Algorithm 3 Learning a kernel for approximating DTW in the case of multiple references and multiple data streams

Inputn reference patterns ${\left\{X_i\right\}}_{i=1\dots n}$, m data streams ${\left\{Y_l\right\}}_{l=1\dots m}$ and parameters $R,L_{min},L_{max},{\beta }^{-},{\beta }^{+},{\sigma }^2$. Output a matrix $\mathbf{K}\in {\mathbb{R}}^{n\times m}$ containing the kernel values. 1:generate a set of R “basis” time series $S=\{s_1,s_2,\dots ,s_R\}$, where each $s_i\sim \mathcal{N}(0,{\sigma }^2)$ 2:for$i=1:n$do 3:    compute ${\mathsf{\Phi }}_S\left(X_i\right)={(DTW(X_i,s_1),\dots ,DTW(X_i,s_R))}^T$ 4:    for $j=1:n$ do 5:        ${\mathbf{M}}_{i,j}=DTW(X_i,X_j)$ 6:    end for 7:end for 8:normalize entries of the matrix $\mathbf{M}$ 9:choose ${\gamma }^{*}={arg\; min}_{\gamma \in {\mathbb{R}}^n}\frac{2}{n(n-1)}{\sum }_{i=1}^{n-1}{\sum }_{j=i+1}^n\left|(1-K_{{\gamma }_i}({\varphi }_S\left(X_i\right),{\varphi }_S\left(X_j\right))-{\mathbf{M}}_{i,j})\right|$ 10:compute every entry of $\mathbf{K}$ as $K_{i,l}=d_{K,{\gamma }^{*}}(X_i,Y_{l,a^{*}:b^{*}})$, where $a^l,b^{*}$ are obtained solving (9).