s_{j+1}}

\begin{aligned} \Delta B^{\ell} & \stackrel{\text { def }}{=}-\frac{\partial C}{\partial B^{\ell}} \\ & =\sum_{n=1}^{N_{T}} \sum_{\ell^{\prime}}\left(\delta_{L_{n}}^{\ell^{\prime}}-f^{\ell^{\prime}}\left(\mathbf{x}_{n}\right)\right) \frac{\partial f^{\ell^{\prime}}\left(\mathbf{x}_{n}\right)}{\partial B^{\ell}} \\ & =\sum_{n=1}^{N_{T}}\left(\delta_{L_{n}}^{\ell}-f^{\ell}\left(\mathbf{x}_{n}\right)\right) \tilde{\Phi}_{n} \end{aligned}

\begin{aligned} \Delta B^\ell & \stackrel{\text{def}}{=} - \frac{\partial C}{\partial B^\ell} \\ & = \sum_{n=1}^{N_T} \sum_{\ell'} (\delta^{\ell'}_{L_n}-f^{\ell'}(\mathbf{x}_n)) \frac{\partial f^{\ell'}(\mathbf{x}_n)}{\partial B^\ell} \\ & = \sum_{n=1}^{N_T} (\delta^\ell_{L_n}-f^\ell(\mathbf{x}_n)) \tilde{\Phi}_n \ . \end{aligned}

B_{s_{j} s_{j+1}}^{\alpha_{j-1} \ell \alpha_{j+1}}=\sum_{\alpha_{j}^{\prime} \alpha_{j}} U_{s_{j} \alpha_{j}^{\prime}}^{\alpha_{j-1}} S^{\alpha_{j}^{\prime}}{ }_{\alpha_{j}} V_{s_{j+1}}^{\alpha_{j} \ell \alpha_{j+1}}

B^{\alpha_{j-1} \ell \alpha_{j+1}}_{s_j s_{j+1}} = \sum_{\alpha^\prime_j \alpha_j} U^{\alpha_{j-1}}_{s_j \alpha^\prime_j} S^{\alpha^\prime_j}\,{}_{\alpha_j} V^{\alpha_j \ell \alpha_{j+1}}_{s_{j+1}} \ ,

\Phi\left(x_{1}, x_{2}\right)=\phi^{s_{1}}\left(x_{1}\right) \otimes \phi^{s_{2}}\left(x_{2}\right)

\Phi(x_1,x_2) = \phi^{s_1}(x_1) \otimes \phi^{s_2}(x_2)

f(\mathbf{x})=\sum_{\{s\}} W_{s_{1} s_{2} \cdots s_{N}} \phi^{s_{1}}\left(x_{1}\right) \otimes \phi^{s_{2}}\left(x_{2}\right) \otimes \cdots \phi^{s_{N}}\left(x_{N}\right) .

f(\mathbf{x}) = \sum_{\{s\}} W_{s_1 s_2 \cdots s_N} \phi^{s_1}(x_1) \otimes \phi^{s_2}(x_2) \otimes \cdots \phi^{s_N}(x_N) \:.

\phi^{s_{1}}\left(x_{1}\right) \otimes \phi^{s_{2}}\left(x_{2}\right) \otimes \cdots \phi^{s_{N}}\left(x_{N}\right)

\phi^{s_1}(x_1) \otimes \phi^{s_2}(x_2) \otimes \cdots \phi^{s_N}(x_N)

\begin{aligned} & \phi(0)=[1,0] \\ & \phi(1)=[0,1] . \end{aligned}

\begin{aligned} \phi(0) &= [1,\ 0] \\ \phi(1) &= [0,\ 1] \ . \end{aligned}

\begin{aligned} & W_{s_{1} s_{2} \cdots s_{N}}^{\ell}= \\ & \quad \sum_{\{\alpha\}} U_{s_{1}}^{\alpha_{1}} \cdots U_{\alpha_{i-1} s_{i}}^{\alpha_{i}} C_{\alpha_{i} \alpha_{i+1}}^{\ell} V_{s_{i+1} \alpha_{i+2}}^{\alpha_{i+1}} \cdots V_{s_{N}}^{\alpha_{N-1}} \end{aligned}

\begin{aligned} W^\ell_{s_1 s_2 \cdots s_N} & = \\ \sum_{\{\alpha\}} & \, U^{\alpha_1}_{s_1} \cdots U_{\alpha_{i-1} s_i}^{\alpha_i} C^{\ell}_{\alpha_i \alpha_{i+1}} V_{s_{i+1} \alpha_{i+2}}^{\alpha_{i+1}} \cdots V^{\alpha_{N-1}}_{s_N} \end{aligned}

\sum_{\ell} \int_{\mathbf{x}}\left|f^{\ell}(\mathbf{x})\right|^{2} d \mu(x)=1

\sum_\ell \int_{\mathbf{x}} |f^\ell(\mathbf{x})|^2 d\mu(x) = 1

\sum_{\ell} \sum_{s_{1}, s_{2}, \ldots, s_{N}} \bar{W}_{s_{1} s_{2} \cdots s_{N}}^{\ell} W_{s_{1} s_{2} \cdots s_{N}}^{\ell}=1 .

\sum_\ell \sum_{s_1, s_2, \ldots, s_N} \bar{W}^\ell_{s_1 s_2 \cdots s_N} W^\ell_{s_1 s_2 \cdots s_N} = 1 \ .

\int_{x} \bar{\phi}^{s}(x) \phi^{s^{\prime}}(x) d \mu(x)=\delta_{s s^{\prime}}

\int_x \bar{\phi}^s(x) \phi^{s^\prime}(x) \,d\mu(x) = \delta_{s s^\prime} ,

\sum_{s}\left|\phi^{s}(x)\right|^{2}=1

\sum_s |\phi^s(x)|^2 = 1

\phi(x)=[\cos (\pi x), \sin (\pi x)] .

\phi(x) = \big[\cos(\pi x), \ \sin(\pi x) \big] \ .

\phi(x)=\left[e^{i(3 \pi / 2) x} \cos \left(\frac{\pi}{2} x\right), e^{-i(3 \pi / 2) x} \sin \left(\frac{\pi}{2} x\right)\right] .

\phi(x) = \Big[e^{i (3\pi/2) x} \cos\Big(\frac{\pi}{2} x\Big), \ e^{-i (3\pi/2) x} \sin\Big(\frac{\pi}{2} x\Big) \Big] \ .

C=-\sum_{n=1}^{N_{s}} \log \left|f^{L_{n}}\left(\mathbf{x}_{n}\right)\right|^{2}

C = - \sum_{n=1}^{N_s} \log{|f^{L_n}(\mathbf{x}_n)|^2}

D_{\mathrm{KL}}=\sum_{\ell} \int_{\mathbf{x}} p(\ell, \mathbf{x}) \log \left(\frac{p(\ell, \mathbf{x})}{\tilde{p}(\ell, \mathbf{x})}\right)

D_\text{KL} = \sum_\ell \int_{\mathbf{x}} p(\ell,\mathbf{x}) \log{\!\Big(\frac{p(\ell,\mathbf{x})}{\tilde{p}(\ell,\mathbf{x})}\Big)}

\phi^{s_{j}}\left(x_{j}\right)=\left[\cos \left(\frac{\pi}{2} x_{j}\right), \sin \left(\frac{\pi}{2} x_{j}\right)\right] .

\phi^{s_j}(x_j) = \Big[\cos\Big(\frac{\pi}{2} x_j\Big),\,\sin\Big(\frac{\pi}{2} x_j\Big)\Big] \:.

\left(\cos ^{2}\left(\theta_{j}\right)+\sin ^{2}\left(\theta_{j}\right)\right)^{d-1}=1 .

(\cos^2(\theta_j)+\sin^2(\theta_j))^{d-1}=1 \ .

\begin{aligned} & \left(\cos ^{2}\left(\theta_{j}\right)+\sin ^{2}\left(\theta_{j}\right)\right)^{d-1}=1 \\ & \quad=\sum_{p=0}^{d-1}\binom{d-1}{p}\left(\cos \theta_{j}\right)^{2(d-1-p)}\left(\sin \theta_{j}\right)^{2 p} . \end{aligned}

\begin{aligned} \MoveEqLeft (\cos^2(\theta_j)+\sin^2(\theta_j))^{d-1} = 1 \\ & = \sum_{p=0}^{d-1} \binom{d-1}{p} (\cos\theta_j)^{2(d-1-p)} (\sin\theta_j)^{2p} \:. \end{aligned}

\phi^{s_{j}}\left(x_{j}\right)=\sqrt{\binom{d-1}{s_{j}-1}}\left(\cos \left(\frac{\pi}{2} x_{j}\right)\right)^{d-s_{j}}\left(\sin \left(\frac{\pi}{2} x_{j}\right)\right)^{s_{j}-1}

\phi^{s_j}(x_j) = \sqrt{\binom{d-1}{s_j-1}}\ (\cos(\frac{\pi}{2} x_j))^{d-s_j} (\sin(\frac{\pi}{2} x_j))^{s_j-1}