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Auto-associative neural network (AANN) is a fully connected feed-forward neural network, trained to reconstruct its input at its output through a hidden compression layer, which has fewer numbers of nodes than the dimensionality of input. AANNs are used to model speakers in speaker verification, where a speaker-specific AANN model is obtained by adapting (or retraining) the universal background model (UBM) AANN, an AANN trained on multiple held out speakers, using corresponding speaker data. When the amount of speaker data is limited, this adaptation procedure may lead to overfitting as all the parameters of UBM-AANN are adapted. In this paper, we introduce and develop the factor analysis theory of AANNs to alleviate this problem. We hypothesize that only the weight matrix connecting the last nonlinear hidden layer and the output layer is speaker-specific, and further restrict it to a common low-dimensional subspace during adaptation. The subspace is learned using large amounts of development data, and is held fixed during adaptation. Thus, only the coordinates in a subspace, also known as i-vector, need to be estimated using speaker-specific data. The update equations are derived for learning both the common low-dimensional subspace and the i-vectors corresponding to speakers in the subspace. The resultant i-vector representation is used as a feature for the probabilistic linear discriminant analysis model. The proposed system shows promising results on the NIST-08 speaker recognition evaluation (SRE), and yields a 23% relative improvement in equal error rate over the previously proposed weighted least squares-based subspace AANNs system. The experiments on NIST-10 SRE confirm that these improvements are consistent and generalize across datasets.