Singular Value Decomposition 편집하기

'''Singular Value Decomposition (SVD)''' is a mathematical technique used to decompose a matrix into three component matrices. It is widely used in data analysis, dimensionality reduction, machine learning, and signal processing.
==Definition==
SVD decomposes a matrix \( A \) into three matrices:
*'''U:''' An orthogonal matrix containing the left singular vectors.
*'''Σ (Sigma):''' A diagonal matrix with singular values sorted in descending order.
*'''V^T:''' An orthogonal matrix containing the right singular vectors.
The decomposition is represented as:
*'''A = U Σ V^T'''
==Steps in Singular Value Decomposition==
#Decompose the matrix into the matrices U, Σ, and V^T.
#Use the singular values (diagonal elements of Σ) to identify the significance of each component.
#Optionally, approximate the matrix by retaining only the top k singular values, effectively reducing its dimensions.
==Applications of SVD==
SVD is a versatile technique with applications in various fields:
*'''Dimensionality Reduction:''' Compresses data by retaining only the most significant singular values.
*'''Recommender Systems:''' Identifies latent factors in user-item interaction matrices for collaborative filtering.
*'''Image Compression:''' Reduces the size of image data while retaining key features.
*'''Noise Reduction:''' Filters out noise by ignoring small singular values.
*'''Natural Language Processing (NLP):''' Powers techniques like Latent Semantic Analysis (LSA) to find relationships between terms and documents.
==Example of SVD in Python==
A simple example using Python’s NumPy library:<syntaxhighlight lang="python">
import numpy as np

# Example matrix
A = np.array([[1, 2, 3],
              [4, 5, 6],
              [7, 8, 9]])

# Perform SVD
U, S, VT = np.linalg.svd(A)

print("U matrix:", U)
print("Singular values:", S)
print("V^T matrix:", VT)
</syntaxhighlight>
==Advantages==
*'''Efficient Data Representation:''' Retains essential patterns in data while reducing dimensionality.
*'''Robust to Noise:''' Ignores less significant components, effectively filtering noise.
*'''Wide Applicability:''' Used across multiple disciplines like machine learning, image processing, and NLP.
==Limitations==
*'''Computational Cost:''' SVD can be slow for very large matrices.
*'''Interpretability:''' Singular vectors and values may not always have clear meanings.
*'''Storage Requirements:''' Requires substantial memory for large datasets.
==Relation to PCA==
SVD is closely related to [[Principal Component Analysis (PCA)]]. In PCA:
*Singular values correspond to the square roots of the eigenvalues of the covariance matrix.
*Left singular vectors correspond to the principal components.
==Related Concepts and See Also==
*[[Principal Component Analysis]]
*[[Latent Semantic Analysis]]
*[[Matrix Decomposition]]
*[[Dimensionality Reduction]]
*[[Eigenvalues and Eigenvectors]]
*[[Recommender Systems]]
[[분류:Data Science]]