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I see that t-SNE can help us reduce dimensions and visualize the data. But what information are we gaining from this visualization? As we know that the new axis don't have a meaning in our context.

Moreover, if we have a class labeled data, then what information can we gain from the visualization? We already know that there are some 'n' classes and that we have to classify new examples in one of these classes.

Or am I wrong to understand t-SNE?

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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.

Dimensionality reduction yields a more compact, more easily interpretable representation of the target concept, focusing the user’s attention on the most relevant variables when you don't know anything about data.

Having said that Dimension Reduction will not help if you have a less features and only some are highly correlated as you can just use that correlated features for model so it all depends on data.

Unlike PCA, t-sne uses the local relationships between points to create a low-dimensional mapping. This allows it to capture non-linear structure.

T-sne also solves the Crowding Problem making the optimization "spread-out" the medium distance points to prevent crowding.

To understand more about t-sne plot refer here

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