A data-science team implements a single-layer perceptron whose decision function is sign(w路x) (no bias term). On a linearly separable dataset, the model converges to weight values that still misclassify several points because the separating hyperplane must cross the origin. The engineers consider adding an explicit bias neuron so that the decision function becomes sign(w路x + b). Why does introducing this bias term usually allow the perceptron to find a weight vector that perfectly separates the same dataset without changing the learning rule?
It converts the classifier from a homogeneous to an affine hyperplane, allowing the decision boundary to shift away from the origin while keeping its orientation.
It introduces a non-linear interaction that enables the perceptron to model non-linearly separable patterns.
It constrains weight growth during gradient descent, thereby preventing over-fitting.
It reduces the dimensionality of the input space, making the weight search easier.
Adding a bias converts the perceptron's decision rule from a homogeneous linear form (w路x = 0) to an affine one (w路x + b = 0). The extra constant allows the learned hyperplane to translate parallel to itself and no longer be constrained to pass through the origin, giving the algorithm the geometric freedom required to separate any linearly separable dataset whose optimal boundary has a non-zero intercept. The other statements are incorrect because:
A bias does not reduce dimensionality; it adds an additional parameter.
The bias is still linear-it does not introduce polynomial or other non-linear terms.
While the bias can influence weight magnitudes, its purpose is not to regularize or prevent over-fitting; that is achieved with separate techniques (e.g., L2 regularization).
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What is the difference between a homogeneous and affine hyperplane?
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Why can't a perceptron without bias separate all linearly separable datasets?
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How does the bias influence the training process of a perceptron?