A linear classification head mixes two signals in each logit: the angle between a feature vector and a class weight vector, and the magnitudes of both vectors. Cosine normalization removes the magnitude terms, so class scores depend on directional alignment instead.

That change is small in code but substantial in interpretation. Feature norm no longer increases every class comparison merely by growing, class-weight norm no longer acts as an implicit class-specific scale, and the overall sharpness of the softmax must be supplied separately.

Linear logits contain magnitude and direction

For a feature vector x and class weight w_k, a bias-free linear head produces:

z_k = w_k^T x

The dot product can be decomposed as:

z_k = ||w_k|| ||x|| cos(theta_k)

where theta_k is the angle between the vectors. A large logit can therefore come from close directional alignment, a large feature norm, a large class-weight norm, or a combination of those factors.

Cosine normalization replaces both vectors with unit-norm versions:

x_hat   = x / ||x||
w_hat_k = w_k / ||w_k||

c_k = w_hat_k^T x_hat

For nonzero vectors, c_k lies between -1 and 1 and equals cos(theta_k). The classifier now compares directions on a unit hypersphere.

Implementations need a defined policy for norms near zero. A small denominator floor is common for numerical stability, but its exact value is an implementation choice rather than part of the geometric definition.

Normalization removes two implicit scales

In an ordinary linear head, increasing ||x|| multiplies all bias-free logits for that example. Their ordering stays fixed, but softmax confidence can become sharper because the gaps between logits grow.

Cosine normalization prevents feature magnitude from having that effect. Two positive scalar multiples of the same nonzero feature vector map to the same normalized feature and therefore receive identical cosine scores.

Weight magnitude also disappears. With an unnormalized head, two class vectors pointing in similar directions can still produce different scores because one has a larger norm. After normalization, only their directions affect the cosine comparison.

This does not make the representation invariant to every transformation. Changing the direction of a feature still changes its scores, and the upstream network can alter that direction through its parameters.

Softmax still needs a useful logit range

Raw cosine scores occupy a bounded interval. Feeding them directly to softmax can limit how separated class probabilities become, especially when many class directions produce similar cosine values.

A common form introduces a positive scale s:

z_k = s * cos(theta_k)

The scale restores control over softmax sharpness without reintroducing feature or weight magnitude. Increasing s expands logit gaps; decreasing it compresses them. Class ordering is unchanged for positive s.

The scale may be fixed or represented as a trainable positive parameter. Those choices are not equivalent operationally. A trainable scale lets optimization select the global logit range, while a fixed scale makes that range an explicit hyperparameter.

The scale should not be confused with class-specific weight norms. One shared scalar changes every class score uniformly, whereas separate weight magnitudes can change relative class comparisons.

Bias changes the geometric interpretation

Adding a conventional class bias gives:

z_k = s * cos(theta_k) + b_k

The result is no longer a purely angular classifier. A bias can move a class score independently of the angle between its weight direction and the feature.

That may be intentional, but it changes the meaning of the decision boundary. With normalized features and weights and no bias, pairwise class decisions compare angular similarity. With class-specific biases, the boundary also reflects additive offsets.

For systems that use cosine scores as an interpretable similarity signal, retaining an unnoticed bias term can therefore defeat the intended geometry even though the code still normalizes both vectors.

Angular scores do not imply calibrated probabilities

Removing norm effects makes the source of each class score more constrained, but it does not guarantee that softmax probabilities match empirical correctness. The scale, representation geometry, training objective, class distribution, and evaluation distribution can all affect probability behavior.

The same distinction applies to confidence derived from feature norm. An ordinary classifier may correlate feature magnitude with certain data properties, but cosine normalization deliberately discards that magnitude at the final comparison. If an application needs feature norm as a separate signal, it must preserve and evaluate it explicitly rather than expecting the normalized head to retain it.

Cosine normalization is therefore best treated as a statement about classifier geometry, not as a general uncertainty mechanism.

The representation must support directional separation

A cosine-normalized head can only separate classes through feature direction. If upstream representations for two classes occupy nearly the same directions, changing vector magnitude cannot rescue the final comparison.

This places more emphasis on the angular structure produced by the encoder. Training losses that operate on the normalized logits can shape that structure, while margin-based variants can impose additional angular constraints. Those variants change the objective and should not be treated as interchangeable with plain cosine normalization.

The useful boundary is clear: normalization removes magnitude from the final class comparison, not from the rest of the network. Developers adopting it should decide deliberately whether magnitude was unwanted variability or a signal the application still needs. Once that decision is explicit, the classifier’s score geometry becomes easier to reason about and test.

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