# Regularized Within-Class Precision Matrix Based PLDA in Text-Dependent Speaker Verification
## 2 Preliminaries
### Deep Speaker Embeddings  
Owing to remarkable development in deep learning, many studies have investigated the use of deep neural networks (DNNs) to extract more discriminative speaker embeddings. The DNNs have the advantages that they can represent complex nonlinear models and be directly optimized to discriminate between classes. Deep speaker embedding is extracted from a hidden layer of a speaker-and-phrase discriminate DNN and is expected to have more discriminative power. The deep speaker embeddings have become the state-of-the-art in the field of ASV [25].  
Early deep speaker embeddings (called d-vectors) were based on a fully connected neural network (FCNN) [26; 27]. The FCNNs were trained to classify frame-level acoustic features in a temporal context to the corresponding speaker and phrase. However, the FCNN cannot properly model the time-dependency of the frame-level features in a context. To overcome this drawback, the methods of using other kinds of DNNs, such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs), were proposed [28; 29; 30]. RNN is a neural network designed to model time-dependency of time-series data using additional recurrent connections. In this paper, we use the d-vector based on long short-term memory (LSTM) [31], which is an extension of RNN for modeling long term time dependencies efficiently by using gates mechanisms [32].  
A residual network (ResNet) [33] is the network that has constant bypass weight connections between layers to optimize very deep networks efficiently. Motivated by the success of ResNet in the field of image recognition, many studies have employed ResNet to model deep speaker embedding (called the r-vector to distinguish from the original d-vector) and have shown remarkable performance [34; 35; 36; 37]. In [38], the squeeze-and-excitation (SE) block was proposed, which is the building block of CNNs to recalibrate channel-wise responses adaptively by explicitly modeling the relationship between channels. The SE block has been successfully adopted in ResNet, which is called a squeeze-and-excitation residual network (SE-ResNet). In this study, we extract the r-vectors from SE-ResNet34 [6; 11].  
The x-vector [39; 40] is the deep speaker embedding extracted from a model based on a time-delay neural network (TDNN) [41]. The TDNN is the network that computes an output at each time stepusing the inputs from a small temporal context window similarly to CNNs. We can model long-term context information of input by stacking multiple TDNN layers, even if each TDNN layer has a small temporal context. Therefore, the TDNN-based model can efficiently capture long-term speaker information. Each layer of the TDNN models a small temporal context of the output from the previous layer. The output frames of the TDNN are aggregated over temporal pooling to capture long-term information. The x-vector/PLDA has shown state-of-the-art performance in TI-SV [25].  
Like the i-vector, deep speaker embeddings have also been used with the PLDA and exhibit performance improvements. One of the main differences between the i-vector and deep speaker embeddings is that no assumption exists regarding the shape, such as the probability distribution and covariance structure, of deep speaker embeddings. This difference also affects our proposed method, which is described later.