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.


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:

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    $\begingroup$ 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 '18 at 13:41

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