Product of Gaussian Mixture Diffusion Models

Abstract In this work, we tackle the problem of estimating the density $$ f_X $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>f</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:math> of a random variable $$ X $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> by successive smoothing, such that the smoothed random variable $$ Y $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> fulfills the diffusion partial differential equation $$ (\partial _t - \Delta _1)f_Y(\,\cdot \,, t) = 0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Δ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mspace/><mml:mo>·</mml:mo><mml:mspace/><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> with initial condition $$ f_Y(\,\cdot \,, 0) = f_X $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mspace/><mml:mo>·</mml:mo><mml:mspace/><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math> . We propose a product-of-experts-type model utilizing Gaussian mixture experts and study configurations that admit an analytic expression for $$ f_Y (\,\cdot \,, t) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mspace/><mml:mo>·</mml:mo><mml:mspace/><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> . In particular, with a focus on image processing, we derive conditions for models acting on filter, wavelet, and shearlet responses. Our construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results for image denoising where our models are competitive while being tractable, interpretable, and having only a small number of learnable parameters. As a by-product, our models can be used for reliable noise level estimation, allowing blind denoising of images corrupted by heteroscedastic noise.