# An Analysis of the Short Utterance Problem for Speaker Characterization +
## 3 Formulation of the Embedding Extraction with Short Utterances
### I-Vector Embeddings  
The previously described formulation also matches with the traditional i-vectors. The i-vector modelling paradigm explains the utterance \(\mathbf{A}\) as the result of sampling from a Gaussian Mixture Model (GMM), specific for the utterance with parameters \(\lambda_{\mathbf{A}}\). This model \(\lambda_{\mathbf{A}}\) is the result of the adaptation from a Universal Background Model (UBM), a large GMM that reflects all possible acoustic conditions. This adaptation process is restricted to only the UBM Gaussian means. Besides, the shift of the GMM Gaussians is tied and explained by means of a hidden variable \(\mathbf{w_{A}}\), located in the Total Variability subspace, described by matrix \(\mathbf{T}\). Mathematically:  
\[\mathbf{\mu}_{\mathbf{A}}=\mathbf{\mu}_{\text{UBM}}+\mathbf{Tw}_{\mathbf{A}} \tag{3}\]  
where \(\mathbf{\mu}_{\mathbf{A}}\) represents the supervector mean, the concatenation of the GMM component means, from the target \(\lambda_{\mathbf{A}}\). \(\mathbf{\mu}_{\text{UBM}}\) is the supervector mean from the Universal Background Model (UBM), the reference model representing the average behaviour. \(\mathbf{w}_{\mathbf{A}}\) is the latent variable for the utterance \(\mathbf{A}\), with a standard normal prior distribution and \(\mathbf{T}\) is a low rank matrix defining the total variability subspace.  
The i-vector estimation looks for the best value for the latent variable \(\mathbf{w_{A}}\) so as to explain the given utterance by means of the adapted model. For this purpose we estimate the posterior distribution of the latent variable \(\mathbf{w_{A}}\) given the utterance \(\mathbf{A}\). The i-vector representation \(\overline{\mathbf{w}}\) corresponds to the mean of this posterior distribution. Defined in Reference [5], the i-vector is formulated as:  
\[\overline{\mathbf{w}} =\left(\sum_{i=1}^{M}\mathbf{T}_{i}^{T}\Sigma_{i}^{-1}N_{i}( \mathbf{A})\mathbf{T}_{i}+\mathbf{I}\right)^{-1}\sum_{i=1}^{M}\mathbf{T}_{i}^ {T}\Sigma_{i}^{-1}\bar{\mathbf{F}}_{i}(\mathbf{A})\] (4) \[= \frac{1}{N(\mathbf{  
respectively. \(N(\mathbf{A})\) symbolizes the total number of frames in the utterance \(\mathbf{A}\). Finally, the term \(\mathbf{\overline{F}}_{i}(\mathbf{A})\) is the average deviation per sample of the utterance for the component \(i\) of the UBM.  
The formulation of i-vectors offers special characteristics. First, the value of \(M\), the number of traced acoustic units to discriminate, is fixed in the UBM. Its value is equal to the number of Gaussian components in the UBM. Therefore, \(G_{i}(\mathbf{A})\) represents the contribution per sample to the i-vector from component \(i\) and the weight \(\boldsymbol{\alpha}_{i}\) is the proportion of frames assumed to be sampled from same \(ith\) component. Furthermore, i-vectors have no speaker awareness in their formulation. They simply store the variations in the acoustic units within a embedding. These deviations from the average behaviour, properly treated by the backend, are responsible for the performance in speaker identification systems.