这是一篇来自美国的关于将您的编码分配命名为“assignment5.py”。使用提供的python模板文件，并仅完成函数（不要编辑函数定义、函数之外的代码，或使用任何其他库）的Python代写，以下是作业具体内容：

Problem 1: Principal Component Analysis. In this problem, we will process face images coming from the Yale Face Dataset: https://www.kaggle.com/datasets/olgabelitskaya/yale-face-database.

This dataset contains images of the faces of 15 individuals. For each individual there are 11 images taken under a variety of conditions e.g., the person makes a happy expression, wears glasses etc.

Download the dataset from the above URL.

(a) [Code] Processing the data and calculating eigen values.

(1) Fill in the function qa1 load, which takes the folder name input, and returns the data (as a tuple). Please use matplotlib.image.imread to read images.

(2) Fill in the function qa2 preprocess that performs a min max scaling on the faces (the X in dataset). Please use preprocessing.MinMaxScaler.

(3) Fill in the function qa3 calc eig val vec, given the dataset and integer k returns the k eigen vectors (PCA components) and the corresponding to the top k eigenvalues. Hint: use PCA already imported from sklearn.

(b) [Written + Code] Plot a curve displaying the first k eigenvalues λ1, …, λk i.e. the energy of the first K principal components. How many components do we need to capture 50% of the energy? Report the curve and the answer to the question in the report. Fill the function qb plot written used to generate the plot. DO NOT place your code in any other function.

(c) [Written + Code] PCA and Eigen Faces

(1) [Code] Fill in the function qc1 reshape images, that returns eigen faces, given the image dimensions, and PCA object. Note: Eigen faces are re-shaped eigen vectors in the shape of the original image.

(2) [Written + Code] Plot any 10 eigen faces for values of k = len(dataset) (as given in the starter code), and fill in the code qc2 plot. There is no specific format for plotting. Place the plots in the report.

(d) [Written + Code] Projection and Reconstruction

(1) [Code] Fill in the function qd1 project that takes the entire dataset and the PCA objects and projects it. Hint: Use PCA.tranform

(2) [Code] Fill in the function qd2 reconstruct that reconstructs the dataset given the projection (obtained from the previous function) and the fitted PCA object. Hint: use PCA.inverse transform.

(3) [Written + Code] Select a couple of images from the data. Use the first k eigenfaces as a basis to reconstruct the images (use functions written in previous sub-questions). Visualize the reconstructed images using 1, 10, 20, 30, 40, 50 components. How many components do we need to achieve a visually good result (report the plot and your answer in the report)? Use function qd3 visualize to complete this subquestion.

(e) Classification with SVM and Lasso regression post PCA on input data. We will also manually read your code for this question.

(1) [Code] Fill in the function qe1 svm that splits the input data into training and testing. Use as input features the transformed feature space that resulted from PCA. Experiment with a different number of PCA components through a 5-fold cross-validation. Uniformly sample components in range [10, 100] (with a gap of 20 for the sake of homework). User outer 5-fold cross-validation to build predictors using support vector machines (using radial basis function kernel). The function returns the best k across folds (average over folds), and the recognition accuracy on test set.

(2) [Code] Fill in the function qe2 lasso that splits the input data into training and testing. Use as input features the transformed feature space that resulted from PCA. Experiment with a different number of PCA components through a 5-fold cross-validation. Uniformly sample components in range [10, 100] (with a gap of 20 for the sake of homework). User outer 5-fold cross-validation to build predictors using lasso regression. The function returns the best k across folds (average over folds), and the recognition accuracy on test set.