How can right Bregman proximal gradient extend multiplicative updates for Poisson inverse problems?

R-BPG moves the smoothness assumption into mirror coordinates

Standard Bregman proximal gradient assumes the data-fidelity term is relatively smooth in the primal variables, meaning L·φ − f is convex for some Legendre-type potential φ. The Poisson negative log-likelihood does not fit this template with the plain negative entropy: the appendix shows that relative smoothness would require ‖y‖₁ ≤ L·n·t for every t > 0, which fails as t decreases toward zero.

The paper's move is to reparameterize the objective in mirror coordinates and require relative smoothness of the reparameterized function f∘∇φ* instead. In primal variables the resulting scheme alternates a preconditioned gradient step, which uses the inverse Hessian of the potential, with a right Bregman proximal update. The authors call this algorithm R-BPG and summarize the contrast with plain proximal gradient and standard BPG in their Table 1.

The Poisson fidelity is not L-smooth near the boundary

For measurements y modeled as Poisson with mean Ax, the paper minimizes the Poisson negative log-likelihood f(x) = Σⱼ [(Ax)ⱼ − yⱼ log(Ax)ⱼ], which equals the generalized Kullback-Leibler divergence between y and Ax up to constants independent of x. Its gradient is locally Lipschitz, but the Lipschitz constant explodes near the boundary of the positive orthant, so the classical quadratic descent lemma does not hold.

A second obstacle is that for non-diagonal A the proximal operator of f generally has no closed form, which limits proximal splitting methods. The classical multiplicative updates x(k+1) = x(k)/s ⊙ Aᵀ(y/Ax(k)), with s = Aᵀ1ₘ the vector of column sums, avoid both problems: they preserve nonnegativity and were recently shown to converge at a sublinear O(1/k) rate in function values. The paper builds on that analysis rather than replacing it.

Weighted negative entropy yields a unit step size

For the Poisson setting the paper uses the weighted negative entropy φ(x) = Σᵢ sᵢ(xᵢ log xᵢ − xᵢ) as the potential, whose Bregman divergence is a weighted Kullback-Leibler divergence. Lemma 3.3 proves that the reparameterized Poisson objective is 1-smooth relative to φ*, and the appendix shows the constant cannot be smaller than 1.

That constant matters because it fixes the admissible step size. With L = 1 the descent step satisfies τ ≤ 1, and Proposition 4.8 verifies the standing assumptions with L = 1 and τ = 1 under the paper's conditions on A, y, and g. This is a different step-size regime from standard BPG applied to the Poisson fidelity, where Table 1 lists the requirement 0 < τ ≤ 1/‖y‖₁.

At step size one the scheme recovers Richardson-Lucy updates

Substituting the inverse Hessian of the weighted negative entropy into the preconditioned gradient step gives x(k+1) = x(k) − τ (x(k)/s) ⊙ ∇f(x(k)), which can be rewritten as a convex combination of the current iterate and the multiplicative update. Setting τ = 1 collapses this to x(k+1) = (x(k)/s) ⊙ Aᵀ(y/Ax(k)), exactly the classical multiplicative updates.

This is the paper's central structural claim: Richardson-Lucy is not an unrelated heuristic but the unit-step, unregularized instance of R-BPG under this potential. The same derivation also explains why the method inherits the nonnegativity preservation of the multiplicative scheme, since the preconditioned gradient point x − [∇²φ(x)]⁻¹∇f(x) equals (x/s) ⊙ Aᵀ(y/Ax) and stays strictly positive under the paper's assumptions on A and y.

Regularization enters as a right Bregman proximal map

With a convex regularizer g and step size 1, the update becomes x(k+1) = →prox^φ_g((x(k)/s) ⊙ Aᵀ(y/Ax(k))), where the arrow denotes the right Bregman proximal map. The paper shows this is equivalent to solving x(k+1) = argmin_{x>0} { g(x) + D_φ(z(k) ‖ x) } with z(k) the multiplicative-update point, and its first-order condition can be written as x(k+1) ⊙ (s + ξ(k+1)) = x(k) ⊙ Aᵀ(y/Ax(k)) for some ξ(k+1) in the subdifferential of g at x(k+1).

The authors connect this implicit form to Green's one-step-late heuristic, which replaces ξ(k+1) by ξ(k) evaluated at the current iterate to get explicit updates. They note that this approximation is inexpensive and widely used in emission tomography and microscopy deconvolution, but does not generally guarantee convergence or descent, whereas the implicit R-BPG update is the object for which they prove monotone decrease and a rate.

Monotone decrease holds generally, the O(1/k) rate is Poisson-specific

In the general convex setting, Proposition 4.7 proves that the composite objective values are non-increasing along the iterates, with Φ(x(k+1)) − Φ(x(k)) ≤ (L − 1/τ) D_φ(x(k) ‖ x(k+1)). The proof combines a primal-coordinate descent inequality with the Bregman three-point identity, and the authors state explicitly that under these general assumptions the argument gives monotonicity but no function-value convergence rate.

The rate comes only after specializing to Poisson. Theorem 4.12 states that for the regularized multiplicative updates, Φ(x(k)) − Φ(x⋆) ≤ KL(x⋆ ⊙ (s + ξ⋆) ‖ x(1) ⊙ (s + ξ(1)))/k for every k ≥ 1, and the same bound holds for the ergodic iterate. The bound is stated in function values, not in iterate distance, and it is derived by telescoping a Lyapunov energy built from a KL divergence between scaled-simplex quantities.

Experiments show competitiveness, not uniform superiority

The paper reports imaging inverse problems with Poisson-distributed observations, including a 2D BrainWeb PET experiment whose figure plots composite objectives, NRMSE in percent, relative distances between successive iterates, and mean recovery coefficients against disk diameter in millimeters. The compared methods include plain multiplicative updates, mirror descent, BPG with L2 or TV regularization, one-step-late with L2 or TV, and R-BPG with L2 or TV.

The discussion states that in the Poisson setting the proposed algorithm is competitive with existing methods such as BPG, using unit-step multiplicative updates and allowing general convex regularization. That is a competitiveness claim across the reported experiments, not a claim that R-BPG dominates every baseline on every metric. The authors also conjecture that the Poisson rate can be improved, since the numerical experiments show the objective decreasing faster than the sublinear bound of Theorem 4.12.

General-setting rates and nonconvex extensions remain open

The general convergence analysis of R-BPG is described as incomplete: establishing convergence rates and convergence of the iterates in the general convex setting is left as future work. The Poisson guarantees also carry conditions, including a nonnegative A with at least one positive entry in each row and column, strictly positive data y, and a regularizer g that is proper, lower semi-continuous, convex, bounded below, with dom φ contained in dom g and bounded sublevel sets for Φ.

Two further boundaries matter for reading the results. The reparameterized objective is not convex in general: the appendix gives a counterexample with A = (1,1) and y = 1 where the Hessian has a negative eigenvalue for t < −log 2 along the diagonal. And the appendix's nonconvex BPG analysis needs extra assumptions, including supercoercivity in mirror coordinates, local regularity, and coercivity of the reparameterized objective, which the authors note differs from coercivity of Φ in the original variable.

Choose R-BPG when you need general convex regularization at unit step size

The decision-relevant distinction in this paper is not whether R-BPG beats multiplicative updates on unregularized Poisson deconvolution, where at τ = 1 it is those updates. It is what happens once a convex regularizer g is added: R-BPG keeps the multiplicative-update point as its preconditioned gradient step, applies a right Bregman proximal map with the weighted negative entropy, and retains a proven O(1/k) function-value rate, while the one-step-late heuristic used in emission tomography and microscopy deconvolution offers no such convergence or descent guarantee.

The trade-off to weigh is computational. The discussion flags efficient computation of the Bregman proximal map for the weighted negative entropy under other convex regularizers as an open item needed for larger-scale problems, and the general-setting rate is still unproven. So the paper's own priority ordering is: use R-BPG when general convex regularization and a unit step size matter and the proximal map is tractable; treat the reported competitiveness against BPG as evidence from the paper's Poisson imaging experiments rather than a blanket performance ranking.

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