Low-light, long-exposure defocus deblurring remains a challenging problem due to the simultaneous presence of severe blur and complex biased noise. Existing methods typically rely on simplified noise assumptions, which limits their effectiveness under realistic imaging conditions. In this work, we propose Physen-Noise2Noise, a self-supervised deblurring framework guided by the physical model of defocus imaging, which leverages noisy multi-frame observations without requiring clean reference images. Unlike conventional Noise2Noise-based approaches that assume zero-mean noise, we derive a frequency-domain constraint inherent to the defocus imaging process and incorporate it into the learning framework via a learnable noise bias parameter. In addition, a multi-frame noisy initialization strategy is introduced to suppress complex biased noise prior to deblurring, providing a more stable starting point for reconstruction. This formulation explicitly models biased noise and enables joint bias correction and high-frequency detail recovery during training. Furthermore, we develop a pretrain-finetune variant to enhance robustness and generalization under challenging noise conditions. Extensive experiments on both simulation and real-world datasets demonstrate that the proposed method consistently outperforms state-of-the-art self-supervised approaches for defocus deblurring in the presence of complex biased noise.

We propose below the proofs of the results presented in the main text. Most of the arguments are adapted from the development proposed in (Zhang, 2013) which goes beyond real or complex-valued RKBS developed in (Zhang et al., 2009; Song et al., 2013) to develop the notion of vector-valued RKBS. In addition, we note that assumptions regarding the properties of the RKBS of interests such as uniform Fréchet differentiability and uniform convexity have been further relaxed in other works (Xu and Ye, 2019; Lin et al., 2022) but are here sufficient for our discussion since they guarantee the unicity of a semi-inner product x.,.yB compatible with the norm ||.||B (Giles, 1967). S.1.1 Theoretical results Theorem 1 Theorem 1 gathers for the sake of compactness the definition of a vector-valued reproducing kernel Banach space with the properties of existence and unicity of the kernel K. Proof. For any v PV and u PU, the mapping OÞÑ xOpvq,uyU is a bounded linear form in LpBq.

While deep reinforcement learning (RL) has been demonstrated effective in solving complex control tasks, sample efficiency remains a key challenge due to the large amounts of data required for remarkable performance. Existing research explores the application of representation learning for data-efficient RL, e.g., learning predictive representations by predicting long-term future states. However, many existing methods do not fully exploit the structural information inherent in sequential state signals, which can potentially improve the quality of long-term decision-making but is difficult to discern in the time domain. To tackle this problem, we propose State Sequences Prediction via Fourier Transform (SPF), a novel method that exploits the frequency domain of state sequences to extract the underlying patterns in time series data for learning expressive representations efficiently. Specifically, we theoretically analyze the existence of structural information in state sequences, which is closely related to policy performance and signal regularity, and then propose to predict the Fourier transform of infinite-step future state sequences to extract such information. One of the appealing features of SPF is that it is simple to implement while not requiring storage of infinite-step future states as prediction targets. Experiments demonstrate that the proposed method outperforms several state-of-the-art algorithms in terms of both sample efficiency and performance.2

In this section, we provide a brief introduction to the concepts from Group Theory which we need in our derivations. A group is a pair (G,)containing a set Gand a binary operation: G G! G,(h,g) 7! h g which satisfies the group axioms: Associativity: 8a,b,c 2 Ga (b c)=( a b) c Identity: 9e 2 G: 8g 2 Gg e = e g = g Inverse: 8g 2 G 9g 1 2 G: g g 1 = g 1 g = e The operation is the group law of G. The inverse elements g 1 of an element g, and the identity element e are unique. In addition, if the group law is also commutative, the group G is an abelian group. To simplify the notation, we commonly write ab instead of a b. It is also common to denote the group (G,) just with the name of its underlying set G. The order of a group G is the cardinality of its set and is indicated by |G|. A group G is finite when |G|2 N, i.e., when it has a finite number of elements. A compact group is a group that is also a compact topological space with continuous group operation. Given a group G, its action on a set X is a map . A simple example of group action is the group law itself: G G! Gwhich defines an action of G on its own elements (X = G). Another important action is the one defined on signals overs the group G. Given a signal x: G! R, the action of an element g 2 G maps x 7! g.x, [g.x](h):= x(g 1h).

Given a T-periodic function f(x), its Fourier series representation is a decomposition into a sum of Fourier basis functions.

Current theoretical results on optimization trajectories of neural networks trained by gradient descent typically have the form of rigorous but potentially loose bounds on the loss values. In the present work we take a different approach and show that the learning trajectory of a wide network in a lazy training regime can be characterized by an explicit asymptotic at large training times. Specifically, the leading term in the asymptotic expansion of the loss behaves as a power law L(t) Ct ξ with exponent ξ expressed only through the data dimension, the smoothness of the activation function, and the class of function being approximated. Our results are based on spectral analysis of the integral operator representing the linearized evolution of a large network trained on the expected loss. Importantly, the techniques we employ do not require a specific form of the data distribution, for example Gaussian, thus making our findings sufficiently universal.