Canonically, the variational principle suggests to prefer an expressive inference model so that the variational approximation is accurate.

Both families ofmodels perturb data with a sequence of noise distributions, and generate samples by learning to reverse this path from noise todata.

Whereas diverse variations of diffusion models exist, extending the linear diffusion into a nonlinear diffusion process is investigated by very few works. The nonlinearity effect has been hardly understood, but intuitively, there would be promising diffusion patterns to efficiently train the generative distribution towards the data distribution. This paper introduces a data-adaptive nonlinear diffusion process for score-based diffusion models. The proposed Implicit Nonlinear Diffusion Model (INDM) learns by combining a normalizing flow and a diffusion process.

Score-based diffusion models synthesize samples by reversing a stochastic process that diffuses data to noise, and are trained by minimizing a weighted combination of score matching losses. The log-likelihood of score-based diffusion models can be tractably computed through a connection to continuous normalizing flows, but log-likelihood is not directly optimized by the weighted combination of score matching losses. We show that for a specific weighting scheme, the objective upper bounds the negative log-likelihood, thus enabling approximate maximum likelihood training of score-based diffusion models. We empirically observe that maximum likelihood training consistently improves the likelihood of score-based diffusion models across multiple datasets, stochastic processes, and model architectures. Our best models achieve negative log-likelihoods of 2.83 and 3.76 bits/dim on CIFAR-10 and ImageNet $32\times 32$ without any data augmentation, on a par with state-of-the-art autoregressive models on these tasks.

Canonically, the variational principle suggests to prefer an expressive inference model so that the variational approximation is accurate.

Whereas diverse variations of diffusion models exist, extending the linear diffusion into a nonlinear diffusion process is investigated by very few works. The nonlinearity effect has been hardly understood, but intuitively, there would be promising diffusion patterns to efficiently train the generative distribution towards the data distribution. This paper introduces a data-adaptive nonlinear diffusion process for score-based diffusion models. The proposed Implicit Nonlinear Diffusion Model (INDM) learns by combining a normalizing flow and a diffusion process. This flow network is key to forming a nonlinear diffusion, as the nonlinearity depends on the flow network.

Score-based diffusion models synthesize samples by reversing a stochastic process that diffuses data to noise, and are trained by minimizing a weighted combination of score matching losses. The log-likelihood of score-based diffusion models can be tractably computed through a connection to continuous normalizing flows, but log-likelihood is not directly optimized by the weighted combination of score matching losses. We show that for a specific weighting scheme, the objective upper bounds the negative log-likelihood, thus enabling approximate maximum likelihood training of score-based diffusion models. We empirically observe that maximum likelihood training consistently improves the likelihood of score-based diffusion models across multiple datasets, stochastic processes, and model architectures. Our best models achieve negative log-likelihoods of 2.83 and 3.76 bits/dim on CIFAR-10 and ImageNet 32\times 32 without any data augmentation, on a par with state-of-the-art autoregressive models on these tasks.

Accurate predictions of when a component will fail are crucial when planning maintenance, and by modeling the distribution of these failure times, survival models have shown to be particularly useful in this context. The presented methodology is based on conventional neural network-based survival models that are trained using data that is continuously gathered and stored at specific times, called snapshots. An important property of this type of training data is that it can contain more than one snapshot from a specific individual which results in that standard maximum likelihood training can not be directly applied since the data is not independent. However, the papers show that if the data is in a specific format where all snapshot times are the same for all individuals, called homogeneously sampled, maximum likelihood training can be applied and produce desirable results. In many cases, the data is not homogeneously sampled and in this case, it is proposed to resample the data to make it homogeneously sampled. How densely the dataset is sampled turns out to be an important parameter; it should be chosen large enough to produce good results, but this also increases the size of the dataset which makes training slow. To reduce the number of samples needed during training, the paper also proposes a technique to, instead of resampling the dataset once before the training starts, randomly resample the dataset at the start of each epoch during the training. The proposed methodology is evaluated on both a simulated dataset and an experimental dataset of starter battery failures. The results show that if the data is homogeneously sampled the methodology works as intended and produces accurate survival models. The results also show that randomly resampling the dataset on each epoch is an effective way to reduce the size of the training data.

(Neal and Hinton, 1998) recast maximum likelihood estimation of any given latent variable model as the minimization of a free energy functional $F$, and the EM algorithm as coordinate descent applied to $F$. Here, we explore alternative ways to optimize the functional. In particular, we identify various gradient flows associated with $F$ and show that their limits coincide with $F$'s stationary points. By discretizing the flows, we obtain practical particle-based algorithms for maximum likelihood estimation in broad classes of latent variable models. The novel algorithms scale to high-dimensional settings and perform well in numerical experiments.