Experiments

In order to illustrate the performance of LTSA algorithm, this article presents several numerical examples. First it tests the LTSA method for 1D manifold with uniformly sampled coordinates in a interval. In all those cases, it adds Gaussian noise to obtain the data set\(\left\{ {{x_i}} \right\}\),which is 
\({x_i} = f\left( {x_i^*} \right) + \eta randn\left( {m,1} \right)\)
where m is the dimension of the input space, and randn is the Matlab’s standard normal distribution, is the noise.
It gave the following three one-variable functions
\(f\left( \tau \right) = {\left( {10\tau ,10{\tau ^3} + 2{\tau ^2} - 10\tau } \right)^T},\tau \in \left[ { - 1,1} \right],\eta = 0.1\)
\(f\left( \tau \right) = {\left( {\tau \cos \left( \tau \right),\tau \sin \left( \tau \right)} \right)^T},\tau \in \left[ {0,4\pi } \right],\eta = 0.2\)
\(f\left( \tau \right) = {\left( {3\cos \left( \tau \right),3\sin \left( \tau \right),3\tau } \right)^T},\tau \in \left[ {0,4\pi } \right],\eta = 0.2\)
then plotted the color-coded sample data points as following Fig1