Diffusion-Based Hypothesis Testing and Change-Point Detection
Authors: Sean Moushegian, Taposh Banerjee, Vahid Tarokh
Organizations: Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708. · Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15213.
Score-based methods have recently seen increasing popularity in modeling and generation. Methods have been constructed to perform hypothesis testing and change-point detection with score functions, but these methods are in general not as powerful as their likelihood-based peers. Recent works consider generalizing the score-based Fisher divergence into a diffusion-divergence by transforming score functions via multiplication with a matrix-valued function or a weight matrix. In this paper, we extend the score-based hypothesis test and change-point detection stopping rule into their diffusion-based analogs. Additionally, we theoretically quantify the performance of these diffusion-based algorithms and study scenarios where optimal performance is achievable. We propose a method of numerically optimizing the weight matrix and present numerical simulations to illustrate the advantages of diffusion-based algorithms.
Figures & tables
Figure 1: Performance of LLR-based, Fisher-based, and diffusion-based Change-Point Detection Stopping Rules. In this simulation, P∞,P1 were chosen to be (a) Gaussian Distributions, (b) Gauss-Bernoulli Restricted Boltzmann Machine Distributions, and (c) Quartic Exponential Distributions. ARL and EDD are averaged over 10,000 sample paths.
Figure 2: ROC curves for diffusion-based, score-based, and LLR-based hypothesis tests, plotted for batch sizes n∈{1,10,100} . P∞,P1 are chosen to follow Gaussian Distributions (top), Gauss-Bernoulli Restricted Boltzmann Machine Distributions (middle), and Quartic Exponential Distributions (bottom). Each ROC curve uses 10,000 batches. Similar plots for n∈{5,25,50} are shown in Figure 3 in the Appendix.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: ROC curves for diffusion-based, score-based, and LLR-based hypothesis tests, plotted for batch sizes n∈{5,25,50} . P∞,P1 are chosen to follow Gaussian Distributions (top), Gauss-Bernoulli Restricted Boltzmann Machine Distributions (middle), and Quartic Exponential Distributions (bottom). Each ROC curve uses 10,000 batches. Note that similar plots for n∈{1,10,100} are provided in Figure 2 in the main text.
Department of Mathematical Sciences, Tsinghua University · Department of Mathematics, City University of Hong Kong, Hong Kong SAR · Artificial Intelligence Research Institute, Shenzhen MSU-BIT University +1