Variational Inference of Correspondence-LDA with Multinomial Generative Process

In original Correspondence LDA (Corr-LDA) [1] the image region is generated by a multivariate Gaussian distribution. We replace this Gaussian distribution by a multinomial distribution, which gives a modified algorithm for parameter estimation. 1 Corr-LDA with Multinomial Generative Process To keep the consistence, we denote visual word and auditory word as sensory word. Thus, A document (a captioned image/audio) consists of a bag of sensory words and a bag of textual words. Our modified Corr-LDA gives a generative process as follows: 1. Draw topic proportions θ ∼ Dirichlet(α). 2. For each sensory word vm,m ∈ {1, · · · ,M}: (a) Draw topic assignment zm|θ ∼ Multinomial(θ). (b) Draw sensory word vm|zm ∼ Multinomial(πzm) 3. For each textual word wn, n ∈ {1, · · · , N} (a) Draw discrete indexing variable yn ∼ Uniform(1, · · · ,M) (b) Draw textual word wn ∼ Multinomial(βzyn ) Observe that step 2.b is differed from original Corr-LDA paper [1], where D.Blei used a multivariate Gaussian to describe the generative process of sensory word v. The graphical representation of our modified Corr-LDA is depicted in Figure 1. The notations used in this paper are summarized in Table 1.