## 1 Introduction  
Speaker verification is the task of verifying the claimed speaker identity in the input speech. The speaker verification process is composed of three steps: acoustic feature extraction, utterance-level feature extraction, and scoring. In the first step, spectral parameters--also referred to as acoustic features--are extracted from short speech frames. One of the most popular acoustic features are the Mel-frequency cepstral coefficients (MFCCs), which represent the spectral envelope of the speech within the given time-frame [1]. However, since most classification or verification algorithms operate on fixed dimensional vectors, acoustic features cannot be directly used with such methods due to the variable duration property of speech [1]. Therefore, in the second step, the frame-level acoustic features are aggregated to obtain a single utterance-level feature, which is also known as an embedding vector. The utterance-level feature is a compact representation of the given speech segment or utterance, conveying information on the variability (_variability_ refers to the spread of the data distribution) caused by various factors (e.g., speaker identity, recording channel, environmental noise). In the final step, the system compares two utterance-level features and measures the similarity or the likelihood of the given utterances produced by the same speaker. Many previous studies on utterance-level features focused on efficiently reducing the dimensionality of a Gaussian mixture model (GMM) supervector, which is a concatenation of the mean vectors of each mixture component [2], while preserving the speaker-relevant information via factorization (e.g., eigenvoice adaptation and joint factor analysis) [3, 4]. Particularly the i-vector framework [5, 6], which projects the variability within the GMM supervector caused by various factors (e.g., channel and speaker) onto a low-dimensional subspace, has become one of the most dominant techniques used in speaker recognition. The i-vector framework is essentially a linear factorization technique which decomposes the variability of the GMM supervector into a total variability matrix and a latent random variable. However, since these linear factorization techniques assume the variability as a linear function of the latent factor, the i-vector framework is not considered to fully capture the whole variability of the given speech utterances. Recently, various studies [7, 8, 9, 10, 11, 12, 13] have been carried out for non-linearly extracting utterance-level features via deep learning. In [7], a deep neural network (DNN) for frame-level speaker classification was trained and the activations of the last hidden layer--namely, the d-vectors--were taken as a non-linear speaker representation. In [8, 9], a time delay neural network (TDNN) utterance embedding technique is proposed where the embedding (i.e., the x-vector) is obtained by statistically pooling the frame-level activations of the TDNN. In [10], the x-vector framework was further improved by incorporating long short-term memory (LSTM) layers to obtain the embedding. In [11], gradient reversal layer and adversarial training were employed to extract an embedding vector robust to channel variation. In [12, 13], the embedding neural networks were trained to directly optimize the verification performance in an end-to-end fashion. However, since most of the previously proposed deep learning-based feature extraction models are trained in a supervised manner (which requires speaker or phonetic labels for the training data), it is impossible to use them when little to no labeled data are available for training. At present, due to the increasing demand for voice-based authentication systems, verifying users with a randomized pass-phrase with constrained vocabulary has become an important task [14]. This particular task is called _random digit-string speaker verification_, where the speakers are enrolled and tested with random sequences of digits.