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We propose a fast algorithm for computing the entire ridge regression regularization path in nearly linear time. Our method constructs a basis on which the solution of ridge regression can be computed instantly for any value of the regularization parameter. Consequently, linear models can be tuned via cross-validation or other risk estimation strategies with substantially better efciency. The algorithm is based on iteratively sketching the Krylov subspace with a binomial decomposition over the regularization path. We provide a convergence analysis with various sketching matrices and show that it improves the state-of-the-art computational complexity. We also provide a technique to adaptively estimate the sketching dimension. This algorithm works for both the over-determined and under-determined problems. We also provide an extension for matrix-valued ridge regression. The numerical results on real medium and large-scale ridge regression tasks illustrate the efectiveness of the proposed method compared to standard baselines which require super-linear computational time. 

Fluorescent protein biomaterials have important applications such as bioimaging in pharmacological studies. Self-assembly of proteins, especially into fibrils, is known to produce fluorescence in the blue band. Capable of self-assembly into nanofibers, we have shown we can modulate its aggregation into mesofibers by encapsulation of a small hydrophobic molecule. Conversely, azobenzenes are hydrophobic small molecules that are virtually non-fluorescent in solution due to their highly efficient photoisomerization. However, they demonstrate fluorogenic properties upon confinement in nanoscale assemblies by reducing the non-radiative photoisomerization. Here, we report the fluorescence of a hybrid protein-small molecule system in which azobenzene is confined in our protein assembly leading to fiber thickening and increased fluorescence. We show our engineered protein Q encapsulates AzoCholine, bearing a photoswitchable azobenzene moiety, in the hydrophobic pore to produce fluorescent mesofibers. This study further investigates the photocontrol of protein conformation as well as fluorescence of an azobenze-containing biomaterial. 

Knowledge graph (KG) representation learning aims to encode entities and relations into dense continuous vector spaces such that knowledge contained in a dataset could be consistently represented. Dense embeddings trained from KG datasets benefit a variety of downstream tasks such as KG completion and link prediction. However, existing KG embedding methods fell short to provide a systematic solution for the global consistency of knowledge representation. We developed a mathematical language for KG based on an observation of their inherent algebraic structure, which we termed as Knowledgebra. By analyzing five distinct algebraic properties, we proved that the semigroup is the most reasonable algebraic structure for the relation embedding of a general knowledge graph. We implemented an instantiation model, SemE, using simple matrix semigroups, which exhibits state-of-the-art performance on standard datasets. Moreover, we proposed a regularization-based method to integrate chain-like logic rules derived from human knowledge into embedding training, which further demonstrates the power of the developed language. As far as we know, by applying abstract algebra in statistical learning, this work develops the first formal language for general knowledge graphs, and also sheds light on the problem of neural-symbolic integration from an algebraic perspective. 

Abstract Porous MXene-polymer composites have gained attention due to their low density, large surface area, and high electrical conductivity, which can be used in applications such as electromagnetic interference shielding, sensing, energy storage, and catalysis. High internal phase emulsions (HIPEs) can be used to template the synthesis of porous polymer structures, and when solid particles are used as the interfacial agent, composites with pores lined with the particles can be realized. Here, we report a simple and scalable method to prepare conductive porous MXene/polyacrylamide structures via polymerization of the continuous phase in oil/water HIPEs. The HIPEs are stabilized by salt flocculated Ti 3 C 2 T x nanosheets, without the use of a co-surfactant. After polymerization, the polyHIPE structure consists of porous polymer struts and pores lined with Ti 3 C 2 T x nanosheets, as confirmed by scanning electron microscopy, energy dispersive x-ray spectroscopy, and x-ray photoelectron spectroscopy. The pore size can be tuned by varying the Ti 3 C 2 T x concentration, and the interconnected Ti 3 C 2 T x network allows for electrical percolation at low Ti 3 C 2 T x loading; further, the electrical conductivity is stable for months indicating that in these composites, the nanosheets are stable to oxidation at ambient conditions. The polyHIPEs also exhibit rapid radio frequency heating at low power (10 °C s −1 at 1 W). This work demonstrates a simple approach to accessing electrically conductive porous MXene/polymer composites with tunable pore morphology and good oxidation stability of the nanosheets. 

Polymer dielectrics have been widely used in electrical and electronic systems for capacitive energy storage and electrical insulation. However, emerging applications such as electric vehicles and hybrid electric aircraft demand improved polymer dielectrics for operation not only under high electric fields and high temperatures, but also extreme conditions, for example, low pressures at high altitudes, with largely increased likelihood of electrical partial discharges. To meet these stringent requirements of grand electrifications for payload efficiency, polymers with enhanced discharge resistance are highly desired. Here, we present a surface-engineering approach for Kapton® coated with self-assembled two-dimensional montmorillonite nanosheets. By suppressing the magnitude of the high-field partial discharges, this nanocoating endows polymers with improved discharge resistance, with satisfactory discharge endurance life of 200 hours at a high electric field of 46 kV mm −1 while maintaining the surface morphology of the polymer. Moreover, the MMT nanocoating can also improve the thermal stability of Kapton®, with significantly suppressed temperature coefficients for both the dielectric constant and dielectric loss over a wide temperature range from 25 to 205 °C. This work provides a practical method of surface nanocoating to explore high-discharge-resistant polymers for harsh condition electrification. 

Pattern formation in plasma–solid interaction represents a great research challenge in many applications from plasma etching to surface treatment, whereby plasma attachments on electrodes (arc roots) are constricted to self-organized spots. Gliding arc discharge in a Jacob’s Ladder, exhibiting hopping dynamics, provides a unique window to probe the nature of pattern formation in plasma–surface interactions. In this work, we find that the existence of negative differential resistance (NDR) across the sheath is responsible for the observed hopping pattern. Due to NDR, the current density and potential drop behave as activator and inhibitor, the dynamic interactions of which govern the surface current density re-distribution and the formation of structured spots. In gliding arc discharges, new arc roots can form separately in front of the existing root(s), which happens periodically to constitute the stepwise hopping. From the instability phase-diagram analysis, the phenomenon that arc attachments tend to constrict itself spontaneously in the NDR regime is well explained. Furthermore, we demonstrate via a comprehensive magnetohydrodynamics (MHD) computation that the existence of a sheath NDR can successfully reproduce the arc hopping as observed in experiments. Therefore, this work uncovers the essential role of sheath NDR in the plasma–solid surface pattern formation and opens up a hitherto unexplored area of research for manipulating the plasma–solid interactions.