# Regularized Within-Class Precision Matrix Based PLDA in Text-Dependent Speaker Verification
###### Abstract  
In the field of speaker verification, probabilistic linear discriminant analysis (PLDA) is the dominant method for back-end scoring. To estimate the PLDA model, the between-class covariance and within-class precision matrices must be estimated from samples. However, the empirical covariance/precision estimated from samples has estimation errors due to the limited number of samples available. In this paper, we propose a method to improve the conventional PLDA by estimating the PLDA model using the regularized within-class precision matrix. We use graphical least absolute shrinking and selection operator (GLASSO) for the regularization. computational cost even with short utterances, for example, shorter than about 3 s. Because TD-SV has no need to compensate for the phrase variability, rather it should distinguish between not only speakers but also phrases. In addition, it is relatively easy to control the phrase variability in TD-SV because of the limitation for the phrase. Due to these advantages, TD-SV has been widely used in many real applications that require both the higher performance and short utterance, such as voice assistant [3, 4]. One drawback of TD-SV is that it can be vulnerable to replay attacks which use recorded voices of the target user. However, currently there have been many active studies to prepare for the attacks and they have achieved high performance [5, 6, 7, 8, 9, 10, 11, 12]. As mentioned above, speaker embedding is the feature vector on fixed-dimension subspace that represents speaker information contained in the utterance. Typically, the speaker embedding not only has class information (corresponding to speaker-and-phrase information in TD-SV) but also undesired information, such as session information [13]. Note that the speaker embedding in TD-SV actually contains not only speaker information and but also phrase information. Throughout this paper, nevertheless, we call it speaker embedding for convenience, rather than speaker-and-phrase embedding. Too much undesired information raises the within-class variability, which can degrade the speaker verification performance. However, it is challenging to completely remove only unwanted information in the embedding. In many cases, the score is computed in a more discriminative subspace to compensate for the within-class variability of the embedding [14]. In the field of ASV, probabilistic linear discriminant analysis (PLDA) [15] has become the dominant method for back-end scoring. It probabilistically models a discriminative subspace that compensates for the within-class variability. To estimate a PLDA model, the between-class covariance matrix and within-class precision matrix (the inverse of the within-class covariance matrix) must be estimated first. In practice, the empirical covariance/precision matrix is used instead of the true covariance/precision matrix because it is unknown. However, the empirical covariance/precision matrix has estimation errors because of the limited number of available samples (corresponding to the embeddings in our case). The error is increased if the number of samples is insufficient compared to the number of parameters, which can degrade performance. In this paper, we propose a method to improve the performance of conventional PLDA by regularizing the within-class precision matrix used to estimate the PLDA model. The regularization of the within-class precision matrix is motivated by the need for the reduction of estimation errors contained in the empirical within-class precision matrix. We use graphical least absolute shrinking and selection operator (GLASSO) [16, 17, 18] to regularize the within-class precision matrix, which makes the precision matrix sparse. The reason for using the GLASSO for regularization, among many kinds of regularization methods, is that the GLASSO makes precision matrix sparse, which is based on our assumption that the true precision matrix has the conditional independence structure.