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GAN-based Gaussian Mixture Model Responsibility Learning

Mixture Model (MM) is a probabilistic framework which allows us to define a dataset containing K different modes. When each of the modes is associated with a Gaussian distribution, we refer it as Gaussian MM, or GMM. Given a data point x, GMM may assume the existence of a random index k ∈ {1, . . . , K } identifying which Gaussian the particular data is associated with. In a traditional GMM paradigm, it is straightforward to compute in closed-form, the conditional like- lihood p(x|k, θ), as well as responsibility probability p(k|x, θ) which describes the distribution index corresponds to the data. Computing the responsibility allows us to retrieve many important statistics of the overall dataset, including the weights of each of the modes. Modern large datasets often contain multiple unlabelled modes, such as paintings dataset containing several styles; fashion images containing several unlabelled categories. In its raw representation, the Euclidean distances between the data do not allow them to form mixtures naturally, nor it’s feasible to compute responsibility distribution, making GMM unable to apply. To this paper, we utilize the Generative Adversarial Network (GAN) framework to achieve an alternative plausible method to compute these probabilities at the data’s latent space z instead of x. Instead of defining p(x|k, θ) explicitly, we devised a modified GAN to allow us to define the distribution using p(z|k, θ), where z is the corresponding latent representation of x, as well as p(k|x, θ) through an additional classification network which is trained with the GAN in an “end-to-end” fashion. These techniques allow us to discover interesting properties of an unsupervised dataset, including dataset segments as well as generating new “out-distribution” data by smooth linear interpolation across any combinations of the modes in a completely unsupervised manner.

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