Dimensionality Reduction Using PCA

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Project due Jul $10,202417$:$29$ IST

PCA finds $($orthogonal$)$ directions of maximal variation in the data. In this problem we're going to project our

data onto the principal components and explore the effects on performance.

You will be working in the files part$1/$main$.$ py and part$1/$features $.$ py in this problem

Project onto Principal Components

$\frac{3\mathrm{.}0}{3\mathrm{.}0}$ points $($graded$)$

Fill in function

in

features.py that implements PCA dimensionality reduction of dataset $x.$

Note that to project a given $nd$ dataset $x$ into its $k-$dimensional PCA representation, one can use matrix

multiplication, after first centering $x$ :

widetilde$(x)V$

where widetilde$(x)$ is the centered version of the original data $x$ using the mean learned from training data and $V$ is

the $dk$ matrix whose columns are the top $k$ eigenvectors of widetilde$(x{)}^T$widetilde$(x).$ This is because the eigenvectors are

of unit$-$norm, so there is no need to divide by their length.

Function input:: You are given the full principal component matrix $V^{\text{'}}$ as pcs and the features mean

computed from the training data set as feature$\_$means in this function. Note that pcs and feature$\_$means

are learned from the training data set, which should not be computed in this function using $x.$

Available Functions: You have access to the NumPy python library as np$.$n$=$nthis function returns a new data array in which each sample in $\}$nmncentered$\_\}$X$=$x$\backslash$mathrm$\{-$ feature$\_$means