Progressively applying Gaussian noise transforms complex data distributions\nto approximately Gaussian. Reversing this dynamic defines a generative model.\nWhen the forward noising process is given by a Stochastic Differential Equation\n(SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the\nassociated reverse-time SDE may be estimated using score-matching. A limitation\nof this approach is that the forward-time SDE must be run for a sufficiently\nlong time for the final distribution to be approximately Gaussian. In contrast,\nsolving the Schr\\"odinger Bridge problem (SB), i.e. an entropy-regularized\noptimal transport problem on path spaces, yields diffusions which generate\nsamples from the data distribution in finite time. We present Diffusion SB\n(DSB), an original approximation of the Iterative Proportional Fitting (IPF)\nprocedure to solve the SB problem, and provide theoretical analysis along with\ngenerative modeling experiments. The first DSB iteration recovers the\nmethodology proposed by Song et al. (2021), with the flexibility of using\nshorter time intervals, as subsequent DSB iterations reduce the discrepancy\nbetween the final-time marginal of the forward (resp. backward) SDE with\nrespect to the prior (resp. data) distribution. Beyond generative modeling, DSB\noffers a widely applicable computational optimal transport tool as the\ncontinuous state-space analogue of the popular Sinkhorn algorithm (Cuturi,\n2013).\n