HEDGE generates hypergraphs via a linear-Gaussian forward diffusion on incidence matrices with a hypergraph-specific heat operator, then learns a permutation-equivariant reverse drift to sample from the Gaussian base.
The general mixture-diffusion sde and its relationship with an uncertain-volatility option model with volatility-asset decorrelation
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abstract
In the present paper, given an evolving mixture of probability densities, we define a candidate diffusion process whose marginal law follows the same evolution. We derive as a particular case a stochastic differential equation (SDE) admitting a unique strong solution and whose density evolves as a mixture of Gaussian densities. We present an interesting result on the comparison between the instantaneous and the terminal correlation between the obtained process and its squared diffusion coefficient. As an application to mathematical finance, we construct diffusion processes whose marginal densities are mixtures of lognormal densities. We explain how such processes can be used to model the market smile phenomenon. We show that the lognormal mixture dynamics is the one-dimensional diffusion version of a suitable uncertain volatility model, and suitably reinterpret the earlier correlation result. We explore numerically the relationship between the future smile structures of both the diffusion and the uncertain volatility versions.
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2026 2verdicts
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An expository tutorial deriving the ELBO for SDE-based generative models and presenting diffusion, score, and flow matching as variational parameterizations illustrated on a 1D example.
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Hypergraph Generation via Structured Stochastic Diffusion
HEDGE generates hypergraphs via a linear-Gaussian forward diffusion on incidence matrices with a hypergraph-specific heat operator, then learns a permutation-equivariant reverse drift to sample from the Gaussian base.