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Reinterpreting EMML as Mirror Descent for Constrained Maximum Likelihood Estimation

Antonin Clerc, Ségolène Martin, Nicolas Papadakis, Gabriele Steidl

TL;DR

The paper addresses Poisson-noise image reconstruction by recasting EMML as a mirror-descent step on a reparametrized objective, enabling convex constraints through Bregman projections while preserving multiplicative update structure. It shows EMML is equivalent to MD on the exponential-family with a canonical parameter $\theta=\log x$ and analyzes convergence, including a sublinear $\mathcal{O}(1/N)$ rate for the unconstrained case and a similar bound for constrained variants. By introducing dual-space projections and a divergence $D_u$, the authors derive a constrained EMML that maintains nonnegativity and achieves faster practical convergence and higher PSNR in hyperspectral unmixing experiments under simplex constraints. The work broadens EMML applicability to general convex constraints with guaranteed convergence and practical efficiency for photon-limited imaging.

Abstract

The Expectation--Maximization Maximum Likelihood (EMML) algorithm belongs to the Expectation--Maximization family and is widely used for image reconstruction problems under Poisson noise.In this paper, we reinterpret EMML as a mirror descent method applied to a reparametrized objective function. This perspective allows us to incorporate convex constraints into the algorithm through appropriately chosen Bregman projections, while preserving the multiplicative structure of the EMML updates to ensure computational efficiency. We then establish the convergence of the resulting algorithm toward a solution of the constrained maximum-likelihood problem. Numerical experiments on hyperspectral unmixing problems demonstrate that the constrained EMML converges in fewer iterations than the classical EMML.

Reinterpreting EMML as Mirror Descent for Constrained Maximum Likelihood Estimation

TL;DR

The paper addresses Poisson-noise image reconstruction by recasting EMML as a mirror-descent step on a reparametrized objective, enabling convex constraints through Bregman projections while preserving multiplicative update structure. It shows EMML is equivalent to MD on the exponential-family with a canonical parameter and analyzes convergence, including a sublinear rate for the unconstrained case and a similar bound for constrained variants. By introducing dual-space projections and a divergence , the authors derive a constrained EMML that maintains nonnegativity and achieves faster practical convergence and higher PSNR in hyperspectral unmixing experiments under simplex constraints. The work broadens EMML applicability to general convex constraints with guaranteed convergence and practical efficiency for photon-limited imaging.

Abstract

The Expectation--Maximization Maximum Likelihood (EMML) algorithm belongs to the Expectation--Maximization family and is widely used for image reconstruction problems under Poisson noise.In this paper, we reinterpret EMML as a mirror descent method applied to a reparametrized objective function. This perspective allows us to incorporate convex constraints into the algorithm through appropriately chosen Bregman projections, while preserving the multiplicative structure of the EMML updates to ensure computational efficiency. We then establish the convergence of the resulting algorithm toward a solution of the constrained maximum-likelihood problem. Numerical experiments on hyperspectral unmixing problems demonstrate that the constrained EMML converges in fewer iterations than the classical EMML.
Paper Structure (8 sections, 6 theorems, 31 equations, 3 figures, 1 algorithm)

This paper contains 8 sections, 6 theorems, 31 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background on Mirror Descent
  3. The EMML Algorithm
  4. EMML as EM algorithm
  5. EMML as MD
  6. Constrained EMML
  7. Numerical Experiments
  8. Conclusion and Perspectives

Key Result

Lemma 2

Suppose that Assumption assumption:functions holds. Let $\tau = 1/L$ and, for all $x \in \operatorname{int}(\operatorname{dom} u)$, where $T_\tau$ is given by eq:Ttau_MD. Then, for all $s \in \operatorname{dom} u$,

Figures (3)

  • Figure 1: Comparison between the observed convergence behavior and the corresponding theoretical bounds for EMML and constrained EMML with $\sigma=40$. In both cases, the empirical convergence is substantially faster than the predicted sublinear rate.
  • Figure 2: PSNR of the reconstructed abundance maps obtained with EMML and constrained EMML at different noise levels. The constrained algorithm consistently achieves higher PSNR values and outperforms EMML even at higher noise levels, while converging in nearly twice fewer iterations.
  • Figure 3: Reconstructed abundance map for Soil with Poisson noise $\sigma = 40$. The reconstructions obtained with the EMML algorithm are visibly noisier than those obtained with the Constrained EMML algorithm.

Theorems & Definitions (8)

  • Lemma 2: Adapted from bauschke2017descent
  • Theorem 3: bauschke2017descent
  • Proposition 4
  • Proposition 5
  • Proposition 7
  • proof
  • Proposition 8
  • proof