# An Analysis of the Short Utterance Problem for Speaker Characterization +
## 3 Formulation of the Embedding Extraction with Short Utterances
### General Case  
The method to compact a variable length utterance into a fixed-length representation is similar for most embedding extraction techniques. Given the utterance \(\mathbf{A}\), an ordered set of \(N\) acoustic features \(\mathbf{A}=\{a_{1},..,a_{j},..,a_{N}\}\), we transform them by function \(F(x)\), obtaining the ordered sequence \(F(\mathbf{A})=\{f_{1},..,f_{j},..,f_{N}\}\). This function maps the original feature vector \(a_{j}\) into the speaker characteristics subspace as the projections \(f_{j}\). Depending on the embedding, projection \(f_{j}\) involves the transformation of the feature vector \(a_{j}\) as well as a small context around (approximately 0.15 s). By means of this mapping we attempt to highlight the speaker particularities in the features applying linear (e.g., i-vectors) or non-linear transformations (as in DNNs). The function \(F(x)\) is learnt from a large data pool by data analysis, for example, by Maximum Likelihood algorithms for i-vectors or Back-Propagation [31] with DNNs. Due to the fact that each one of these projections \(f_{j}\) only covers a small period of time, they only have information about few acoustic units. The complete characterization of a speaker requires the study of his/her particularities for all the phonemes. These acoustic units are widespread along the utterance, thus we must combine the effect of all these projections \(f_{j}\). The usual method to combine the projections is its temporal average. The result is the compact representation \(G(\mathbf{A})\), defined as:  
\[G(\mathbf{A})=\frac{1}{N}\sum_{j=1}^{N}f_{j} \tag{1}\]  
This embedding \(G(\mathbf{A})\) keeps track of the phonetic content in the utterance \(\mathbf{A}\). However, we can also treat each acoustic unit independently. Many state-of-the-art embeddings, such as i-vectors, can be interpreted as the sum of \(M\) representations \(G_{i}(\mathbf{A})\), one per acoustic unit, each one estimated according to \(N_{i}\) projections \(f_{j}\). According to this reasoning we can express the embedding as:  
\[G(\mathbf{A})=\sum_{i=1}^{M}\mathbf{\kappa}_{i}G_{i}(\mathbf{A}) \tag{2}\]The obtained expression describes embeddings as a weighed sum of \(M\) estimations \(G_{i}(\mathbf{A})\), each one representing the estimated particularities of the speaker in a single acoustic unit. \(G_{i}(\mathbf{A})\) can also be interpreted as the resulting embedding only taking into account the data related to the phoneme \(i\). All the contributions are weighted by the term \(\mathbf{\alpha}_{i}\), the proportion of this acoustic unit in the utterance.  
Therefore, embeddings are conditioned to two main parts: On the one hand the stability of the distribution of weights \(\mathbf{\alpha}_{=}\{\mathbf{\alpha}_{1},\mathbf{\alpha}_{i},\mathbf{\alpha}_{M}\}\). On the other hand the estimations \(G_{i}(\mathbf{A})\), the particularities per phoneme. Both benefit from large utterances. Every language has its own reference phonetic distribution. Hence the longer the utterance the more its phonetic distribution becomes like this reference. Concerning the estimations \(G_{i}(\mathbf{A})\), the more available data, the less uncertain is the estimation.  
The average stage is the last step in which we keep track of the phoneme distribution. As a consequence, we cannot distinguish between speaker and phonetic variability afterwards. Further steps in the embedding post-processing or the backend may transform the embedding but all phonemes are equally treated.