In the fascinating world of Artificial Intelligence, Machine Learning stands out as one of the most transformative technologies; and at its very core lies mathematics.
From processing large volumes of data to training intelligent models, matrix operation for ML is decoded through linear algebra to make sense of complex patterns.
Every image recognised, every prediction made, and every decision generated by an algorithm is, in essence, the result of algebraic equations working behind the scenes.
Matrices and vectors are the fundamental tools that help represent datasets, perform computations, and structure the flow of information within neural networks.
These networks, no matter how intricate, are built upon layers of mathematical operations with algebra as the very foundation of such huge dataset operations.
Matrix Operations in Machine Learning, and Why They Are Important?
In matrix operation for ML, real-world data like images, texts, or numerical entries is first converted into numerical formats, often organised as vectors and matrices.
These matrices enable algorithms to perform large-scale computations in a structured and parallel manner.
For example, a dot product helps measure the similarity between data points, while eigenvalues and eigenvectors are crucial in understanding the directions of maximum variance in the data. They are used in dimensionality reduction techniques like Principal Component Analysis (PCA).
Singular Value Decomposition (SVD) is another powerful tool that breaks down complex matrices to extract meaningful patterns.
These operations enable machine learning models to compress data, identify relationships, and learn from neural matrices.
Linear Algebra Simplifies Data Representation & Feature Extraction
Raw data is often messy, unstructured, and hard to interpret. Linear algebra, with geometric interpretation of vectors and matrices helps visualise data points and effective pattern recognition.
Each data point, whether it’s an image pixel, a word in a sentence, or a user rating, can be encoded as a vector.
When many such data points are combined, they form a MATRIX – a structured grid of values that can be easily manipulated. The matrix operation in ML not only helps in storing data but also in transforming it for better understanding and performance.
For example, through matrix multiplication and projections, we can reduce high-dimensional data into lower dimensions while preserving essential features — a process known as feature extraction.
This dimensionality reduction simplifies the learning task for models, enhancing both speed and accuracy.
Matrix Multiplication and Activation Layers Forming the Neural Network
Matrix multiplication lies at the heart of every neural network, enabling us to learn, adapt, and make data predictions.
Each layer in a neural network receives an input often a vector or a matrix, which is multiplied by a weight matrix and then adjusted with a bias vector. Thus, matrix operation for ML helps the network combine and weigh different input features.
However, to capture complex and non-linear patterns in data, a non-linear activation function (like ReLU, sigmoid, or tanh) is applied to the result.
This combination of linear transformations (through matrices) and non-linear activation enables neural networks to approximate intricate functions and solve tasks ranging from image recognition to natural language processing.
As data flows forward through these layers — a process known as forward propagation; the network gradually learns how to map inputs to desired outputs by adjusting weights and biases through training.
Gradients and Jacobians of Model Training in Matrix Operation for ML
During model training in deep learning, the goal is to minimise the error between the predicted output and the actual result — and this is achieved through a process called optimisation.
Central to this process are gradients and Jacobians, both rooted in linear algebra and calculus.
Gradients, which are essentially vectors of partial derivatives, indicate the direction and rate of change of the loss function concerning each parameter (weight or bias) in the model. The model learns to adjust its parameters to minimise errors through a process known as gradient descent.
In more complex networks involving multiple inputs and outputs, Jacobian matrices help track how changes in input variables simultaneously influence multiple outputs. Matrix operation for ML enables efficient backpropagation — the algorithm that updates all weights layer by layer.
At OmDayal Group of Institutions, we firmly believe that a firm grasp of linear algebra is essential for any student aspiring to excel in matrix operation for ML.
As neural networks grow more complex and data-driven solutions become the norm, it is the algebraic foundations that empower us to understand, build and optimise intelligent systems with confidence.
Through a curriculum that blends theory with hands-on application, we encourage our learners with the mathematical tools needed to navigate AI and evolve in machine learning through meaningful innovation.
Sources:
- https://medium.com/@megha24asma/understanding-matrices-in-data-science-and-machine-learning-a22867845da8
- https://www.geeksforgeeks.org/machine-learning/matrices-and-matrix-arithmetic-for-machine-learning/
- https://www.analyticsvidhya.com/blog/2021/07/12-matrix-operations-you-should-know-while-starting-your-deep-learning-journey/#
- https://www.ijisrt.com/assets/upload/files/IJISRT24NOV378.pdf
