Linear Algebra:The Mathematical Foundation of AI Large Models
6/13/25...About 6 min
Linear Algebra: The Mathematical Foundation of AI Large Models
The application of linear algebra in AI large models is so widespread that almost every aspect is inseparable from it. Below, I will elaborate on the core concepts of linear algebra and their applications and functions in AI large models.
Core Concepts and Applications
1. Vectors
- Concept: A vector is the most basic element in linear algebra, which can be understood as a quantity with both direction and magnitude. In AI, vectors are often used to represent a single "point" or feature set in data.
- Mathematical representation: Usually as a column or row vector, e.g., .
- Applications and Functions:
- Data point representation: A word in text can be represented as a word vector (e.g., Word2Vec, GloVe, FastText), where each dimension represents a semantic feature. A pixel in an image can be represented as a vector of its RGB values.
- Feature vectors: Model inputs are often transformed into feature vectors, with each dimension corresponding to a feature. For example, a house's feature vector may include area, number of bedrooms, geographic location, etc.
- Word embeddings: This is the foundation of LLMs. By mapping words into high-dimensional vector space, similar words are closer together, capturing semantic relationships.
- Probability distributions: In classification problems, the model's output probabilities for each class are often represented as a vector, with each element representing the probability for a class.
- Direction and distance: The distance between vectors (e.g., Euclidean distance, cosine similarity) measures similarity between data points or features, which is crucial in recommendation systems, information retrieval, and clustering.
2. Matrices
- Concept: A matrix is a rectangular array of numbers arranged in rows and columns, which can be seen as a collection of vectors or as a representation of linear transformations.
- Mathematical representation:
- Applications and Functions:
- Dataset representation: A dataset is often represented as a matrix, with each row as a sample and each column as a feature.
- Neural network weights: The weights between layers in a neural network are stored as matrices. The input vector is multiplied by the weight matrix, which is the core of information propagation in neural networks.
- Image representation: A grayscale image can be directly represented as a matrix of pixel values, while a color image can be represented by multiple matrices (e.g., RGB channels) or tensors.
- Covariance matrix: In statistics and machine learning, the covariance matrix describes the linear relationships between different features in a dataset and is the basis for concepts like multivariate Gaussian distributions and PCA.
- Attention matrix: In Transformers, the core output of the attention mechanism is an attention weight matrix, representing the strength of relationships between different parts of the input sequence.
3. Tensors
- Concept: A tensor is a generalization of vectors and matrices. A 0th-order tensor is a scalar, a 1st-order tensor is a vector, and a 2nd-order tensor is a matrix. Higher-order tensors are used to represent more complex, multi-dimensional data.
- Mathematical representation: Usually as multi-dimensional arrays, e.g., a 3D tensor can be represented as .
- Applications and Functions:
- Multi-dimensional data representation: A color image is usually represented as a 3rd-order tensor (height × width × color channels). Video data can be a 4th-order tensor (frames × height × width × color channels).
- Data flow in deep learning: In deep learning frameworks (e.g., TensorFlow, PyTorch), all data and model parameters are operated and propagated as tensors.
- Batch data processing: During model training, data is often processed in batches. A batch of images can be organized as a 4th-order tensor (batch size × height × width × color channels).
4. Matrix Multiplication
- Concept: Matrix multiplication is one of the core operations in linear algebra, following the "row times column" rule.
- Mathematical representation: If , then .
- Applications and Functions:
- Neural network layer operations: This is the most basic computation in neural networks. Each layer's output is the result of multiplying the previous layer's activation matrix by the weight matrix, plus a bias. For example, in a fully connected layer, the input vector is transformed by , where is the weight matrix.
- Feature extraction: By multiplying with different weight matrices, different levels and types of features can be extracted from raw data.
- Transformations: Matrix multiplication can implement linear transformations of data, such as rotation, scaling, and projection, which are widely used in computer graphics and computer vision.
- Attention mechanism: In Transformers, Query, Key, and Value vectors are multiplied by matrices to compute attention scores and weighted values. For example, Q × K^T gives the attention score matrix.
5. Linear Transformations
- Concept: A linear transformation is a special function that maps vectors from one vector space to another while preserving vector addition and scalar multiplication. Every linear transformation can be represented by a matrix.
- Applications and Functions:
- Feature space mapping: Each layer in a neural network can be seen as a linear transformation (followed by a non-linear activation), mapping data from one feature space to another, more abstract or discriminative space.
- Dimensionality changes: By multiplying with matrices of different shapes, the dimensionality of data can be changed, such as dimensionality reduction or expansion.
- Data augmentation: In image processing, linear transformations (such as rotation, translation, scaling) are used for data augmentation, improving model generalization.
6. Eigenvalues and Eigenvectors
- Concept: For a square matrix , if there exists a nonzero vector and scalar such that , then is an eigenvalue of , and is the corresponding eigenvector. Eigenvectors remain in the same direction after the transformation, only scaled by .
- Applications and Functions:
- Principal Component Analysis (PCA): PCA is a classic application of linear algebra in dimensionality reduction. By computing the eigenvalues and eigenvectors of the covariance matrix, the directions of maximum variance (principal components) are found, allowing dimensionality reduction while retaining the most information.
- Spectral clustering: Some clustering algorithms use the eigenvectors of the graph Laplacian matrix for clustering.
- Data compression: Similar to PCA, important eigenvectors are retained to compress data.
7. Singular Value Decomposition (SVD)
- Concept: Any matrix can be decomposed as , where and are orthogonal matrices, and is a diagonal matrix with singular values on the diagonal.
- Applications and Functions:
- Dimensionality reduction: SVD is a more general dimensionality reduction method than PCA and can be applied to non-square matrices. By retaining the largest singular values and corresponding vectors, effective data dimensionality reduction and denoising can be achieved.
- Latent Semantic Analysis (LSA): In NLP, LSA uses SVD to discover latent semantic relationships in document-term matrices, useful for information retrieval and document classification.
- Recommendation systems: SVD is used in collaborative filtering recommendation systems to find latent factors in user-item rating matrices.
- Image compression: SVD can efficiently compress image data by retaining the main information.
- Pseudoinverse: SVD can be used to compute the pseudoinverse of a matrix, which is useful in solving least squares problems and under/overdetermined linear systems.
8. Gradient and Jacobian Matrix
- Concept:
- Gradient: For a multivariate function, the gradient is a vector pointing in the direction of the greatest increase of the function. In optimization, we usually move in the opposite direction of the gradient (gradient descent).
- Jacobian matrix: For a vector-valued function from -dimensional to -dimensional space, the Jacobian matrix consists of all partial derivatives and describes the local linear approximation of the function at a point.
- Applications and Functions:
- Backpropagation: The core algorithm for training deep learning models. Backpropagation essentially uses the chain rule to compute the gradient of the loss function with respect to model parameters. These gradients (usually high-dimensional vectors or matrices) guide parameter updates.
- Optimization algorithms: Gradient descent, Adam, RMSProp, etc., all rely on gradient computation and updates. Linear algebra provides the tools for computing and manipulating these gradients.
- Automatic differentiation: Modern deep learning frameworks (e.g., TensorFlow, PyTorch) have efficient automatic differentiation, which can automatically compute gradients for complex functions (like neural networks), and their underlying implementation relies heavily on linear algebra rules.
9. Least Squares Method
- Concept: The least squares method is an optimization technique used to find a set of parameters that minimize the sum of squared differences between predicted and observed values.
- Applications and Functions:
- Linear regression: In simple linear regression, the least squares method is used to find the best-fit line's parameters (weights and bias).
- Solving underdetermined/overdetermined systems: When a linear system has no unique solution, the least squares method provides the "best approximate solution."
- Model fitting: In many machine learning models, parameter estimation boils down to a least squares or its variant.
Summary
Linear algebra is an indispensable foundation for AI large models. It provides:
- Data representation and structuring: Vectors, matrices, and tensors are the universal language for representing all AI data.
- Core operations and transformations: Matrix multiplication, addition, transposition, etc., form the basic operations of neural networks and various AI algorithms.
- Organization and updating of model parameters: Model weights and biases are matrices and vectors, and their training (gradient descent) relies on linear algebra operations.
- Feature extraction and dimensionality reduction: Techniques like PCA and SVD use linear algebra principles to extract meaningful features and reduce dimensionality, improving efficiency.
- Efficient computation: The high parallelism of linear algebra operations allows full use of hardware like GPUs, accelerating model training and inference.
- Theoretical foundation: Many complex AI algorithms and model architectures are deeply rooted in linear algebra principles.