Risk-Sensitive Diffusion: Robustly Optimizing Diffusion Models with Noisy Samples

TL;DR

This paper proposes risk-sensitive SDE, a type of stochastic differential equation (SDE) parameterized by the risk vector that permits a robust optimization of diffusion models with noisy samples and significantly outperforms previous baselines.

Abstract

Diffusion models are mainly studied on image data. However, non-image data (e.g., tabular data) are also prevalent in real applications and tend to be noisy due to some inevitable factors in the stage of data collection, degrading the generation quality of diffusion models. In this paper, we consider a novel problem setting where every collected sample is paired with a vector indicating the data quality: risk vector. This setting applies to many scenarios involving noisy data and we propose risk-sensitive SDE, a type of stochastic differential equation (SDE) parameterized by the risk vector, to address it. With some proper coefficients, risk-sensitive SDE can minimize the negative effect of noisy samples on the optimization of diffusion models. We conduct systematic studies for both Gaussian and non-Gaussian noise distributions, providing analytical forms of risk-sensitive SDE. To verify the effectiveness of our method, we have conducted extensive experiments on multiple tabular and time-series datasets, showing that risk-sensitive SDE permits a robust optimization of diffusion models with noisy samples and significantly outperforms previous baselines.

Figures (6)

• Figure 1: A segment of noisy time series from MIMIC johnson2016mimic. The data points outside the orange region (i.e., $95\%$ confidence intervals) are observed, and a Gaussian process interpolates the ones within the area.
• Figure 2: Comparison between the diffusion process of a standard VP SDE for clean samples (i.e., the upper $5$ subfigures) and its alternative: risk-sensitive SDE, for Gaussian-corrupted samples (i.e., the lower $5$ subfigures). With the proper risk-sensitive coefficients, the clean and noisy samples will have the same marginal densities in the stability interval: $t\in[0.26, 1]$.
• Figure 3: Comparison on a Gaussian mixture data (Fig. \ref{['subfig: training data']}, three-sigma regions as ellipses), with part of Gaussian-corrupted samples. Our model (Fig. \ref{['subfig: samples from our model']}) mostly samples within the ellipses, while the samples from standard diffusion model (Fig. \ref{['subfig: samples from standard']}) typically fall out of them, and conditional generation leads to an unbalanced generation distribution (Fig. \ref{['subfig: samples from cond']}).
• Figure 4: Comparison on Gaussian mixture data (Fig. \ref{['subfig: Cauchy data']}), with part of Cauchy-corrupted samples. Despite minimal instability, our model still recovers the potential sample distribution (Fig. \ref{['subfig: Cauchy risk']}), while both baselines (Fig. \ref{['subfig: Cauchy normal']} and Fig. \ref{['subfig: Cauchy cond']}) incorrectly produce many outliers.
• Figure 5: PRD curves (i.e., precision and recall scores) of our model and baselines on Telemonitoring dataset.

Theorems & Definitions (38)

• Remark 2.1
• Definition 3.1: Risk Vectors
• Remark 3.1
• Remark 3.2
• Definition 3.2: Risk-sensitive SDE
• Remark 3.3
• Remark 3.4
• Definition 3.3: Measure of Perturbation Instability
• Remark 3.5
• Remark 3.6