Applications of Linear Algebra in AI
Exploring how linear algebra concepts are applied in various artificial intelligence and machine learning algorithms.
Linear algebra forms the mathematical foundation for many modern AI and machine learning techniques. This section explores practical applications and implementations of linear algebraic concepts in AI systems.
Neural Networks
Forward Propagation
- Matrix representation of layers
- Weight matrices and biases
- Activation functions
- Batch processing
Backpropagation
- Gradient computation
- Chain rule as matrix operations
- Weight updates
- Optimization techniques
Dimensionality Reduction
Principal Component Analysis
- Covariance matrix computation
- Eigendecomposition
- Feature selection
- Visualization techniques
Matrix Factorization
- SVD in recommender systems
- NMF for topic modeling
- Tensor decomposition
- Applications in deep learning
Computer Vision
Image Processing
- Image as matrices
- Convolution operations
- Feature extraction
- Transformation matrices
Face Recognition
- Eigenfaces
- Matrix similarity measures
- Dimensionality reduction
- Deep learning approaches
Natural Language Processing
Word Embeddings
- Word-context matrices
- Word2Vec implementation
- GloVe vectors
- Transformers and attention
Topic Modeling
- Document-term matrices
- LSA/LSI
- Matrix factorization
- Probabilistic approaches
Optimization in AI
Gradient Descent
- Matrix calculus
- Hessian matrices
- Newton's method
- Optimization algorithms
Regularization
- Matrix norms
- Sparse solutions
- Ridge regression
- LASSO implementation
Advanced Applications
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Quantum Computing
- Quantum states as vectors
- Unitary transformations
- Quantum algorithms
- Simulation
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Robotics
- Transformation matrices
- Inverse kinematics
- Path planning
- Control systems
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Graph Neural Networks
- Adjacency matrices
- Graph Laplacians
- Message passing
- Spectral methods