1 SNR - INVARIANT PLDA 1 Lecture Notes on SNR - Invariant PLDA

This document provides the derivations of the equations in the paper: Na Li and M.W. Mak, “SNR-Invariant PLDA Modeling in Nonparametric Subspace for Robust Speaker Verification”, IEEE/ACM Trans. on Audio Speech and Language Processing, vol. 23, no. 10, pp. 1648-1659, Oct. 2015. Please cite this document as: M.W. Mak, Lecture Notes on SNR-Invariant PLDA, Technical Report and Lecture Note Series, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, April 2015. 1 SNR-Invariant PLDA 1.1 Generative Model Denote xij as the j-th D-dimensional i-vector from speaker i, where the i-vector is obtained from an utterance with SNR falling into the k-th SNR group. Then, we have xij = m + Vhi + Uwk + k ij (1) where m is a D × 1 vector representing the global mean of i-vectors, hi is a P × 1 vector denoting the speaker factor with the prior distributionN (0, I), wk is a Q×1 vector denoting the latent SNR factor with prior distributionN (0, I), ij is a D×1 vector denoting the residual with distributionN (0,Σ), V is a D× P matrix whose columns span the speaker subspace, and U is a D ×Q matrix whose columns span the SNR subspace. 1.2 EM Formulation Denote all of the training i-vectors as X = {xij |i = 1, . . . , S; j = 1, . . . ,Hi(k); k = 1, . . . ,K}, where S is the number of training speakers, Hi(k) is the number of utterances from speaker i at the k-th SNR group, and K is the number of SNR groups. Eq. 1 can be written as: xij = m + [V U] [ hi wk ] + ij = m + Bẑik + k ij , where B = [V U] and ẑik = [hi w T k ] T. The model parameters θ = {m,V,U,Σ} are estimated by an EM algorithm, in which the loading matrices V and U can be estimated either separately or jointly. 1.2.1 Decoupled Estimation of Loading Matrices In this approach, we focus on one latent factor at a time and marginalize over the other latent factors. For example, to estimate V, we compute the posterior expectation of hi by marginalizing over wk. Thus, the posterior density of hi is written as: p(hi|xij ,θ) ∝ p(xij |hi,θ)p(hi) = ∫ p(xij ,wk|hi,θ)p(hi)dwk = ∫ p(xij |hi,wk,θ)p(wk)p(hi)dwk 1 SNR-INVARIANT PLDA 2 = ∫ N (xij |m + Vhi + Uwk,Σ)N (wk|0, I)N (hi|0, I)dwk = N (xij |m + Vhi,Φ)N (hi|0, I) ∝ exp { hi V Φ(xij −m)− 1 2 hi (I + V ΦV)hi } (2) where Φ = UUT + Σ. Comparing this posterior density with a standard Gaussian, we have 〈hi|xij〉 = ( I + VTΦ−1V )−1 VΦ(xij −m) 〈hihi |xij〉 = ( I + VTΦ−1V )−1 + 〈hi|xij〉〈hi|xij〉. (3) If all of the i-vectors of speaker i are given, we evaluate the joint posterior p(hi|xij ∀j and k,θ) ∝ K ∏