Algorithm 1: FFT-accelerated Interpolation-based t-SNE (FIt-SNE)

$$\begin{array}{cc}\hfill & \mathbf{Input}:\text{Collection of points}\{y_i{\}}_{i=1}^N,\text{source strengths}\{q_i{\}}_{i=1}^N,\text{number of intervals}{N}_{\text{int}},\hfill \\ \hfill & \text{number of interpolation points per interval}p\hfill \\ \hfill & \mathbf{Output}:\varphi (y_i)=\sum \begin{array}{c}N\hfill \\ j=1\hfill \end{array}K(y_i,y_j)q_j\text{for}i=1,2,\dots N\hfill \\ \hfill & 1\text{For each interval}{I}_{\ell },\text{form the equispaced nodes}{\tilde{y}}_{j,\ell },j=1,2,\dots p\text{given by eq.}(7)\hfill \\ \hfill & 2\mathbf{for}I\leftarrow 1\mathbf{to}{N}_{int}\mathbf{do}\hfill \\ \hfill & \begin{array}{c}\hfill 3\hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}\mid \begin{array}{c}\text{Compute the coefficients}{w}_{m,\ell }\text{given by}\hfill \\ w_{m,\ell }=\sum _{y_i\in I_{\ell }}{L}_{m,{\tilde{y}}^{\ell }}(y_i)q_i,\hfill \\ m=1,2,\dots p.\hfill \end{array}\phantom{\mid }\hfill \\ \hfill & 4\mathbf{end}\hfill \\ \hfill & 5\text{Use the fast-Fourier transform to compute the values of}{v}_{m,n}\text{given by}\hfill \\ \hfill & (10)\left[\begin{array}{c}\hfill v_{1,1}\hfill \\ \hfill v_{2,1}\hfill \\ \hfill ⋮\hfill \\ \hfill v_{p-1},N_{\text{int}}\hfill \\ \hfill v_p,N_{\text{int}}\hfill \end{array}\right]=\tilde{K}\cdot \left[\begin{array}{c}\hfill w_{1,1}\hfill \\ \hfill w_{2,1}\hfill \\ \hfill ⋮\hfill \\ \hfill w_{p-1},N_{\text{int}}\hfill \\ \hfill w_p,N_{\text{int}}\hfill \end{array}\right],\hfill \\ \hfill & \text{where}\tilde{K}\text{is the Toeplitz matrix given by}\hfill \\ \hfill & (11){\tilde{K}}_{i,j}=K({\tilde{y}}_i,{\tilde{y}}_j),\hfill \\ \hfill & i,j=1,2,\dots N_{\text{int}}\cdot p.\hfill \\ \hfill & 6\mathbf{for}I\leftarrow 1\mathbf{to}{N}_{int}\mathbf{do}\hfill \\ \hfill & \begin{array}{c}\hfill 7\hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}\mid \begin{array}{c}\text{Compute}\varphi (y_i)\text{at all points}{y}_i\in I_{\ell }\text{via}\hfill \\ \varphi (y_i)=\sum _{j=1}^p{L}_{j,{\tilde{y}}^{\ell }}(y_i)v_{j,\ell }\hfill \\ \hfill \end{array}\phantom{\mid }\hfill \\ \hfill & 8\mathbf{end}\hfill \end{array}$$