High dimensional space
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potentially millions of dimensions
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Hypercube
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higher dimensional cube
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equal width in every dimensions
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line -> square -> cube -> hypercube
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unit hypercube is \(\epsilon\) in each dimension where \(\epsilon\) is the unit vector
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hyper rectangle has unequal length in each dimension
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\(H_d(\ell) = \{(x_1, x_2 ...) | 0 \le x \le \ell\}\)
Volume
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\(V(H_d(\ell)) = \ell^d\)
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Unit hypercube has volume 1
hypersphere
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equal radius in each dimension
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point -> line -> circle -> sphere
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hollow
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\(S_d(\gamma) = \{(x_1 ...) | \sum(s_i)^2 = \gamma^2\}\)
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hyperball is not hollow
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\(B_d(\gamma) = \{ \vec x | \|x\|^2 \le \gamma^2\}\)
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Volume of hyperball and hypersphere are the same
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\(V(S_d(r)) = \int\limits^r_02\pi r dr = k_d r^d\)
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Volume of hypersphere goes to 0 as dimensionality increases
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\(\frac{2 \pi e}{d}^{d/2} \frac{1}{\sqrt{\pi d}}r^d\)
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All points are in the corners
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Diameter = \(1 - (1-\epsilon/r)^d\)
minimum bounding
Multidimensional gaussiens
Density
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Points around the mean
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\(f(x|\vec0,I) = \frac{1}{\sqrt{2\pi}^d}e^{-\frac{x^T x}{2} \)
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Peak density is \(1/\sqrt{2\pi}\)
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\(f(0)\) is the peak density
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as \(d \rightarrow \infty \quad p (\frac{f(x)}{f(0)} \le \infty) = 0\)