# A Simple Unsupervised Knowledge-Free Domain Adaptation for Speaker Recognition
## 3 Methodology
### Normalization by Full-Rank LDA  
LDA has been widely adopted in the back-end for preprocessing embedding vectors before scoring. The primary objective of LDA is to identify directions that offer the most discrimination for speakers and to eliminate those with weaker correlations to speakers, thereby enhancing the discriminability among speaker vectors [47]. In recent years, deep embedding models based on the maximum margin criterion have simplified the paradigm of speaker recognition systems. By using these 'advanced' models, all the dimensions in the embedding vectors contain speaker information, so a simple cosine scoring on raw speaker embeddings can achieve good performance. As a result, the utilization of LDA has declined. In this paper, we re-discover the merit of LDA from another perspective: instead of using it to identify discriminative dimensions, we use it to 'normalize' the distribution of embeddings from unknown domains. The theory is established as follows. Intuitively, LDA aims to identify directions \(W\) that maximize the discrimination between embedding vectors from different speakers. The pursuit of the directions entails taking into account both the between-class variance and the within-class variance. Therefore,the criterion for determining the optimal direction is to maximize the ratio of between-class variance to within-class variance. This objective can be formulated as follows:  
\[S_{B}W=\lambda S_{W}W \tag{1}\]  
where \(S_{B}\) and \(S_{W}\) are the between-class covariance matrix and the within-class covariance matrix, respectively. The solution to this objective is to find the eigenvectors of \(S_{W}{}^{-1}S_{B}\), with \(\lambda\) representing the corresponding eigenvalues. By arranging all the eigenvectors of \(S_{W}{}^{-1}S_{B}\) in descending order according to their eigenvalues, we obtain the principal directions of discriminability, ranked from strongest to weakest. In speaker recognition systems, it is customary to select the top 125 or 150 eigenvectors to construct the eigenspace of LDA. This helps mitigate the influence of less discriminative directions, thus enhancing speaker discriminability. In this work, we utilize a full-rank LDA, and there is no eigenvector selection process as previously described. Thus, it does not impact the discriminability of speaker vectors; instead, it serves as a linear transform to normalize the distribution of speaker vectors. Previous studies have demonstrated that a full-rank LDA can be decomposed into the following steps [32]:  
1. **Global shift**: Perform centering (mean subtraction) to normalize the data to a zero-mean distribution. 2. **Rotation (first)**: Rotate the coordinate to align the axes to the principal directions of the within-class covariance. 3. **Scaling**: Scale the coordinate axes to unify the within-class variance along each axis, known as whitening. 4. **Rotation (second)**: Rotate the coordinate again to align the axes to the principal directions of between-class variance (note that since within-class variance has been whitened, the subsequent rotation of the coordinate will not affect the within-class variance). From these steps, it is evident that the effect of a full-rank LDA is to align the principal directions of within-class variance and between-class variance with the coordinate axes. In other words, after the transformation, the within-speaker covariance and between-class covariance matrices are all diagonalized, which means the dimensions of the transformed embeddings are mutually uncorrelated. In summary, this full-rank LDA results in a normalized distribution with elegant coordinate alignment. In theory, step (2) and step (4) do not change the cosine score as they are both orthogonal transformations. Step (1) is important, as the location of the origin severely impacts cosine scores. Step (3) is important because, without such scaling, the between-class distance will be affected by the within-class variation that varies across dimensions.