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# tft.pca

Computes PCA on the dataset using biased covariance.

The PCA analyzer computes output_dim orthonormal vectors that capture directions/axes corresponding to the highest variances in the input vectors of `x`. The output vectors are returned as a rank-2 tensor with shape `(input_dim, output_dim)`, where the 0th dimension are the components of each output vector, and the 1st dimension are the output vectors representing orthogonal directions in the input space, sorted in order of decreasing variances.

The output rank-2 tensor (matrix) serves a useful transform purpose. Formally, the matrix can be used downstream in the transform step by multiplying it to the input tensor `x`. This transform reduces the dimension of input vectors to output_dim in a way that retains the maximal variance.

Below are a couple intuitive examples of PCA.

Consider a simple 2-dimensional example:

Input x is a series of vectors `[e, e]` where `e` is Gaussian with mean 0, variance 1. The two components are perfectly correlated, and the resulting covariance matrix is

``````[[1 1],
[1 1]].
``````

Applying PCA with `output_dim = 1` would discover the first principal component `[1 / sqrt(2), 1 / sqrt(2)]`. When multipled to the original example, each vector `[e, e]` would be mapped to a scalar `sqrt(2) * e`. The second principal component would be `[-1 / sqrt(2), 1 / sqrt(2)]` and would map `[e, e]` to 0, which indicates that the second component captures no variance at all. This agrees with our intuition since we know that the two axes in the input are perfectly correlated and can be fully explained by a single scalar `e`.

Consider a 3-dimensional example:

Input `x` is a series of vectors `[a, a, b]`, where `a` is a zero-mean, unit variance Gaussian and `b` is a zero-mean, variance 4 Gaussian and is independent of `a`. The first principal component of the unnormalized vector would be `[0, 0, 1]` since `b` has a much larger variance than any linear combination of the first two components. This would map `[a, a, b]` onto `b`, asserting that the axis with highest energy is the third component. While this may be the desired output if `a` and `b` correspond to the same units, it is not statistically desireable when the units are irreconciliable. In such a case, one should first normalize each component to unit variance first, i.e. `b := b / 2`. The first principal component of a normalized vector would yield `[1 / sqrt(2), 1 / sqrt(2), 0]`, and would map `[a, a, b]` to `sqrt(2) * a`. The second component would be `[0, 0, 1]` and map `[a, a, b]` to `b`. As can be seen, the benefit of normalization is that PCA would capture highly correlated components first and collapse them into a lower dimension.

`x` A rank-2 `Tensor`, 0th dim are rows, 1st dim are indices in row vectors.
`output_dim` The PCA output dimension (number of eigenvectors to return).
`dtype` Tensorflow dtype of entries in the returned matrix.
`name` (Optional) A name for this operation.

`ValueError` if input is not a rank-2 Tensor.

A 2D `Tensor` (matrix) M of shape (input_dim, output_dim).