Sparse SVD (svds) in SciPy - Mini Project: Build & Apply

Sparse SVD with svds from SciPy

📖 Scenario: You work as a data analyst for a movie streaming service. You have a large matrix showing user ratings for movies, but most users have rated only a few movies, so the matrix is mostly empty (sparse). You want to find patterns in this data using a technique called Sparse Singular Value Decomposition (Sparse SVD).

🎯 Goal: Build a Python program that creates a sparse matrix of user ratings, configures the number of singular values to find, applies Sparse SVD using svds from SciPy, and prints the singular values.

📋 What You'll Learn

Create a sparse matrix using scipy.sparse.csr_matrix with given data

Set a variable k for the number of singular values to compute

Use svds from scipy.sparse.linalg to compute the sparse SVD

Print the singular values array

💡 Why This Matters

🌍 Real World

Sparse SVD is used in recommendation systems to find hidden patterns in large, sparse user-item rating data.

💼 Career

Data scientists and machine learning engineers use sparse matrix decompositions to reduce data size and improve model performance.

1

Create a sparse matrix of user ratings

Create a sparse matrix called ratings using scipy.sparse.csr_matrix with the exact data: rows = [0, 0, 1, 2, 2], cols = [0, 2, 2, 0, 1], and data = [5, 3, 4, 1, 2].

SciPy

from scipy.sparse import csr_matrix
# Create the sparse matrix 'ratings' with given rows, cols, and data
# Your code here

Hint

Use csr_matrix((data, (rows, cols)), shape=(3, 3)) to create the sparse matrix.

2

Set the number of singular values to compute

Create a variable called k and set it to 2 to specify the number of singular values to compute.

SciPy

from scipy.sparse import csr_matrix
rows = [0, 0, 1, 2, 2]
cols = [0, 2, 2, 0, 1]
data = [5, 3, 4, 1, 2]
ratings = csr_matrix((data, (rows, cols)), shape=(3, 3))
# Set the variable 'k' to 2
# Your code here

Hint

Just write k = 2 to set the number of singular values.

3

Compute Sparse SVD using svds

Import svds from scipy.sparse.linalg and use it to compute the sparse SVD of ratings with k singular values. Store the results in variables u, s, and vt.

SciPy

from scipy.sparse import csr_matrix
rows = [0, 0, 1, 2, 2]
cols = [0, 2, 2, 0, 1]
data = [5, 3, 4, 1, 2]
ratings = csr_matrix((data, (rows, cols)), shape=(3, 3))
k = 2
# Import svds and compute u, s, vt from ratings with k
# Your code here

Hint

Use from scipy.sparse.linalg import svds and then u, s, vt = svds(ratings, k=k).

4

Print the singular values

Write a print statement to display the singular values stored in the variable s.

SciPy

from scipy.sparse import csr_matrix
from scipy.sparse.linalg import svds
rows = [0, 0, 1, 2, 2]
cols = [0, 2, 2, 0, 1]
data = [5, 3, 4, 1, 2]
ratings = csr_matrix((data, (rows, cols)), shape=(3, 3))
k = 2
u, s, vt = svds(ratings, k=k)
# Print the singular values 's'
# Your code here

Hint

Use print(s) to show the singular values.

Practice

1. What is the main purpose of using svds from scipy.sparse.linalg in data science?

easy

A. To efficiently compute singular value decomposition on large sparse matrices

B. To perform dense matrix multiplication

C. To sort data in ascending order

D. To calculate the determinant of a matrix

Solution

1. Step 1: Understand the function purpose

   svds is designed for sparse matrices, which are mostly empty, to find singular values and vectors efficiently.

2. Step 2: Compare options with function use

   Options A, B, and C describe unrelated matrix operations. Only To efficiently compute singular value decomposition on large sparse matrices matches the purpose of svds.

3. Final Answer:

   To efficiently compute singular value decomposition on large sparse matrices -> Option A

4. Quick Check:

   svds = sparse SVD computation [OK]

Hint: Remember svds is for sparse matrices, not dense operations [OK]

Common Mistakes:

• Confusing svds with dense SVD functions
• Thinking svds sorts or multiplies matrices
• Assuming svds calculates determinants

2. Which of the following is the correct way to import the svds function from SciPy?

easy

A. from scipy.sparse.linalg import svds

B. import svds from scipy.linalg

C. from scipy.linalg import svds

D. import svds from scipy.sparse

Solution

1. Step 1: Identify the correct module for svds

   The svds function is part of scipy.sparse.linalg, which handles sparse linear algebra.

2. Step 2: Check import syntax

   Python import syntax requires 'from module import function'. from scipy.sparse.linalg import svds matches this correctly.

3. Final Answer:

   from scipy.sparse.linalg import svds -> Option A

4. Quick Check:

   Correct import syntax = from scipy.sparse.linalg import svds [OK]

Hint: Use 'from scipy.sparse.linalg import svds' to import correctly [OK]

Common Mistakes:

• Using wrong module like scipy.linalg instead of sparse.linalg
• Incorrect import syntax like 'import svds from ...'
• Importing from scipy.sparse which lacks svds

3. Given the following code, what will be the shape of the matrix U returned by svds?

import numpy as np
from scipy.sparse.linalg import svds
from scipy.sparse import csr_matrix

A = csr_matrix(np.array([[1, 0, 0], [0, 2, 0], [0, 0, 3]]))
U, S, Vt = svds(A, k=2)

medium

A. (3, 3)

B. (3, 2)

C. (2, 3)

D. (2, 2)

Solution

1. Step 1: Understand svds output shapes

   For an input matrix of shape (m, n) and parameter k, svds returns U with shape (m, k), S with length k, and Vt with shape (k, n).

2. Step 2: Apply to given matrix

   Matrix A is 3x3, k=2, so U shape is (3, 2).

3. Final Answer:

   (3, 2) -> Option B

4. Quick Check:

   U shape = (rows, k) = (3, 2) [OK]

Hint: U shape is (rows, k) where k is number of singular values [OK]

Common Mistakes:

• Confusing U shape with Vt shape
• Assuming U is square matrix
• Mixing up k with matrix dimensions

4. What is wrong with the following code snippet that tries to compute sparse SVD?

from scipy.sparse.linalg import svds
import numpy as np

A = np.array([[1, 0], [0, 1]])
U, S, Vt = svds(A, k=1)

medium

A. svds does not return three outputs

B. Parameter k cannot be 1

C. Matrix A is not a sparse matrix

D. Import statement is incorrect

Solution

1. Step 1: Check matrix type requirement

   svds expects a sparse matrix input, but A is a dense numpy array.

2. Step 2: Validate other parts

   Parameter k=1 is valid, svds returns three outputs, and import is correct. So only matrix type is wrong.

3. Final Answer:

   Matrix A is not a sparse matrix -> Option C

4. Quick Check:

   Input must be sparse matrix [OK]

Hint: Convert dense arrays to sparse before svds [OK]

Common Mistakes:

• Passing dense numpy arrays directly to svds
• Thinking k=1 is invalid
• Misunderstanding svds output count

5. You have a large sparse user-item rating matrix with shape (10000, 5000). You want to reduce its dimensionality to 50 features using svds. Which of the following code snippets correctly performs this and returns the reduced user features matrix?

hard

A. from scipy.sparse.linalg import svds
U, S, Vt = svds(ratings_sparse, k=50)
user_features = np.diag(S) @ Vt

B. from scipy.sparse.linalg import svds
U, S, Vt = svds(ratings_sparse, k=50)
user_features = Vt.T @ np.diag(S)

C. from scipy.linalg import svd
U, S, Vt = svd(ratings_sparse)
user_features = U[:, :50]

D. from scipy.sparse.linalg import svds
U, S, Vt = svds(ratings_sparse, k=50)
user_features = U @ np.diag(S)

Solution

1. Step 1: Understand svds output and dimensionality reduction

   svds returns U (users x k), S (k,), and Vt (k x items). Multiplying U by diag(S) gives user features in reduced space.

2. Step 2: Analyze options for correct user features

   from scipy.sparse.linalg import svds U, S, Vt = svds(ratings_sparse, k=50) user_features = U @ np.diag(S) correctly computes user_features = U @ diag(S). from scipy.sparse.linalg import svds U, S, Vt = svds(ratings_sparse, k=50) user_features = np.diag(S) @ Vt mixes user and item matrices. from scipy.linalg import svd U, S, Vt = svd(ratings_sparse) user_features = U[:, :50] uses dense svd, not sparse. from scipy.sparse.linalg import svds U, S, Vt = svds(ratings_sparse, k=50) user_features = Vt.T @ np.diag(S) computes item features, not user features.

3. Final Answer:

   from scipy.sparse.linalg import svds U, S, Vt = svds(ratings_sparse, k=50) user_features = U @ np.diag(S) -> Option D

4. Quick Check:

   User features = U * S diagonal [OK]

Hint: Multiply U by diag(S) for user features after svds [OK]

Common Mistakes:

• Using Vt for user features instead of U
• Using dense svd on sparse data
• Not multiplying U by singular values