## 2 I-Vector Framework  
Given a universal background model (UBM), which is a GMM representing the utterance-independent distribution of the frame-level features, an utterance-dependent model can be obtained by adapting the parameters of the UBM via a Bayesian adaptation algorithm [22]. The GMM supervector is obtained by concatenating the mean vectors of each mixture component, summarizing the overall pattern of the frame-level feature distribution. However, since the GMM supervector is known to have high dimensionality and contains variability caused by many different factors, various studies have focused on reducing the dimensionality and compensating the irrelevant variability inherent in the GMM supervector. Among them, the i-vector framework is now widely used to represent the distinctive characteristics of the utterance in the field of speaker and language recognition [23]. Similar to the eigenvoice decomposition [3] or joint factor analysis (JFA) [4] techniques, i-vector extraction can be understood as a factorization process decomposing the GMM supervector as  
\[\mathbf{m}(\mathbf{X})=\mathbf{u}+\mathbf{Tw}(\mathbf{X}), \tag{1}\]  
where \(\mathbf{m}(\mathbf{X})\), \(\mathbf{u}\), \(\mathbf{T}\), and \(\mathbf{w}(\mathbf{X})\) indicate the ideal GMM supervector dependent on a given speech utterance \(\mathbf{X}\), UBM supervector, total variability matrix, and i-vector, respectively. Hence, the i-vector framework aims to find the optimal \(\mathbf{w}(\mathbf{X})\) and \(\mathbf{T}\) to fit the UBM parameters to a given speech utterance. Given an utterance \(\mathbf{X}\), the \(0^{\text{th}}\) and the \(1^{\text{st}}\)-order Baum-Welch statistics are obtained as  
\[n_{c}(\mathbf{X})=\sum_{l=1}^{L}\gamma_{l}(c), \tag{2}\]  
\[\tilde{\mathbf{f}}_{c}(\mathbf{X})=\sum_{l=1}^{L}\gamma_{l}(c)(\mathbf{x}_{l}- \mathbf{u}_{c}), \tag{3}\]  
where for each frame \(l\) within \(\mathbf{X}\) with \(L\) frames, \(\gamma_{l}(c)\) denotes the posterior probability that the \(l^{\text{th}}\) frame feature \(\mathbf{x}_{l}\) is aligned to the \(c^{\text{th}}\) Gaussian component of the UBM, \(\mathbf{u}_{c}\) is the mean vector of the \(c^{\text{th}}\) mixture component of the UBM, and \(n_{c}(\mathbf{X})\) and \(\tilde{\mathbf{f}}_{c}(\mathbf{X})\) are the \(0^{\text{th}}\) and the centralized \(1^{\text{st}}\)-order Baum-Welch statistics, respectively. The extraction of the i-vector can be thought of as an adaptation process where the mean of each Gaussian component in the UBM is altered to maximize the likelihood with respect to a given utterance. Let \(\mathbf{\Sigma}_{c}\) denote the covariance matrix of the \(c^{\text{th}}\) mixture component of the UBM and \(F\) be the dimensionality of the frame-level features. Then, the log-likelihood given an utterance \(\mathbf{X}\) conditioned on \(\mathbf{w}(\mathbf{X})\) can be computed as \[ \begin{split}\text{log}P(\mathbf{X}|\mathbf{T},\mathbf{w}(\mathbf{X}))= \sum_{c=1}^{\text{C}}(n_{c}(\mathbf{X})\log\frac{1}{(2\pi)^{F/2}|\mathbf{ \Sigma}_{c}|^{1/2}}\\ -\frac{1}{2}\sum_{l=1}^{L}\gamma_{l}(c)(\mathbf{x}_{l}-\mathbf{m }_{c}(\mathbf{X}))^{t}\mathbf{\Sigma}_{c}^{-1}(\mathbf{x}_{l}-\mathbf{m}_{c}( \mathbf{X}))),\end{split}  \tag{4}\]  
where \(\mathbf{m}_{c}(\mathbf{X})\) is the mean of the \(c^{\text{th}}\) mixture component of \(\mathbf{m}(\mathbf{X})\) and the superscript \(t\) indicates matrix transpose. From lemma 1 in [3], the log-likelihood given \(\mathbf{X}\) obtained by marginalizing (4) over \(\mathbf{w}(\mathbf{X})\) turns out to be  
\[ \begin{split}\text{log}P(\mathbf{X}|\mathbf{T})=& \text{log}\mathbb{E}_{\mathbf{w}}[P(\mathbf{X}|\mathbf{T},\mathbf{w})]\\ =&\text{log}\int P(\mathbf{X}|\mathbf{T},\mathbf{w}) \mathcal{N}(\mathbf{w}|0,\mathbf{I})d\mathbf{w}\\ =& G(\mathbf{X})-\frac{1}{2}\text{log}|\xi(\mathbf{ X})|\\ &+\frac{1}{2}\sum_{c=1}^{\text{C}}(\mathbf{\tilde{m}}_{c}( \mathbf{X})-\mathbf{u}_{c})^{t}\mathbf{\Sigma}_{c}{}^{-1}\mathbf{q}_{c}( \mathbf{X}),\end{split}