  Deep denoisers have shown excellent performance in solving inverse problems
in signal and image processing. In order to guarantee the convergence, the
denoiser needs to satisfy some Lipschitz conditions like non-expansiveness.
However, enforcing such constraints inevitably compromises recovery
performance. This paper introduces a novel training strategy that enforces a
weaker constraint on the deep denoiser called pseudo-contractiveness. By
studying the spectrum of the Jacobian matrix, relationships between different
denoiser assumptions are revealed. Effective algorithms based on gradient
descent and Ishikawa process are derived, and further assumptions of strict
pseudo-contractiveness yield efficient algorithms using half-quadratic
splitting and forward-backward splitting. The proposed algorithms theoretically
converge strongly to a fixed point. A training strategy based on holomorphic
transformation and functional calculi is proposed to enforce the
pseudo-contractive denoiser assumption. Extensive experiments demonstrate
superior performance of the pseudo-contractive denoiser compared to related
denoisers. The proposed methods are competitive in terms of visual effects and
quantitative values.
