Diffusion models are generative models that have recently demonstrated
impressive performances in terms of sampling quality and density estimation in
high dimensions. They rely on a forward continuous diffusion process and a
backward continuous denoising process, which can be described by a
time-dependent vector field and is used as a generative model. In the original
formulation of the diffusion model, this vector field is assumed to be the
score function (i.e. it is the gradient of the log-probability at a given time
in the diffusion process). Curiously, on the practical side, most studies on
diffusion models implement this vector field as a neural network function and
do not constrain it be the gradient of some energy function (that is, most
studies do not constrain the vector field to be conservative). Even though some
studies investigated empirically whether such a constraint will lead to a
performance gain, they lead to contradicting results and failed to provide
analytical results. Here, we provide three analytical results regarding the
extent of the modeling freedom of this vector field. {Firstly, we propose a
novel decomposition of vector fields into a conservative component and an
orthogonal component which satisfies a given (gauge) freedom. Secondly, from
this orthogonal decomposition, we show that exact density estimation and exact
sampling is achieved when the conservative component is exactly equals to the
true score and therefore conservativity is neither necessary nor sufficient to
obtain exact density estimation and exact sampling. Finally, we show that when
it comes to inferring local information of the data manifold, constraining the
vector field to be conservative is desirable.

扩散模型是一种生成模型，通过前向连续扩散过程和后向连续去噪过程，使用一个时变向量场作为生成模型。尽管大多数研究将该向量场实现为神经网络函数，并不加限制为某种能量函数的梯度，限制该向量场具有保守性在推断数据流形的局部信息时是可取的。同时，通过分解向量场为保守组件和正交组件，并展示了具有保守组件等于真实得分时实现精确密度估计和精确抽样的情况，从而表明保守性既不是必要条件也不是充分条件。