PCA
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allows us to visualize the algorithms better.) Here is our dataset: | allows us to visualize the algorithms better.) Here is our dataset: | ||
| - | + | [[File:PCA-rawdata.png|600px]] | |
This data has already been pre-processed so that each of the features <math>\textstyle x_1</math> and <math>\textstyle x_2</math> | This data has already been pre-processed so that each of the features <math>\textstyle x_1</math> and <math>\textstyle x_2</math> | ||
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variation of the data, and <math>\textstyle u_2</math> the secondary direction of variation: | variation of the data, and <math>\textstyle u_2</math> the secondary direction of variation: | ||
| - | + | [[File:PCA-u1.png | 600px]] | |
I.e., the data varies much more in the direction <math>\textstyle u_1</math> than <math>\textstyle u_2</math>. | I.e., the data varies much more in the direction <math>\textstyle u_1</math> than <math>\textstyle u_2</math>. | ||
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as follows: | as follows: | ||
:<math>\begin{align} | :<math>\begin{align} | ||
| - | \Sigma = \sum_{i=1}^m (x^{(i)})(x^{(i)})^T. | + | \Sigma = \frac{1}{m} \sum_{i=1}^m (x^{(i)})(x^{(i)})^T. |
\end{align}</math> | \end{align}</math> | ||
| - | If <math>\textstyle x</math> has zero mean, then <math>\textstyle | + | If <math>\textstyle x</math> has zero mean, then <math>\textstyle \Sigma</math> is exactly the covariance matrix of <math>\textstyle x</math>. (The symbol "<math>\textstyle \Sigma</math>", pronounced "Sigma", is the standard notation for denoting the covariance matrix. Unfortunately it looks just like the summation symbol, as in <math>\sum_{i=1}^n i</math>; but these are two different things.) |
It can then be shown that <math>\textstyle u_1</math>---the principal direction of variation of the data---is | It can then be shown that <math>\textstyle u_1</math>---the principal direction of variation of the data---is | ||
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the second eigenvector. | the second eigenvector. | ||
| - | + | Note: If you are interested in seeing a more formal mathematical derivation/justification of this result, see the CS229 (Machine Learning) lecture notes on PCA (link at bottom of this page). You won't need to do so to follow along this course, however. | |
You can use standard numerical linear algebra software to find these eigenvectors (see Implementation Notes). | You can use standard numerical linear algebra software to find these eigenvectors (see Implementation Notes). | ||
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<math>\textstyle x_{\rm rot}</math>, we get: | <math>\textstyle x_{\rm rot}</math>, we get: | ||
| - | + | [[File:PCA-rotated.png|600px]] | |
This is the training set rotated into the <math>\textstyle u_1</math>,<math>\textstyle u_2</math> basis. In the general | This is the training set rotated into the <math>\textstyle u_1</math>,<math>\textstyle u_2</math> basis. In the general | ||
case, <math>\textstyle U^Tx</math> will be the training set rotated into the basis | case, <math>\textstyle U^Tx</math> will be the training set rotated into the basis | ||
| - | <math>\textstyle u_1</math>,<math>\textstyle u_2</math>, | + | <math>\textstyle u_1</math>,<math>\textstyle u_2</math>, ...,<math>\textstyle u_n</math>. |
| - | One of the properties of <math>\textstyle U</math> is that it is an orthogonal | + | One of the properties of <math>\textstyle U</math> is that it is an "orthogonal" matrix, which means |
| - | So if you ever need to go | + | that it satisfies <math>\textstyle U^TU = UU^T = I</math>. |
| + | So if you ever need to go from the rotated vectors <math>\textstyle x_{\rm rot}</math> back to the | ||
original data <math>\textstyle x</math>, you can compute | original data <math>\textstyle x</math>, you can compute | ||
:<math>\begin{align} | :<math>\begin{align} | ||
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because <math>\textstyle U x_{\rm rot} = UU^T x = x</math>. | because <math>\textstyle U x_{\rm rot} = UU^T x = x</math>. | ||
| - | == Reducing the Data == | + | == Reducing the Data Dimension == |
We see that the principal direction of variation of the data is the first | We see that the principal direction of variation of the data is the first | ||
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In our example, this gives us the following plot of <math>\textstyle \tilde{x}</math> (using <math>\textstyle n=2, k=1</math>): | In our example, this gives us the following plot of <math>\textstyle \tilde{x}</math> (using <math>\textstyle n=2, k=1</math>): | ||
| - | + | [[File:PCA-xtilde.png | 600px]] | |
However, since the final <math>\textstyle n-k</math> components of <math>\textstyle \tilde{x}</math> as defined above would | However, since the final <math>\textstyle n-k</math> components of <math>\textstyle \tilde{x}</math> as defined above would | ||
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Now, <math>\textstyle \tilde{x} \in \Re^k</math> is a lower-dimensional, "compressed" representation | Now, <math>\textstyle \tilde{x} \in \Re^k</math> is a lower-dimensional, "compressed" representation | ||
of the original <math>\textstyle x \in \Re^n</math>. Given <math>\textstyle \tilde{x}</math>, how can we recover an approximation <math>\textstyle \hat{x}</math> to | of the original <math>\textstyle x \in \Re^n</math>. Given <math>\textstyle \tilde{x}</math>, how can we recover an approximation <math>\textstyle \hat{x}</math> to | ||
| - | the original value of <math>\textstyle x</math>? From | + | the original value of <math>\textstyle x</math>? From an [[#Rotating the Data|earlier section]], we know that <math>\textstyle x = U x_{\rm rot}</math>. Further, |
we can think of <math>\textstyle \tilde{x}</math> as an approximation to <math>\textstyle x_{\rm rot}</math>, where we have | we can think of <math>\textstyle \tilde{x}</math> as an approximation to <math>\textstyle x_{\rm rot}</math>, where we have | ||
set the last <math>\textstyle n-k</math> components to zeros. Thus, given <math>\textstyle \tilde{x} \in \Re^k</math>, we can | set the last <math>\textstyle n-k</math> components to zeros. Thus, given <math>\textstyle \tilde{x} \in \Re^k</math>, we can | ||
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= \sum_{i=1}^k u_i \tilde{x}_i. | = \sum_{i=1}^k u_i \tilde{x}_i. | ||
\end{align}</math> | \end{align}</math> | ||
| - | The final equality above comes from the definition of <math>\textstyle U</math> given | + | The final equality above comes from the definition of <math>\textstyle U</math> [[#Example and Mathematical Background|given earlier]]. |
(In a practical implementation, we wouldn't actually zero pad <math>\textstyle \tilde{x}</math> and then multiply | (In a practical implementation, we wouldn't actually zero pad <math>\textstyle \tilde{x}</math> and then multiply | ||
by <math>\textstyle U</math>, since that would mean multiplying a lot of things by zeros; instead, we'd just | by <math>\textstyle U</math>, since that would mean multiplying a lot of things by zeros; instead, we'd just | ||
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Applying this to our dataset, we get the following plot for <math>\textstyle \hat{x}</math>: | Applying this to our dataset, we get the following plot for <math>\textstyle \hat{x}</math>: | ||
| - | + | [[File:PCA-xhat.png | 600px]] | |
We are thus using a 1 dimensional approximation to the original dataset. | We are thus using a 1 dimensional approximation to the original dataset. | ||
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approximation to the data. | approximation to the data. | ||
| - | To decide how to set <math>\textstyle k</math>, we will usually look at the | + | To decide how to set <math>\textstyle k</math>, we will usually look at the '''percentage of variance retained''' |
| - | + | for different values of <math>\textstyle k</math>. Concretely, if <math>\textstyle k=n</math>, then we have | |
| - | an exact approximation to the data, and we say that 100 | + | an exact approximation to the data, and we say that 100% of the variance is |
retained. I.e., all of the variation of the original data is retained. | retained. I.e., all of the variation of the original data is retained. | ||
Conversely, if <math>\textstyle k=0</math>, then we are approximating all the data with the zero vector, | Conversely, if <math>\textstyle k=0</math>, then we are approximating all the data with the zero vector, | ||
| - | and thus 0 | + | and thus 0% of the variance is retained. |
More generally, let <math>\textstyle \lambda_1, \lambda_2, \ldots, \lambda_n</math> be the eigenvalues | More generally, let <math>\textstyle \lambda_1, \lambda_2, \ldots, \lambda_n</math> be the eigenvalues | ||
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In our simple 2D example above, <math>\textstyle \lambda_1 = 7.29</math>, and <math>\textstyle \lambda_2 = 0.69</math>. Thus, | In our simple 2D example above, <math>\textstyle \lambda_1 = 7.29</math>, and <math>\textstyle \lambda_2 = 0.69</math>. Thus, | ||
by keeping only <math>\textstyle k=1</math> principal components, we retained <math>\textstyle 7.29/(7.29+0.69) = 0.913</math>, | by keeping only <math>\textstyle k=1</math> principal components, we retained <math>\textstyle 7.29/(7.29+0.69) = 0.913</math>, | ||
| - | or 91.3 | + | or 91.3% of the variance. |
A more formal definition of percentage of variance retained is beyond the scope | A more formal definition of percentage of variance retained is beyond the scope | ||
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and for which we would incur a greater approximation error if we were to set them to zero. | and for which we would incur a greater approximation error if we were to set them to zero. | ||
| - | In the case of images, one common heuristic is to choose <math>\textstyle k</math> so as to retain 99 | + | In the case of images, one common heuristic is to choose <math>\textstyle k</math> so as to retain 99% of |
the variance. In other words, we pick the smallest value of <math>\textstyle k</math> that satisfies | the variance. In other words, we pick the smallest value of <math>\textstyle k</math> that satisfies | ||
:<math>\begin{align} | :<math>\begin{align} | ||
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\end{align}</math> | \end{align}</math> | ||
Depending on the application, if you are willing to incur some | Depending on the application, if you are willing to incur some | ||
| - | additional error, values in the 90-98 | + | additional error, values in the 90-98% range are also sometimes used. When you |
| - | describe to others how you applied PCA, saying that you chose <math>\textstyle k</math> to retain 95 | + | describe to others how you applied PCA, saying that you chose <math>\textstyle k</math> to retain 95% of |
the variance will also be a much more easily interpretable description than saying | the variance will also be a much more easily interpretable description than saying | ||
that you retained 120 (or whatever other number of) components. | that you retained 120 (or whatever other number of) components. | ||
| + | == PCA on Images == | ||
For PCA to work, usually we want each of the features <math>\textstyle x_1, x_2, \ldots, x_n</math> | For PCA to work, usually we want each of the features <math>\textstyle x_1, x_2, \ldots, x_n</math> | ||
to have a similar range of values to the others (and to have a mean close to | to have a similar range of values to the others (and to have a mean close to | ||
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suppose we are training our algorithm on '''natural images''', so that <math>\textstyle x_j</math> is | suppose we are training our algorithm on '''natural images''', so that <math>\textstyle x_j</math> is | ||
the value of pixel <math>\textstyle j</math>. By "natural images," we informally mean the type of image that | the value of pixel <math>\textstyle j</math>. By "natural images," we informally mean the type of image that | ||
| - | a typical animal or person might see over their lifetime. | + | a typical animal or person might see over their lifetime. |
| - | images of outdoor scenes with grass, trees, etc., and cut out small (say 16x16) image | + | |
| - | patches randomly from these to train the algorithm. But in practice most | + | Note: Usually we use images of outdoor scenes with grass, trees, etc., and cut out small (say 16x16) image patches randomly from these to train the algorithm. But in practice most feature learning algorithms are extremely robust to the exact type of image it is trained on, so most images taken with a normal camera, so long as they aren't excessively blurry or have strange artifacts, should work. |
| - | feature learning algorithms are extremely robust to the exact type of image | + | |
| - | it is trained on, so most images taken with a normal camera, so long as they | + | When training on natural images, it makes little sense to estimate a separate mean and |
| - | aren't excessively blurry or have strange artifacts, should work. | + | |
| - | + | ||
variance for each pixel, because the statistics in one part | variance for each pixel, because the statistics in one part | ||
of the image should (theoretically) be the same as any other. | of the image should (theoretically) be the same as any other. | ||
| - | This property of images is called '''stationarity''' | + | This property of images is called '''stationarity.''' |
In detail, in order for PCA to work well, informally we require that (i) The | In detail, in order for PCA to work well, informally we require that (i) The | ||
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a 16x16 image patch (<math>\textstyle n=256</math>), we might normalize the intensity of each image | a 16x16 image patch (<math>\textstyle n=256</math>), we might normalize the intensity of each image | ||
<math>\textstyle x^{(i)}</math> as follows: | <math>\textstyle x^{(i)}</math> as follows: | ||
| - | + | ||
| - | \mu^{(i)} | + | <math>\mu^{(i)} := \frac{1}{n} \sum_{j=1}^n x^{(i)}_j</math> |
| - | x^{(i)}_j | + | |
| - | + | <math>x^{(i)}_j := x^{(i)}_j - \mu^{(i)}</math>, for all <math>\textstyle j</math> | |
| + | |||
Note that the two steps above are done separately for each image <math>\textstyle x^{(i)}</math>, | Note that the two steps above are done separately for each image <math>\textstyle x^{(i)}</math>, | ||
and that <math>\textstyle \mu^{(i)}</math> here is the mean intensity of the image <math>\textstyle x^{(i)}</math>. In particular, | and that <math>\textstyle \mu^{(i)}</math> here is the mean intensity of the image <math>\textstyle x^{(i)}</math>. In particular, | ||
| - | this is not the same thing as estimating a mean value separately for each pixel <math>\textstyle x_j</math>. | + | this is not the same thing as estimating a mean value separately for each pixel <math>\textstyle x_j</math>. |
| - | If you are training your algorithm on images other than natural images (for | + | If you are training your algorithm on images other than natural images (for example, images of handwritten characters, or images of single isolated objects centered against a white background), other types of normalization might be worth considering, and the best choice may be application dependent. But when training on natural images, using the per-image mean normalization method as given in the equations above would be a reasonable default. |
| - | example, images of handwritten characters, or images of single isolated objects | + | |
| - | centered against a white background), other types of normalization might be | + | |
| - | worth considering, and the best choice may be application dependent. | + | |
| - | when training on natural images, using the per-image mean normalization | + | |
| - | as in | + | |
| - | would be a reasonable default. | + | |
== References == | == References == | ||
http://cs229.stanford.edu | http://cs229.stanford.edu | ||
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| + | {{PCA}} | ||
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| + | {{Languages|主成分分析|中文}} | ||
