Linear algebra is a cornerstone of machine learning, providing the essential tools needed to represent and manipulate data for machine learning concepts. Understanding machine learning concepts is crucial for anyone looking to excel in this field. In this article, we will explore the core machine learning concepts that every aspiring engineer should know, with simplified explanations and practical examples that you can apply directly to machine learning projects.
1. Understanding Machine Learning and Linear Algebra:- Scalars, Vectors, Matrices, and Tensors
Scalars, vectors, matrices, and tensors represent data in increasingly complex structures. These objects are the basic building blocks of data manipulation in ML.
- Scalar: A scalar is a single numerical value, representing a constant or measurement in ML
For example: a = 3
- Vector: A vector is an ordered list of numbers, representing a data point with multiple features
For example: v = [170, 65, 25] # height, weight, age
- Matrix: A matrix is a two-dimensional array of numbers, where rows represent samples and columns represent features
For example:
A = [[5, 8],
[3, 7],
[1, 2]]
Tensor: A tensor generalizes matrices to higher dimensions, often used to represent multi-dimensional data like images, where dimensions could represent width, height, and color channels.
2. Vector and Matrix Operations
Vector and matrix operations allow transformations of data, which are crucial in ML algorithms.
Addition and Subtraction
You can add or subtract vectors or matrices element-wise if they have the same dimensions.
Example: u = [2, 5, 8] v = [1, 3, 4] u + v = [3, 8, 12]
Scalar Multiplication
This operation scales each element in a vector or matrix by multiplying it by a scalar.
Example: 2 * u = [4, 10, 16]
Dot Product
The dot product of two vectors results in a single number, calculated by multiplying corresponding elements and adding the results.
Example: u = [2, 5, 8] v = [1, 3, 4] u . v = 2*1 + 5*3 + 8*4 = 49
Cross Product (for 3D vectors)
The cross-product results in a new vector that is perpendicular to both input vectors (in 3D space).
Example: If u = [a, b, c] and v = [d, e, f], the cross product u x v is [bf - ce, cd - af, ae - bd]
3. Matrix Multiplication
Matrix multiplication involves combining rows from the first matrix with columns from the second matrix. This operation is foundational in ML, especially for transforming data through layers in neural networks.
Example:
A = [[1, 2],
[3, 4]]
B = [[5, 6],
[7, 8]]
AB = [[(1*5 + 2*7), (1*6 + 2*8)],
[(3*5 + 4*7), (3*6 + 4*8)]]
= [[19, 22],
[43, 50]]
4. Vector Spaces and Linear Independence
A vector space is a collection of vectors that can be scaled and added while remaining within the same space. Linear independence occurs when no vector in a set can be represented as a combination of the others.
Example: v1 = [1, 0] v2 = [0, 1]
v1 and v2 are linearly independent because neither can be represented as a combination of the other. Linear independence is key in ML for feature selection, helping to remove redundant data.
5. Norms and Distance Metrics
Norms measure the size or length of a vector, which is essential in ML for calculating distances and penalties in regularization.
L1 Norm (Manhattan Distance)
The L1 norm is the sum of the absolute values of the elements in a vector.
Example: v = [2, 5, 8] L1 norm of v = |2| + |5| + |8| = 15
L2 Norm (Euclidean Distance)
The L2 norm is the square root of the sum of squared elements, commonly used in ML to minimize error.
Example: L2 norm of v = sqrt(2^2 + 5^2 + 8^2) = sqrt(93) ≈ 9.64
6. Eigenvalues and Eigenvectors
An eigenvector of a matrix is a vector that only changes in magnitude, not direction when transformed by that matrix. The scalar factor is called the eigenvalue.
- Example: For a matrix A, if applying A to a vector v scales v but doesn’t change its direction, then v is an eigenvector, and the scaling factor is the eigenvalue
Eigenvalues and eigenvectors are crucial in Principal Component Analysis (PCA), which is used for dimensionality reduction in ML.
7. Singular Value Decomposition (SVD)
SVD is a matrix decomposition technique that breaks a matrix into three matrices (U, Σ, and V^T), revealing the structure of the data and reducing its dimensionality.
SVD is particularly useful in applications like recommendation systems and natural language processing, where it simplifies complex data.
8. Orthogonality and Orthogonalization
Orthogonal vectors have a dot product of zero, representing independent directions. Orthogonalization is the process of making a set of vectors orthogonal.
Example: u = [1, 0] v = [0, 1] u . v = 0 # hence u and v are orthogonal
Orthogonalization helps reduce redundancy and improve interpretability in ML.
9. Projections
A projection maps one vector onto another, showing how much of one vector aligns with another. This is commonly used in linear regression to minimize the error between predicted and actual values.
- Example: Projecting vector a = [3, 4] onto b = [1, 2] gives a new vector along the direction of b, which represents the alignment of a with b
10. Determinants and Inverse of Matrices
The determinant is a scalar value that indicates whether a matrix is invertible. The inverse of a matrix, if it exists, is a matrix that, when multiplied with the original, yields the identity matrix.
- Determinant: For a 2×2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad – bc
- Inverse: For the matrix A = [[a, b], [c, d]], if the determinant is non-zero, the inverse is:
A^(-1) = (1 / det(A)) * [[d, -b], [-c, a]]
Matrix inverses are essential in solving systems of linear equations in ML, particularly in optimization tasks.
11. Principal Component Analysis (PCA)
PCA is a dimensionality reduction technique that transforms data by finding new axes (principal components) representing the directions of maximum variance. This is especially useful for reducing the number of features in a dataset.
PCA involves computing the eigenvalues and eigenvectors of the covariance matrix to find the principal components. By projecting data onto a smaller number of components, PCA reduces complexity without significant information loss.
Definitions
- Scalar: A single numerical value
- Vector: An ordered set of numbers representing a data point or model parameters
- Matrix: A two-dimensional array representing structured data
- Tensor: A multidimensional array used for complex data representations
- Norm: A function that measures the “length” or “size” of a vector
- Dot Product: The sum of the products of corresponding entries in two vectors
- Cross Product: A vector operation that produces a new vector perpendicular to the original two (in 3D)
- Eigenvalue/Eigenvector: A scalar and vector pair that describes directional transformations within a matrix
- SVD: A decomposition method that simplifies data for analysis and reduces dimensionality
- Orthogonality: A property where vectors have a dot product of zero, representing independence
- Projection: A mapping of one vector onto another, measuring alignment
- Determinant: A scalar that indicates whether a matrix is invertible
- Inverse: A matrix when multiplied by the original, yields the identity matrix
- PCA: A technique to reduce data dimensionality by projecting data onto principal components that capture the most variance
References
- Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
- Shalev-Shwartz, S. & Ben-David, S. (2014). Understanding Machine Learning: From Theory to Algorithms. Cambridge University Press.
- Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.
- Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT Press.
