Whitening
From Ufldl
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which is needed for some algorithms. If we are training on images, | which is needed for some algorithms. If we are training on images, | ||
the raw input is redundant, since adjacent pixel values | the raw input is redundant, since adjacent pixel values | ||
| - | are highly correlated. The goal of whitening is to make the input less redundant, | + | are highly correlated. The goal of whitening is to make the input less redundant; more formally, |
| - | + | our desiderata are that our learning algorithms sees a training input where (i) the features are less | |
correlated with each other, and (ii) the features all have the same variance. | correlated with each other, and (ii) the features all have the same variance. | ||
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Further, | Further, | ||
the off-diagonal entries are zero; thus, | the off-diagonal entries are zero; thus, | ||
| - | <math>\textstyle x_{{\rm rot},1}</math> and <math>\textstyle x_{{\rm rot},2}</math> are uncorrelated, satisfying one of | + | <math>\textstyle x_{{\rm rot},1}</math> and <math>\textstyle x_{{\rm rot},2}</math> are uncorrelated, satisfying one of our desiderata |
| - | our desiderata for whitened data. | + | for whitened data (that the features be less correlated). |
To make each of our input features have unit variance, we can simply rescale | To make each of our input features have unit variance, we can simply rescale | ||
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nearly zero anyway, and thus can safely be dropped. | nearly zero anyway, and thus can safely be dropped. | ||
| - | == | + | == ZCA Whitening == |
Finally, it turns out that this way of getting the | Finally, it turns out that this way of getting the | ||
data to have covariance identity <math>\textstyle I</math> isn't unique. | data to have covariance identity <math>\textstyle I</math> isn't unique. | ||
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\end{align}</math> | \end{align}</math> | ||
Plotting <math>\textstyle x_{\rm ZCAwhite}</math>, we get: | Plotting <math>\textstyle x_{\rm ZCAwhite}</math>, we get: | ||
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| - | + | [[File:ZCA-whitened.png | 600px]] | |
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It can be shown that out of all possible choices for <math>\textstyle R</math>, | It can be shown that out of all possible choices for <math>\textstyle R</math>, | ||
this choice of rotation causes <math>\textstyle x_{\rm ZCAwhite}</math> to be as close as possible to the | this choice of rotation causes <math>\textstyle x_{\rm ZCAwhite}</math> to be as close as possible to the | ||
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When using ZCA whitening (unlike PCA whitening), we usually keep all <math>\textstyle n</math> dimensions | When using ZCA whitening (unlike PCA whitening), we usually keep all <math>\textstyle n</math> dimensions | ||
| - | of the data, and do not try to reduce its dimension. | + | of the data, and do not try to reduce its dimension. |
== Regularizaton == | == Regularizaton == | ||
When implementing PCA whitening or ZCA whitening in practice, sometimes some | When implementing PCA whitening or ZCA whitening in practice, sometimes some | ||
of the eigenvalues <math>\textstyle \lambda_i</math> will be numerically close to 0, and thus the scaling | of the eigenvalues <math>\textstyle \lambda_i</math> will be numerically close to 0, and thus the scaling | ||
| - | step | + | step where we divide by <math>\sqrt{\lambda_i}</math> would involve dividing by a value close to zero; this |
| - | the data to blow up (take on large values) or otherwise be numerically unstable. In practice, we | + | may cause the data to blow up (take on large values) or otherwise be numerically unstable. In practice, we |
| - | implement | + | therefore implement this scaling step using |
a small amount of regularization, and add a small constant <math>\textstyle \epsilon</math> | a small amount of regularization, and add a small constant <math>\textstyle \epsilon</math> | ||
to the eigenvalues before taking their square root and inverse: | to the eigenvalues before taking their square root and inverse: | ||
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\end{align}</math> | \end{align}</math> | ||
When <math>\textstyle x</math> takes values around <math>\textstyle [-1,1]</math>, a value of <math>\textstyle \epsilon \approx 10^{-5}</math> | When <math>\textstyle x</math> takes values around <math>\textstyle [-1,1]</math>, a value of <math>\textstyle \epsilon \approx 10^{-5}</math> | ||
| - | might be typical. | + | might be typical. |
For the case of images, adding <math>\textstyle \epsilon</math> here also has the effect of slightly smoothing (or low-pass | For the case of images, adding <math>\textstyle \epsilon</math> here also has the effect of slightly smoothing (or low-pass | ||
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performed by ZCA. This results in a less redundant representation of the input | performed by ZCA. This results in a less redundant representation of the input | ||
image, which is then transmitted to your brain. | image, which is then transmitted to your brain. | ||
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| + | {{PCA}} | ||
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| + | {{Languages|白化|中文}} | ||
