Create two new variables trainnmat and testnmat which contain the L2-normalized versions of train and test respectively. L2-normalization should be performed across each row r of the train and test matrices, so that:
Vr^21 +r^22+..+r^2n = 1 However, you must compute the L2-normalized matrices using numpy and scipy.sparse operations, using the scikit-learn normalize function to generate your answer is not valid. Also ensure that your output matrix is still sparse. Many Numpy operations will either fail or convert CSR matrices to dense matrices. We prefer to keep this matrix in CSR format for the duration of this lab. The documentation for the following sparse matrix function may useful for you: • scipy.sparse.csr_matrix Docs • scipy.sparse.csr_matrix.sum Docs
• scipy.sparse.csr_matrix. power Docs • scipy.sparse.spdiags Docs Above we have already imported scipy.sparse as sp HINT: For both train and test you need to structure your normalization as a matrix multiply operation. Start by creating a diagonal matrix where the squared sums of each row are along the diagonal elements. [46]: [47]: trainnmat - testnmat = |

The results in the L2-normalized versions of the train and test matrices, stored in the trainnmat and testnmat variables, respectively. These matrices should still be in CSR format, as requested.

Here is a possible implementation of the requested code:

import numpy as np

import scipy.sparse as sp

# assuming train and test are already defined and are sparse CSR matrices

train_sum = sp.csr_matrix(np.array(train.power(2).sum(axis=1)).flatten())

test_sum = sp.csr_matrix(np.array(test.power(2).sum(axis=1)).flatten())

train_diag = sp.spdiags(1 / np.sqrt(train_sum), 0, train.shape[0], train.shape[0])

test_diag = sp.spdiags(1 / np.sqrt(test_sum), 0, test.shape[0], test.shape[0])

trainnmat = train_diag.dot(train)

testnmat = test_diag.dot(test)

Explanation:

First, we calculate the L2 norm of each row of the train and test matrices by summing their squared values using the power and sum methods of sparse matrices, and flattening the resulting array using NumPy's flatten function. The power method squares all elements of the matrix, while the sum method sums the elements along the rows, resulting in a 1D array of row sums.

Then, we create diagonal matrices where the diagonal elements are the reciprocal of the square root of the row sums, as per the L2 normalization formula. We use the spdiags function from SciPy to create sparse diagonal matrices, specifying the diagonal offset as 0 and the matrix dimensions as the number of rows in the original matrices.

Finally, we perform a matrix multiplication between each diagonal matrix and its respective original matrix, using the dot method of the sparse matrices. This results in the L2-normalized versions of the train and test matrices, stored in the trainnmat and testnmat variables, respectively. These matrices should still be in CSR format, as requested.

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To create the L2-normalized versions of the train and test matrices using sparse operations in NumPy and SciPy, you can follow the steps below:

import numpy as np

from scipy.sparse import csr_matrix, spdiags

# Compute L2-normalized train matrix

squared_sum_train = np.square(train).sum(axis=1)  # Compute the squared sum of each row

diag_train = spdiags(1 / np.sqrt(squared_sum_train), 0, train.shape[0], train.shape[0])  # Create diagonal matrix

trainnmat = csr_matrix(diag_train train)  # Perform matrix multiplication

# Compute L2-normalized test matrix

squared_sum_test = np.square(test).sum(axis=1)  # Compute the squared sum of each row

diag_test = spdiags(1 / np.sqrt(squared_sum_test), 0, test.shape[0], test.shape[0])  # Create diagonal matrix

testnmat = csr_matrix(diag_test test)  # Perform matrix multiplication

The initial step of the code involves calculating the sum of squares of every row in both the train and test matrices, utilizing np.square and np.sum while taking into account the relevant axis.

Next, diag_train and diag_test are formed using spdiags, wherein the diagonal cells are populated with the inverted square root of the sum of squares.

In conclusion, we utilize the "at" symbol to conduct matrix multiplication on diag_train (or diag_test) and matched matrix (train or test), which results in obtaining trainnmat and testnmat - the sparse matrices that have been L2-normalized.

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