A structure we conceptually called &quot;<a href="/en/Spiny%20spikes%20in%20higher%20dimensional%20space">Spiny spikes in higher dimensional space</a>&quot;.
 - This is the reason for the <a href="/en/110%2C000%20UMAP">110,000 UMAP</a> here.
 - <img src="https://gyazo.com/9ffbc15d64803a29d328244ea591a147/thumb/1000" alt="image">
<ul>
<li>This means that in this of [Public Opinion Map 3970 UMAP<ul>
<li><img src="https://gyazo.com/30c9d0303e0de3e01269148c72184c1e/thumb/1000" alt="image"></li>
</ul>
</li>
<li>Mathematically.<ul>
<li>$N((0, ...) , \sigma)$ and a small number of $N((4\sigma, 0, ...) , \sigma)$ and a small number of data (representation of one spike).</li>
<li>$4\sigma$ means &quot;far enough away&quot;.</li>
<li>Since this is tedious, we will write this as 1 (just make $\sigma$ that much smaller).</li>
<li>K spikes are (1, 0, 0, 0, ...) , (0, 1, 0, 0, ...) , (0, 0, 1, 0, ...) , ... like this</li>
</ul>
</li>
</ul>
experiment
SD=0.1
<ul>
<li><img src="https://gyazo.com/4c4036bfb0383007baa9e66e4a65ce22/thumb/1000" alt="image">
SD=0.2</li>
<li><img src="https://gyazo.com/7d9d089ca4db78a451d22fbfb3c762a6/thumb/1000" alt="image">
SD=0.3</li>
<li><img src="https://gyazo.com/cd61931f4c4677188428ec8994527616/thumb/1000" alt="image"></li>
</ul>
When it&#39;s clearly separated, it&#39;s an enclave like UMAP with SD=0.1.
If it&#39;s too mixed, you can&#39;t tell the boundary, like SD=0.3.
It becomes spiky at its boundaries.
<img src="https://gyazo.com/75f8004dc6cbb748f1d0e81d22c24c77/thumb/1000" alt="image">
<hr>
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Mathematical multidimensional spiky spikes