22,317 research outputs found
Provably Convergent Plug-and-Play Quasi-Newton Methods
Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as non-expansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2--8x faster convergence as compared to other provable PnP methods with similar reconstruction quality
Provably Convergent Plug-and-Play Quasi-Newton Methods
Plug-and-Play (PnP) methods are a class of efficient iterative methods that
aim to combine data fidelity terms and deep denoisers using classical
optimization algorithms, such as ISTA or ADMM. Provable PnP methods are a
subclass of PnP methods with convergence guarantees, such as fixed point
convergence or convergence to critical points of some energy function. Many
existing provable PnP methods impose heavy restrictions on the denoiser or
fidelity function, such as non-expansiveness or strict convexity, respectively.
In this work, we propose a novel algorithmic approach incorporating
quasi-Newton steps into a provable PnP framework based on proximal denoisers,
resulting in greatly accelerated convergence while retaining light assumptions
on the denoiser. By characterizing the denoiser as the proximal operator of a
weakly convex function, we show that the fixed points of the proposed
quasi-Newton PnP algorithm are critical points of a weakly convex function.
Numerical experiments on image deblurring and super-resolution demonstrate
significantly faster convergence as compared to other provable PnP methods with
similar convergence results
Provably Convergent Plug-and-Play Quasi-Newton Methods
Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as non-expansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2--8x faster convergence as compared to other provable PnP methods with similar reconstruction quality
Learning Robust Deep Equilibrium Models
Deep equilibrium (DEQ) models have emerged as a promising class of implicit
layer models in deep learning, which abandon traditional depth by solving for
the fixed points of a single nonlinear layer. Despite their success, the
stability of the fixed points for these models remains poorly understood.
Recently, Lyapunov theory has been applied to Neural ODEs, another type of
implicit layer model, to confer adversarial robustness. By considering DEQ
models as nonlinear dynamic systems, we propose a robust DEQ model named LyaDEQ
with guaranteed provable stability via Lyapunov theory. The crux of our method
is ensuring the fixed points of the DEQ models are Lyapunov stable, which
enables the LyaDEQ models to resist minor initial perturbations. To avoid poor
adversarial defense due to Lyapunov-stable fixed points being located near each
other, we add an orthogonal fully connected layer after the Lyapunov stability
module to separate different fixed points. We evaluate LyaDEQ models on several
widely used datasets under well-known adversarial attacks, and experimental
results demonstrate significant improvement in robustness. Furthermore, we show
that the LyaDEQ model can be combined with other defense methods, such as
adversarial training, to achieve even better adversarial robustness
On the Bourbaki-Witt Principle in Toposes
The Bourbaki-Witt principle states that any progressive map on a
chain-complete poset has a fixed point above every point. It is provable
classically, but not intuitionistically.
We study this and related principles in an intuitionistic setting. Among
other things, we show that Bourbaki-Witt fails exactly when the trichotomous
ordinals form a set, but does not imply that fixed points can always be found
by transfinite iteration. Meanwhile, on the side of models, we see that the
principle fails in realisability toposes, and does not hold in the free topos,
but does hold in all cocomplete toposes
Iterated reflection principles over full disquotational truth
Iterated reflection principles have been employed extensively to unfold
epistemic commitments that are incurred by accepting a mathematical theory.
Recently this has been applied to theories of truth. The idea is to start with
a collection of Tarski-biconditionals and arrive by finitely iterated
reflection at strong compositional truth theories. In the context of classical
logic it is incoherent to adopt an initial truth theory in which A and 'A is
true' are inter-derivable. In this article we show how in the context of a
weaker logic, which we call Basic De Morgan Logic, we can coherently start with
such a fully disquotational truth theory and arrive at a strong compositional
truth theory by applying a natural uniform reflection principle a finite number
of times
Intuitionistic fixed point theories over Heyting arithmetic
In this paper we show that an intuitionistic theory for fixed points is
conservative over the Heyting arithmetic with respect to a certain class of
formulas. This extends partly the result of mine. The proof is inspired by the
quick cut-elimination due to G. Mints
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