L2normalized

L2normalized refers to the process of L2 normalization, a method of scaling a vector so that its L2 norm equals one. The L2 norm of a vector x = (x1, x2, ..., xn) is defined as ||x||2 = sqrt(sum_i xi^2). The L2 normalization of x is x_hat = x / ||x||2, provided ||x||2 is nonzero. In practice, a small epsilon may be added to the denominator to avoid division by zero.

The result of L2 normalization is a unit vector with length one. This makes the vector’s direction preserved while its magnitude is fixed. A common corollary is that the dot product of two L2-normalized vectors equals their cosine similarity: if x and y are normalized, x_hat · y_hat = cos(theta), where theta is the angle between them. L2 normalization is therefore scale-invariant with respect to the direction of the vector.

In many applications, normalization is performed per vector, per row, or per feature depending on the data layout. It is widely used in machine learning and data processing to ensure consistent magnitudes across inputs, stabilize training, and improve distance or similarity computations. It is also common when working with embeddings, features for neural networks, or before computing cosine similarity.

L2 normalization differs from other normalization schemes such as L1 normalization (sum of absolute values equals one) and max (L-infinity) normalization (maximum absolute value equals one). L2 normalization is more sensitive to larger components due to squaring, and it emphasizes the overall energy of the vector rather than individual components.

Example: the vector [3, 4] has ||x||2 = 5, so its L2-normalized form is [0.6, 0.8].