Gaussian Interpolation Flows
Authors: Yuan Gao, Jian Huang, and Yuling Jiao
JMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | To demonstrate the bounds presented in Propositions 6.1 and 6.2, we conducted further experiments with a mixture of eight two-dimensional Gaussian distributions. These propositions provide bounds for the stability of the flow when subjected to perturbations in either the source distribution or the velocity field. Let the target distribution be the following two-dimensional Gaussian mixture j=1 φ(x; µj, Σj), where φ(x; µj, Σj) is the probability density function for the Gaussian distribution with mean µj = 12(sin(2(j 1)π/8), cos(2(j 1)π/8)) and covariance matrix Σj = 0.032I2 for j = 1, , 8. For Gaussian mixtures, the velocity field has an explicit formula, which facilitates the perturbation analysis. To illustrate the bound in Proposition 6.1, we consider a perturbation of the source distribution for the following model: Xt = at Z + bt X with at = 1 t + ζ 1 + ζ , bt = t + ζ where ζ [0, 0.3] is a value controlling the perturbation level. It is easy to see a0 = 1/(1 + ζ), b0 = ζ/(1 + ζ). Thus, the source distribution Law(a0Z + b0X) is a mixture of Gaussian distributions. Practically, we can use a Gaussian distribution γ2,a2 0 to replace this source distribution. In Proposition 6.1, we bound the error between the distributions of generated samples due to the replacement, that is, W2(X1#γd,a2 0, ν) Cb0, where C is a constant. We illustrate this theoretical bound using the mixture of Gaussian distributions and the Gaussian interpolation flow given above. We consider a mesh for the variable ζ and plot the curve for b0 and W2(X1#γd,a2 0, ν) in Figure 5. Through Figure 5, an approximate linear relation between b0 and W2(X1#γd,a2 0, ν) is observed, which supports the results of Proposition 6.1. We now consider perturbing the velocity field vt by adding random noise. Let ϵ [0.5, 5.5]. The random noise is generated using a Bernoulli random variable supported on { ϵ, ϵ}. Let vt denote the perturbed velocity field. Then we can compute vt := vt vt 2 = 2ϵ2. We use the velocity field vt and the perturbed velocity field t to generate samples and compute the squared Wasserstein-2 distance between the sample distributions. According to Proposition 6.2, the squared Wasserstein-2 distance should be linearly upper bounded as O( vt), that is, W 2 2 (Y1#µ, ν) C Z 1 R2 ϵ2pt(x)dxdt = Cϵ2, where C is a constant. This theoretical insight is illustrated in Figure 6, where a linear relationship between these two variables is observed. |
| Researcher Affiliation | Academia | Yuan Gao EMAIL Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong SAR, China Jian Huang EMAIL Departments of Data Science and AI, and Applied Mathematics The Hong Kong Polytechnic University Hong Kong SAR, China Yuling Jiao EMAIL School of Mathematics and Statistics and Hubei Key Laboratory of Computational Science Wuhan University, Wuhan, China |
| Pseudocode | No | The paper does not contain any structured pseudocode or algorithm blocks. The methodologies are described through mathematical formulations and theorems. |
| Open Source Code | No | The implementation is based on the Git Hub repository at https://github.com/suxuann/ddib. This link refers to the code for DDIBs (Dual Diffusion Implicit Bridges), which is a related work by Su et al. (2023), used for replicating their figures 3 and 4, not for the Gaussian Interpolation Flows methodology described in this paper. |
| Open Datasets | Yes | To demonstrate the bounds presented in Propositions 6.1 and 6.2, we conducted further experiments with a mixture of eight two-dimensional Gaussian distributions. These propositions provide bounds for the stability of the flow when subjected to perturbations in either the source distribution or the velocity field. Let the target distribution be the following two-dimensional Gaussian mixture j=1 φ(x; µj, Σj), where φ(x; µj, Σj) is the probability density function for the Gaussian distribution with mean µj = 12(sin(2(j 1)π/8), cos(2(j 1)π/8)) and covariance matrix Σj = 0.032I2 for j = 1, , 8. |
| Dataset Splits | No | The paper uses synthetic datasets like a 'mixture of eight two-dimensional Gaussian distributions', 'Concentric Rings data', 'Moons data', and 'Concentric Squares data'. For these synthetic datasets, standard training, validation, and test splits are not typically defined or relevant, and the paper does not specify any such splits. |
| Hardware Specification | No | The paper does not provide any specific details about the hardware used for running experiments, such as GPU models, CPU models, or memory specifications. |
| Software Dependencies | No | The paper mentions a GitHub repository (https://github.com/suxuann/ddib) for replicating figures 3 and 4, which is code from a related work (Su et al., 2023). However, it does not specify any software dependencies (e.g., programming languages, libraries, frameworks with version numbers) for its own described methodology or illustrations. |
| Experiment Setup | No | The paper focuses on theoretical analysis and illustrations of theoretical bounds. While it describes parameters for generating synthetic data and perturbation levels for its demonstrations in Section 6, it does not provide specific experimental setup details such as hyperparameters (e.g., learning rate, batch size, number of epochs) or training configurations for any machine learning model developed within this paper. |