What is linear algebra for machine learning?

Powerful modern machine learning algorithms and generative AI, at their core, are powered by linear algebra. Whether training a neural network , building a recommendation system or applying principal component analysis (PCA) to a complex and high-dimensional dataset, practitioners are using linear algebra to perform massive calculations.

From its earliest days to recent advancements in deep learning, linear algebra has been ubiquitous in the ML landscape. Many core machine learning models are fundamentally expressed and solved using linear algebra principles. In practice, data is rarely a simple, single number; instead, data often comes in the form of datasets: collections of often messy data points. Linear algebra provides the tools to organize, manipulate and analyze this data efficiently.

It allows practitioners to manipulate objects like vectors, matrices and tensors to represent structured (often tabular data) and unstructured data like images or videos. These seemingly abstract concepts are the language of data for computer science and data scientists. For instance, an image can be represented as a matrix of pixel values, and a collection of features describing a house (such as neighborhood, age and square footage) can be represented as a vector in a linear regression model. Linear regression models the output as a linear combination of the input features, serving as a classic example of how linear algebra works in the real world.  

In machine learning and data science, linear algebra is the framework used to describe and work with data. It explains how numbers are arranged, combined and transformed—whether that’s multiplying matrices in a neural network, finding eigenvalues in PCA or reducing dimensions with singular value decomposition (SVD).  

At its most basic level, linear algebra gives the tools to represent and work with data in structured forms. Most machine learning workflows start by organizing data into numerical formats, and each structure—scalar, vector, matrix and tensor—serves a different purpose.

• A scalar is the simplest building block, which is a single numerical value, like 5 or 2.3. Scalars often represent parameters, scaling factors or single measurements.
  
  
• A vector is an ordered array of numbers, usually written as a column or row. Vectors can represent anything from a list of features describing a single data point to the coordinates of a position in space. For example, the vector [3,5,7] might represent the number of visits, purchases and returns for a customer.
  
  
• A matrix is a two-dimensional array of numbers arranged in rows and columns. A dataset where each row is a data point and each column is a feature naturally forms a matrix. Matrices are central to linear algebra because they allow for efficient storage of data. Operations like scalar multiplication (multiplying every element of a matrix by a constant number) and matrix multiplication (combining two matrices to apply a transformation or compute relationships) are pervasive in algorithms.
  
  
• A tensor is a generalization of scalars, vectors and matrices to higher dimensions. For instance, a color image might be stored as a 3D tensor where height, width and color channels form three separate axes. In deep learning, tensors are the standard data structure for feeding information into neural networks.

The dot product is a way to multiply two vectors to produce a single scalar. It is widely used to calculate similarities between vectors, which is a crucial step in many recommendation systems. The transpose of a matrix, which flips its rows and columns, is another fundamental operation that enables one to align dimensions for multiplication and uncover structural patterns in data.

Linear algebra enables the expression of complex datasets in a way that algorithms can understand and process, therefore allowing the construction of complex models using a plethora of data collected from the real world. 

Many machine learning algorithms are built upon a system of linear equations. Linear regression is a simple yet powerful algorithm used for predicting continuous values. The process of finding the “best fit” line or plane that minimizes the error between predicted and actual values often boils down to solving a system of linear equations. For example,when predicting house prices based on square footage and number of bedrooms, coefficients (weights) must be found to satisfy equations like:

 $price=w_1*squar{e}_{f}ootage+w_2*numbe{r}_o{f}_{b}edrooms+b$

...where  $w_1$, $w_2$ and  $b$  are the unknown coefficients to solve for. This can be represented and solved using matrices. Techniques like “least squares” are used to find the approximate solutions to these systems when an exact solution doesn’t exist, which is often the case with real-world, noisy data. In other words, approximating a loss function, is represented as a collection of linear equations that solved for with calculus.

More complex algorithms, such as those found in deep learning and neural networks, heavily rely on operations like massive matrix multiplication for processing information through different layers. Each layer in a neural network performs a linear transformation on its input data, which is essentially a matrix transformation where the input vector is multiplied by a weight matrix. This allows the network to learn complex patterns and relationships within the data.

Many real-world datasets contain a large number of features (or variables) for each data point: sometimes in the hundreds, thousands or even millions. This is called high-dimensional data. While more features might seem like they should make models more accurate, they often make learning harder. High-dimensional data can be computationally expensive to process, memory-intensive to store and prone to overfitting, where a model memorizes noise instead of learning meaningful patterns.

Another challenge is the curse of dimensionality. As the number of dimensions grows, data points become increasingly sparse in the feature space, and the notion of “closeness” between points becomes less meaningful. This sparsity makes it difficult for algorithms to reliably detect relationships. Therefore, having the right tools to reduce the amount of features and extract the signals from the noise is pivotal. Dimensionality reduction is the process of transforming data from a high-dimensional space into a lower-dimensional one while preserving as much of the original structure and important information as possible. By reducing the number of features, practitioners can simplify models, improve generalization, speed up computations and often make helpful data visualizations.

Linear algebra is at the core of many dimensionality reduction techniques. For example, principal component analysis uses concepts like eigenvalues and eigenvectors to find new axes (principal components) that capture the maximum variance in the data, representing a meaningful attribute in the high dimensional dataset. By projecting the data onto the first few principal components, practitioners keep the most important patterns while discarding less useful variations.

For example, imagine a dataset describing thousands of customers with 100 different features each (age, income, spending in various product categories, etc.). Analyzing all 100 features at once would be slow and complex, and many of them may be redundant (for example, interest in “sports gear” often overlaps with “outdoor equipment”). PCA can reduce the dataset to just 2 or 3 components that summarize most of the variation in customer behavior, making it easier to visualize and run downstream algorithms more efficiently.

In short, dimensionality reduction is a way to distill complex data into its most informative parts, and linear algebra provides the mathematical machinery to make it possible.

Eigenvalues, eigenvectors and eigendecomposition together describe the fundamental modes of behavior of a linear transformation or system:

• Eigenvector: Imagine a linear transformation (like stretching or rotating a vector space). An eigenvector of a square matrix is a non-zero vector that, when that transformation is applied to it, only changes by a scalar factor. It doesn’t change its direction. It’s a special direction in the data that remains stable under the transformation.

• Eigenvalue: This is the scalar factor by which an eigenvector is scaled. It tells you how much the eigenvector is stretched or compressed during the transformation. In PCA, larger eigenvalues correspond to principal components that capture more variance in the data.

• Eigendecomposition: This is the process of breaking down a square matrix into a set of its eigenvectors and eigenvalues. For a given matrix, if one can find its eigenvectors and eigenvalues, one can reconstruct the original matrix from them. In PCA, eigendecomposition of the covariance matrix of the data allows for the  identification of the principal components (eigenvectors) that best represent the variance in the data, ordered by their corresponding eigenvalues.

Another powerful technique, singular  value decomposition (SVD), also plays a crucial role in dimensionality reduction and is fundamental to areas like matrix factorization in recommendation systems. While related to eigendecomposition, SVD can be applied to any matrix (not just square matrices) and offers a more general way to decompose a matrix into its constituent parts, revealing underlying structures and reducing dimensions effectively. For instance, in recommendation systems, SVD helps decompose a user-item interaction matrix into lower-dimensional matrices representing latent features of users and items, which are then used to predict new recommendations.

Many machine learning models involve optimization problems, where the goal is to find the best set of parameters for a model that minimize an error function or maximize a likelihood function. Algorithms like gradient descent, used extensively in training neural networks and other machine learning algorithms, rely on linear algebra to calculate gradients (vectors pointing in the direction of the steepest ascent of a function) and update model parameters iteratively.

Understanding optimization also means understanding the properties of the matrices involved in these calculations. This is where concepts like the determinant and the identity matrix become relevant. The determinant of a square matrix is a single number that provides crucial information about the matrix. For example, a non-zero determinant indicates that the matrix is invertible (meaning it has a corresponding matrix inversion operation), which is critical for solving systems of linear equations uniquely. If the determinant is zero, the system might have no unique solution or infinitely many, indicating issues like linear independence (where one vector in a set can be expressed as a linear combination of others). An identity matrix (a square matrix with ones on the main diagonal and zeros elsewhere) is special because when you multiply any matrix by the identity matrix, the original matrix remains unchanged, acting like the number ‘1’ in scalar multiplication.

The good news is that ML practitioners don’t need to manually perform these complex calculations. Libraries like NumPy in Python provide highly optimized functions for all these linear algebra concepts, making it the de facto standard for numerical computing in machine learning. For example, numpy.linalg.eig() can compute eigenvalues and eigenvectors, and numpy.dot() handles dot products and matrix multiplications with ease. Frameworks like TensorFlow (popular in deep learning) also heavily leverage linear algebra under the hood, abstracting away the low-level details so users can focus on building models.

This introduction to linear algebra for machine learning barely scratches the surface. Concepts such as linear transformation and matrix transformation describe how data can be manipulated and reshaped, for instance, rotating an image or scaling its features. Understanding types of matrices like the identity matrix (which leaves vectors unchanged when multiplied) and orthogonal matrix (where the inverse is simply the transpose, simplifying calculations) is also beneficial. While one won’t typically be performing gaussian elimination (an algorithm for solving system of linear equations) by hand in ML, understanding its principles illuminates how these systems are solved computationally. Linear independence is also critical for understanding the uniqueness of solutions and the basis of a vector space (the set of all possible linear combinations of a set of vectors).

Ultimately, a solid grasp of linear algebra concepts empowers ML practitioners to not only use pre-built machine learning algorithms but also to truly understand their inner workings, debug them effectively and even develop novel solutions. It’s the silent workhorse that has driven ML for decades and will continue to be essential in the future of artificial intelligence.