{"url":"/method/affine-coupling","slug":"affine-coupling","name":"Affine Coupling","full_name":"Affine Coupling","full_name_withheld":false,"description_markdown":"**Affine Coupling** is a method for implementing a normalizing flow (where we stack a sequence of invertible bijective transformation functions). Affine coupling is one of these bijective transformation functions. Specifically, it is an example of a reversible transformation where the forward function, the reverse function and the log-determinant are computationally efficient. For the forward function, we split the input dimension into two parts:\r\n\r\n$$ \\mathbf{x}\\_{a}, \\mathbf{x}\\_{b} = \\text{split}\\left(\\mathbf{x}\\right) $$\r\n\r\nThe second part stays the same $\\mathbf{x}\\_{b} = \\mathbf{y}\\_{b}$, while the first part  $\\mathbf{x}\\_{a}$ undergoes an affine transformation, where the parameters for this transformation are learnt using the second part $\\mathbf{x}\\_{b}$ being put through a neural network. Together we have:\r\n\r\n$$ \\left(\\log{\\mathbf{s}, \\mathbf{t}}\\right) = \\text{NN}\\left(\\mathbf{x}\\_{b}\\right) $$\r\n\r\n$$ \\mathbf{s} = \\exp\\left(\\log{\\mathbf{s}}\\right) $$\r\n\r\n$$ \\mathbf{y}\\_{a} = \\mathbf{s} \\odot \\mathbf{x}\\_{a} + \\mathbf{t}  $$\r\n\r\n$$ \\mathbf{y}\\_{b} = \\mathbf{x}\\_{b} $$\r\n\r\n$$ \\mathbf{y} = \\text{concat}\\left(\\mathbf{y}\\_{a}, \\mathbf{y}\\_{b}\\right) $$\r\n\r\nImage: [GLOW](https://paperswithcode.com/method/glow)","description_state":"present","introduced_year":null,"introduced_by":{"title":null,"paper":null,"first_author":null,"n_authors":0,"url_abs":null,"archive_paper_url":null},"source":{"url":"http://arxiv.org/abs/1410.8516v6","title":"NICE: Non-linear Independent Components 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