Can t-SNE help feature selection? I'm training a fully connected feed forward neural network for regression. 
Given one training example $(x_i, y_i)$, I need to convert the raw representation $x_i$ into an invariant representation $x_i^{'}$ of higher dimensions by some non-linear transformation 
$$x_i^{'} = f_{\theta}(x_i)$$
$\theta$ is the hyperparameter determines such a transformation, the dimensionalities of $x_i$ and $x_i^{'}$ are about $150$ and $4500$ respectively.
This transformation step cannot be accomplished by a neural network layer because the original representation is not invariant under translation. The original representation has to be transformed by some map controlled by $\theta$, and the map is constructed based on domain knowledge.   
To choose a good hyperparameter $\theta$ is not easy, so I wonder whether dimensionality reduction can help me to make a good choice.
My idea is:
If a resulting representation $\{x_i^{'}\}_{i = 1}^N$ preserves much of the topological structures of the raw representation $\{x_i\}_{i = 1}^{N}$, then it is reasonable to believe it will be good for regression. The topological structure of the raw input can be probed by the domain knowledge, to probe the structure of the resulting representation, I decided to use t-SNE because of its non-linear nature.
However, by playing with t-SNE on this beautiful site, I found the result given by t-SNE is quite sensitive to the choice of perplexity, iterations and learning rate, and it seems only qualitative conclusions can be made by plotting the result. 
Is my idea reasonable? Is there a better choice for a quick dimensionality reduction?  
 A: No, I'm afraid your idea is not reasonable, for two reasons. Firstly, t-sne is a nice tool for decorative pictures, but it doesn't give you any reliable information to base decisions on. As you already noticed, it is very sensitive to the choice of parameters. Different parameters lead to different images which in turn lead to different conclusions regarding virtually all properties you might want to read off such a plot, and you have no way of knowing which of these parameter choices is the one leading to a correct conclusion (if any). You don't have to take my word for it; you can find a sufficient number of very simple examples here.
If you'd like a better alternative, you may want to think about why you need to evaluate the choice of $\theta$ at all. Your mapping goes from a lower-dimensional space into a higher-dimensional one. So it is plausible to assume that you're not loosing information, i.e., different choices for $x_i$ lead to different choices for $x_i'$ (of course, you have to check that with your concrete mapping). 
This means that you might even be able to choose an arbitrary value for $\theta$ and just trust that your feedforward network will figure out the meaning of the resulting representation. So, in all likelihood we're not talking about a situation where you have to find one of very few values for $\theta$ to make your regression work at all. It is much more likely that the choice of $\theta$ has more of a gradual influence on the outcome: With some choices, the neural network will perform better than with others. 
Of course choosing just one arbitrary value is just a thought experiment. In practice, I would recommend not to try and choose $\theta$ in a separate step before fitting the neural network, but rather treat it as an additional hyperparameter which has to be optimized along with the neural network parameters (such as number of layers etc.).
