I am trying to visualize a higher dimensional vector field. Is there a way to do this. I asked this question here, I was told to post it here.

As an example, one can use

$$\begin{eqnarray} \dot{x}&=& \left(J-R\right)x \end{eqnarray}$$

where $x\in \mathbb{R}^5$ $J $ is a skew-symmetric matrix and $R$ is a positive semi-definite matrix.

PS: The only idea I came so far is: somehow I have to use t-sne, but I really don't know how.


1 Answer 1


Somehow I have to use t-sne, but I really don't know how.

Since you have a PhD my answer will be brief, it's quite a lengthy subject.

The aim of dimensionality reduction is to preserve as much of the significant structure of the high-dimensional data as possible in the low-dimensional map. For high-dimensional data that lies on or near a low-dimensional, non-linear manifold it is usually more important to keep the low-dimensional representations of very similar datapoints close together, which is typically not possible with a linear mapping.

T-distributed Stochastic Neighbor Embedding (t-SNE) is a way of converting a high-dimensional data set into a matrix of pair-wise similarities. t-SNE is capable of capturing much of the local structure of the high-dimensional data very well, while also revealing global structure such as the presence of clusters at several scales.

Stochastic Neighbor Embedding (SNE) starts by converting the high-dimensional Euclidean distances between datapoints into conditional probabilities that represent similarities. The similarity of datapoint $x_j$ to datapoint $x_i$ is the conditional probability, $p_{j|i}$, that $x_i$ would pick $x_j$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at $x_i$. For nearby datapoints, $p_{j|i}$ is relatively high, whereas for widely separated datapoints, $p_{j|i}$ will be almost infinitesimal (for reasonable values of the variance of the Gaussian, $σ_i$).

The cost function used by t-SNE differs from the one used by SNE in two ways: (1) it uses a symmetrized version of the SNE cost function with simpler gradients that was briefly introduced by Cook et al. (2007) and (2) it uses a Student-t distribution rather than a Gaussian to compute the similarity between two points in the low-dimensional space. t-SNE employs a heavy-tailed distribution in the low-dimensional space to alleviate both the crowding problem and the optimization problems of SNE.

A few t-SNE tutorials:

  • 1
    $\begingroup$ Thanks for the answer! SNE/t-SNE might not work as there is no dimensionality reduction in my problem. I can, however, try a model-reduction technique together with this. I wanted to use this to demonstrate my results. However, this itself will be a new research direction. $\endgroup$
    – kosa
    Aug 31, 2018 at 13:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.