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This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as with inexact Newton methods—depending on which variant is implemented. We prove theoretically and verify experimentally, on a variety of problems from simulation experiments to deep learning applications, that our method is a powerful accelerated iterative algorithm. The code is available at https://github.com/Data-driven-numerical-methods/Nonlinear-Truncated-Conjugate-Residual. 

NADPH-dependent assimilatory sulfite reductase (SiR) from Escherichia coli performs a six-electron reduction of sulfite to the bioavailable sulfide. SiR is composed of a flavoprotein (SiRFP) reductase subunit and a hemoprotein (SiRHP) oxidase subunit. There is no known high-resolution structure of SiR or SiRFP, thus we do not yet fully understand how the subunits interact to perform their chemistry. Here, we used small-angle neutron scattering to understand the impact of conformationally restricting the highly mobile SiRFP octamer into an electron accepting (closed) or electron donating (open) conformation, showing that SiR remains active, flexible, and asymmetric even with these conformational restrictions. From these scattering data, we model the first solution structure of SiRFP. Further, computational modeling of the N-terminal 52 amino acids that are responsible for SiRFP oligomerization suggests an eight-helical bundle tethers together the SiRFP subunits to form the SiR core. Finally, mass spectrometry analysis of the closed SiRFP variant show that SiRFP is capable of inter-molecular domain crossover, in which the electron donating domain from one polypeptide is able to interact directly with the electron accepting domain of another polypeptide. This structural characterization suggests that SiR performs its high-volume electron transfer through both inter- and intramolecular pathways between SiRFP domains and, thus, cis or trans transfer from reductase to oxidase subunits. Such highly redundant potential for electron transfer makes this system a potential target for designing synthetic enzymes. 

Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting function classes or model parameters and the inverse transformation is often approximated by root-finding algorithms as a closed-form inverse is unavailable. In this paper, we introduce a new integral-based approach termed: Atomic Unrestricted Time Machine (AUTM), equipped with unrestricted integrands and easy-to-compute explicit inverse. AUTM offers a versatile and efficient way to the design of normalizing flows with explicit inverse and unrestricted function classes or parameters. Theoretically, we present a constructive proof that AUTM is universal: all monotonic normalizing flows can be viewed as limits of AUTM flows. We provide a concrete example to show how to approximate any given monotonic normalizing flow using AUTM flows with guaranteed convergence. The result implies that AUTM can be used to transform an existing flow into a new one equipped with explicit inverse and unrestricted parameters. The performance of the new approach is evaluated on high dimensional density estimation, variational inference and image generation. Experiments demonstrate superior speed and memory efficiency of AUTM.