Vector Spaces

Vector spaces form the mathematical playground where ML lives. This file covers vector addition, scalar multiplication, closure axioms, subspaces, and why nearly everything in AI is represented as vectors.

• Think of a Vector Space as a specific kind of playground where mathematical objects live, and each object is called a vector.

• For geometric intuition in machine learning (ML), we will always think of vectors as a point in Euclidean space, represented by it's coordinates.

• The vector \(\mathbf{a}\) (denoted mathematically as lowercase letters in bold) has \(n\) coordinates, each representing a position along an axis.

\[\mathbf{a} = [a_1, a_2, a_3]\]

• The vectors in the vector space live under a very specific, unbreakable set of rules:

  • Vector Addition (Combining): You can take any two vectors and combine them to create a new one. Think of vectors as instructions for movement. If vector A means "walk 3 steps forward" and vector B means "walk 2 steps right," adding them (A + B) creates a new, single instruction: "walk 3 steps forward and 2 steps right."

  • Scalar Multiplication (Scaling): You can take any vector and scale it using a regular number (a "scalar"). You can stretch it, shrink it, or reverse it. If vector A is "walk 3 steps forward," scaling it by 2 makes it "walk 6 steps forward." Scaling it by -1 flips it entirely to "walk 3 steps backward."

• The dimension of a vector space is the number of independent directions it contains. \(\mathbb{R}^2\) is 2-dimensional (needs 2 coordinates), while \(\mathbf{a}\) above lives in \(\mathbb{R}^3\).

• We can for instance represent any object, say, a human, as a vector, where \(h_1\) = height in cm, \(h_2\) = weight in kg, \(h_3\) = age.

\[\mathbf{h} = [185, 75, 30]\]

• We have now created a vector space with a vector representing a human.

• We can represent multiple humans, and see how close or apart they are!

• We can add more features, creating a rich representation of a human, often called feature vectors in ML.

• The more unique and meaningful features you have, the more descriptive the feature vector is, an important factor to remember.

• Beyond 3 dimensions, vectors become very difficult to visually inspect, inspiring a field of mathematics called Linear Algebra.

• Now, Linear algebra is the study of vectors, vector spaces and mappings between vectors.

• We represent almost every thing in AI/ML as vectors, making linear algebra the bedrock of the field.

• Vector addition can be performed by placing one vector on the tail of the other visually, and drawing from the origin to the endpoint.

• For two vectors \(\mathbf{a} = (a_1, a_2)\) and \(\mathbf{b} = (b_1, b_2)\): \(\mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2)\)

• Vectors can also be subtracted, with all addition rules applying too.

• Multiplying a vector by a scalar scales the vector by that factor in the same direction.

• For a scalar \(c\) and vector \(\mathbf{v} = (v_1, v_2)\): \(c\mathbf{v} = (cv_1, cv_2)\)

• Closure under Addition: If you add any two vectors from the vector space, the result is also a vector within the same space: If \(\mathbf{u} \in V\) and \(\mathbf{v} \in V\), then \(\mathbf{u} + \mathbf{v} \in V\)

• Closure under Scalar Multiplication: If you multiply any vector from the vector space by a scalar, the result is a vector within the same space: If \(\mathbf{v} \in V\) and \(c \in F\), then \(c\mathbf{v} \in V\)

• Associativity of Addition: For any three vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\): \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\)

• Commutativity of Addition: For any two vectors \(\mathbf{u}\) and \(\mathbf{v}\): \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\)

• Both paths through the parallelogram arrive at the same point.

• (Zero Vector): There exists a vector \(\mathbf{0}\) such that for any vector \(\mathbf{v}\): \(\mathbf{v} + \mathbf{0} = \mathbf{v}\)

• Additive Inverse: For every vector \(\mathbf{v}\), there exists a vector \(-\mathbf{v}\) such that: \(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\)

• Distributivity 1: For any scalar \(c\) and vectors \(\mathbf{u}\), \(\mathbf{v}\): \(c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}\)

• Scaling the sum (gold) gives the same result as summing the scaled vectors.

• Distributivity 2: For any scalars \(c\), \(d\) and vector \(\mathbf{v}\): \((c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}\)

• Associativity: For any scalars \(c\), \(d\) and vector \(\mathbf{v}\): \((cd)\mathbf{v} = c(d\mathbf{v})\)

• Identity Element: For any vector \(\mathbf{v}\): \(1\mathbf{v} = \mathbf{v}\), where \(1\) is the multiplicative identity in the field of scalars.

• A subspace is just a smaller playground inside the bigger one. Imagine 3D space as a room. A flat sheet of paper passing through the centre of the room is a subspace, and so is a single straight wire through the centre.

• The key requirement is that the subspace must pass through the origin. If you shift that sheet of paper off-centre, it stops being a subspace because the zero vector is no longer on it.

• All the same rules from the vector space (addition, scaling, closure) still work inside a subspace. You can add or scale vectors within it and never "fall off" into the larger space.

• A line through the origin is a 1-dimensional subspace, a plane through the origin is a 2-dimensional subspace, and the full space is a subspace of itself.

• In ML, subspaces appear naturally. High-dimensional data often has structure that lives on a lower-dimensional subspace. Techniques like PCA find that subspace so we can work with the data more efficiently.

Coding Tasks (use CoLab or notebook)

1. Run code to verify the distributivity property, then Modify and play around to test other rules!

   import jax.numpy as jnp
   
   u = jnp.array([1, 2])
   v = jnp.array([3, 0])
   c = 2
   
   lhs = c * (u + v)
   rhs = c*u + c*v
   
   print(f"LHS: {lhs}")
   print(f"RHS: {rhs}")
   

2. Run code to visualise different vectors, then modify values for different coordinates to understand how each axis affects position.

   import jax.numpy as jnp
   import matplotlib.pyplot as plt
   
   # Try changing these vectors!
   a = jnp.array([3, 2, 4])
   b = jnp.array([1, 4, 2])
   c = jnp.array([4, 1, 3])
   
   fig = plt.figure()
   ax = fig.add_subplot(111, projection="3d")
   
   for vec, name, color in [(a, "a", "red"), (b, "b", "blue"), (c, "c", "green")]:
       ax.quiver(0, 0, 0, *vec, color=color, arrow_length_ratio=0.1, linewidth=2, label=name)
   
   lim = int(jnp.abs(jnp.stack([a, b, c])).max()) + 1
   ax.set_xlim([0, lim]); ax.set_ylim([0, lim]); ax.set_zlim([0, lim])
   ax.set_xlabel("X"); ax.set_ylabel("Y"); ax.set_zlabel("Z")
   ax.legend()
   plt.show()