When it comes to data classification, one of the key concepts is the decision boundary: how an algorithm separates different classes in a dataset. In this article, we’ll explore what these boundaries look like, from simple lines to the complex hyperplanes of neural networks.
Linear Decision Boundaries
The simplest type of decision boundary is linear. Imagine a 2D dataset with two input variables. In this case, we can separate two classes using a straight line that divides the points.
This approach works well for binary classes in two dimensions. However, in practice, we rarely use a neural network for such simple problems: linear boundaries are easily handled with simpler methods like logistic regression or decision trees.
From Lines to Hyperplanes
As we increase the number of dimensions (the number of input variables), the concept of a linear decision boundary extends:
In 3D, the boundary becomes a plane.
In higher dimensions, it becomes a hyperplane, a concept that is hard to visualize but mathematically straightforward.
For a dataset with n variables, a linear decision boundary will be an (n-1)-dimensional hyperplane. Even though we can’t easily visualize beyond 3D, we can think of it as an extension of a plane that divides space.
The limitation of linear boundaries is that they cannot handle complex or non-linear data. When classes are not separable by a line or plane, more advanced methods are required.
Neural Network Decision Boundaries
Neural networks shine in cases where the data is non-linear and high-dimensional. Each neuron introduces a small hyperplane, and by combining these hyperplanes across the network, we get complex decision boundaries, often in hundreds of dimensions.
These boundaries are no longer linear, allowing the network to fit very intricate datasets. However, their high dimensionality makes them nearly impossible to visualize directly.
Visualizing Decision Boundaries
To better understand how neural networks separate classes, we can use dimensionality reduction techniques like t-SNE (t-Distributed Stochastic Neighbor Embedding).
t-SNE allows us to project high-dimensional data into 2D or 3D while preserving the class structure, giving us an approximate idea of the network’s decision boundaries.
When Large Neural Networks Are Needed
If the dataset is particularly complex, the required decision boundary will also be complex. In such cases, we need a larger neural network, with more neurons and layers, capable of modeling the intricate structure of the data.
In summary:
Linear boundaries are simple and interpretable but limited.
Neural networks create non-linear, high-dimensional boundaries, ideal for complex data.
Visualization is challenging but can be approximated using techniques like t-SNE.
Understanding decision boundaries is crucial for designing effective neural networks and choosing the right architecture for the problem you want to solve.