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chayantpaulha
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Fixes tildes in rst files (scikit-learn#8751)
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doc/tutorial/statistical_inference/supervised_learning.rst

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@@ -109,21 +109,21 @@ The curse of dimensionality
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For an estimator to be effective, you need the distance between neighboring
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points to be less than some value :math:`d`, which depends on the problem.
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In one dimension, this requires on average :math:`n ~ 1/d` points.
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In one dimension, this requires on average :math:`n \sim 1/d` points.
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In the context of the above :math:`k`-NN example, if the data is described by
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just one feature with values ranging from 0 to 1 and with :math:`n` training
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observations, then new data will be no further away than :math:`1/n`.
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Therefore, the nearest neighbor decision rule will be efficient as soon as
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:math:`1/n` is small compared to the scale of between-class feature variations.
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If the number of features is :math:`p`, you now require :math:`n ~ 1/d^p`
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If the number of features is :math:`p`, you now require :math:`n \sim 1/d^p`
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points. Let's say that we require 10 points in one dimension: now :math:`10^p`
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points are required in :math:`p` dimensions to pave the :math:`[0, 1]` space.
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As :math:`p` becomes large, the number of training points required for a good
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estimator grows exponentially.
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For example, if each point is just a single number (8 bytes), then an
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effective :math:`k`-NN estimator in a paltry :math:`p~20` dimensions would
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effective :math:`k`-NN estimator in a paltry :math:`p \sim 20` dimensions would
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require more training data than the current estimated size of the entire
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internet (±1000 Exabytes or so).
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