Here, we include implementation details omitted from the main paper for brevity. Upon acceptance, a deanonymized repository will be released. The last layer's dimension depended upon the exact The feature extractors and decoders varied by domain. In particular, we found that if we did not apply this linear transformation (i.e., pass the raw encodings For VQ-based methods, use a large enough codebook to have at least one element per class. Other differences simply reflected differences in architecture (e.g., For iNat, we trained all models with batch size 256, using the hyperparameters specified in Table 3.

We introduce Statistical Flow Matching (SFM), a novel and mathematically rigorous flow-matching framework on the manifold of parameterized probability measures inspired by the results from information geometry. We demonstrate the effectiveness of our method on the discrete generation problem by instantiat-ing SFM on the manifold of categorical distributions whose geometric properties remain unexplored in previous discrete generative models. Utilizing the Fisher information metric, we equip the manifold with a Riemannian structure whose intrinsic geometries are effectively leveraged by following the shortest paths of geodesics. We develop an efficient training and sampling algorithm that overcomes numerical stability issues with a diffeomorphism between manifolds. Our distinctive geometric perspective of statistical manifolds allows us to apply optimal transport during training and interpret SFM as following the steepest direction of the natural gradient. Unlike previous models that rely on variational bounds for likelihood estimation, SFM enjoys the exact likelihood calculation for arbitrary probability measures. We manifest that SFM can learn more complex patterns on the statistical manifold where existing models often fail due to strong prior assumptions. Comprehensive experiments on real-world generative tasks ranging from image, text to biological domains further demonstrate that SFM achieves higher sampling quality and likelihood than other discrete diffusion or flow-based models. Our code is available at https://github.com/ccr-cheng/

InfoGAN [10] in particular, learns an unsupervised disentangled representation by maximizing the mutual information between the discrete or continuous latent variables and the corresponding generated samples.

We assume Satt 6=, and Sbbox 6=, meaning that for every image, we attend toatleast1location, and we haveabounding box with area greater than0.

Gradient-based optimization with categorical variables typically relies on score-function estimators, which are unbiased but noisy, or on continuous relaxations that replace the discrete distribution with a smooth surrogate admitting a pathwise (reparameterized) gradient, at the cost of optimizing a biased, temperature-dependent objective. In this paper, we extend this family of relaxations by introducing a diffusion-based soft reparameterization for categorical distributions. For these distributions, the denoiser under a Gaussian noising process admits a closed form and can be computed efficiently, yielding a training-free diffusion sampler through which we can backpropagate. Our experiments show that the proposed reparameterization trick yields competitive or improved optimization performance on various benchmarks.

Categorical random variables can faithfully represent the discrete and uncertain aspects of data as part of a discrete latent variable model. Learning in such models necessitates taking gradients with respect to the parameters of the categorical probability distributions, which is often intractable due to their combinatorial nature. A popular technique to estimate these otherwise intractable gradients is the Log-Derivative trick. This trick forms the basis of the well-known REINFORCE gradient estimator and its many extensions. While the Log-Derivative trick allows us to differentiate through samples drawn from categorical distributions, it does not take into account the discrete nature of the distribution itself. Our first contribution addresses this shortcoming by introducing the CatLog-Derivative trick -- a variation of the Log-Derivative trick tailored towards categorical distributions. Secondly, we use the CatLog-Derivative trick to introduce IndeCateR, a novel and unbiased gradient estimator for the important case of products of independent categorical distributions with provably lower variance than REINFORCE. Thirdly, we empirically show that IndeCateR can be efficiently implemented and that its gradient estimates have significantly lower bias and variance for the same number of samples compared to the state of the art.