1D Path Research Articles

On the number of regions of piecewise linear neural networks

Many feedforward neural networks (NNs) generate continuous and piecewise-linear (CPWL) mappings. Specifically, they partition the input domain into regions on which the mapping is affine. The number of these so-called linear regions offers a natural metric to characterize the expressiveness of CPWL NNs. The precise determination of this quantity is often out of reach in practice, and bounds have been proposed for specific architectures, including for ReLU and Maxout NNs. In this work, we generalize these bounds to NNs with arbitrary and possibly multivariate CPWL activation functions. We first provide upper and lower bounds on the maximal number of linear regions of a CPWL NN given its depth, width, and the number of linear regions of its activation functions. Our results rely on the combinatorial structure of convex partitions and confirm the distinctive role of depth which, on its own, is able to exponentially increase the number of regions. We then introduce a complementary stochastic framework to estimate the average number of linear regions produced by a CPWL NN. Under reasonable assumptions, the expected density of linear regions along any 1D path is bounded by the product of depth, width, and a measure of activation complexity (up to a scaling factor). This yields an identical role to the three sources of expressiveness: no exponential growth with depth is observed anymore.

Lacing topological orders in two dimensions: Exactly solvable models for Kitaev's sixteen-fold way

A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nuν species of Majorana fermions coupled to a static \mathbb{Z}_2ℤ2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1𝒞=1, yielding a total Chern number \mathcal{C}=\nu𝒞=ν. It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nuν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.

Effect of confinement of SIA cluster diffusion by impurities on radiation defect accumulation due to 14 MeV neutrons in tungsten

Impurities acting as traps reduce the self-interstitial atom (SIA) cluster mobility, thereby increasing recombination and, in turn, enhancing radiation-resistance of polycrystalline materials. However, at elevated temperatures, depending on SIA-trap (impurity) binding energy, SIA clusters may trap and detrap (detach) numerous times. In tungsten, SIA clusters of all sizes glide one-dimensionally (1D) along their Burgers vector direction. This can result in detrapped SIA clusters to retrace their original 1D path, confining their 1D-glide between traps. However, SIA clusters can change the direction of their 1D-glide by overcoming a rotation barrier and impurities or solutes are known to reduce this barrier. A lower rotation barrier can result in detrapped SIA clusters to diffuse in a random 1D direction, effectively leading to 3D (net-3D) diffusion. Here we investigate the effect of two cases of SIA diffusion, namely the confined-1D diffusion and the net-3D diffusion, on the damage accumulation in polycrystalline tungsten under 14 MeV neutron irradiation using the object kinetic Monte Carlo method. Simulations were performed using cascade debris obtained from molecular dynamics simulations with primary knock-on atom (PKA) energies corresponding to 14 MeV neutrons. In the simulations with net-3D diffusion, SIA clusters of all sizes diffuse in a random 1D direction after the clusters detrap. While for the confined 1D diffusion case, clusters larger than size 5 retain their original direction of 1D glide. In both types of simulations, trapped vacancies are assumed to be permanently immobilized. Unexpectedly, the damage accumulation was lower with confined 1D diffusion than that with net-3D. We present a systematic comparison of the influence of the two cases of SIA diffusion on the radiation damage accumulation as a function of dose rate, detrapping barrier, and impurity concentration at 1025 K.

Ultrasound-Guided Wireless Tubular Robotic Anchoring System.

Untethered miniature robots have significant poten-tial and promise in diverse minimally invasive medical applications inside the human body. For drug delivery and physical contra-ception applications inside tubular structures, it is desirable to have a miniature anchoring robot with self-locking mechanism at a target tubular region. Moreover, the behavior of this robot should be tracked and feedback-controlled by a medical imaging-based system. While such a system is unavailable, we report a reversible untethered anchoring robot design based on remote magnetic actuation. The current robot prototype's dimension is 7.5 mm in diameter, 17.8 mm in length, and made of soft polyurethane elastomer, photopolymer, and two tiny permanent magnets. Its relaxation and anchoring states can be maintained in a stable manner without supplying any control and actuation input. To control the robot's locomotion, we implement a two-dimensional (2D) ultrasound imaging-based tracking and control system, which automatically sweeps locally and updates the robot's position. With such a system, we demonstrate that the robot can be controlled to follow a pre-defined 1D path with the maximal position error of 0.53 ± 0.05 mm inside a tubular phantom, where the reversible anchoring could be achieved under the monitoring of ultrasound imaging.

The metallic 1T-phase WS2 nanosheets as cocatalysts for enhancing the photocatalytic hydrogen evolution of g-C3N4 nanotubes

Herein, we report that 2D metallic 1T-WS2 acts as co-catalyst in assisting g-C3N4 nanotubes (NTs) obtained via a simple grinding method for promoting photocatalytic H2 production. The 1T-WS2/g-C3N4 composite with optimum 27% 1T-WS2 displays a significantly improved photocatalytic H2 production rate (1021 μmol h−1 g−1), 17.6 times higher than g-C3N4 NTs. Besides, the apparent quantum efficiency (AQE) of 1T-WS2/g-C3N4 composite (27%) achieves 11.23% under light at λ = 370 nm. Meanwhile, the 1T-WS2/g-C3N4 composites present an excellent photocatalytic H2 production stability. The possible photocatalytic mechanism over 1T-WS2/g-C3N4 composites is proposed. Specifically, the photogenerated electrons from 1D g-C3N4 tubular nanostructure with open mesoporous morphology are migrated to conductive 1T-WS2 NSs quickly as electron acceptors along the 1D path. 1T-WS2 NSs can also offer abundant active sites on the edge and basal planes.

Neural-Network-Based Path Collective Variables for Enhanced Sampling of Phase Transformations.

The investigation of the microscopic processes underlying structural phase transformations in solids is extremely challenging for both simulation and experiment. Atomistic simulations of solid-solid phase transitions require extensive sampling of the corresponding high-dimensional and often rugged energy landscape. Here, we propose a rigorous construction of a 1D path collective variable that is used in combination with enhanced sampling techniques for efficient exploration of the transformation mechanisms. The path collective variable is defined in a space spanned by global classifiers that are derived from local structural units. A reliable identification of the local structural environments is achieved by employing a neural-network-based classification scheme. The proposed path collective variable is generally applicable and enables the investigation of both transformation mechanisms and kinetics.

Oriented-linear-tree based cost aggregation for stereo matching

Matching cost aggregation is one of the most important steps in dense stereo correspondence, and non-local cost aggregation methods based on tree structures have been widely studied recently. In this paper, we analyze the shortcomings of both the local window-based aggregation methods and the non-local tree-based aggregation methods, and propose a novel oriented linear tree structure for each pixel to perform the non-local cost aggregation strategy. Firstly, each pixel in the image has an oriented linear tree rooted on it and each oriented linear tree consists of multiple 1D paths from different directions. Compared to other spanning trees, our oriented linear trees don’t need to be additionally constructed beforeh and since they are naturally embedded in the original image. Moreover, each root pixel not only gets supports from adjacent pixels within its local support window, but also receives supports from the other pixels along all 1D paths. Secondly, for each pixel lying on the same 1D path, we can at the same time calculate their aggregated cost along their path by traversing the path back and forth twice. Finally, the final aggregated cost for each root pixel can be calculated by summing the aggregated costs from all 1D paths. Performance evaluation on the Middlebury and KITTI datasets shows that the proposed method outperforms the current state-of-the-art aggregation methods.

Information ratchets exploiting spatially structured information reservoirs.

Fully mechanized Maxwell's demons, also called information ratchets, are an important conceptual link between computation, information theory, and statistical physics. They exploit low-entropy information reservoirs to extract work from a heat reservoir. Previous models of such demons have either ignored the cost of delivering bits to the demon from the information reservoir or assumed random access or infinite-dimensional information reservoirs to avoid such an issue. In this work we account for this cost when exploiting information reservoirs with physical structure and show that the dimensionality of the reservoir has a significant impact on the performance and phase diagram of the demon. We find that for conventional one-dimensional tapes the scope for work extraction is greatly reduced. An expression for the net-extracted work by demons exploring information reservoirs by means of biased random walks on d-dimensional, Z^{d}, information reservoirs is presented. Furthermore, we derive exact probabilities of recurrence in these systems, generalizing previously known results. We find that the demon is characterized by two critical dimensions. First, to extract work at zero bias the dimensionality of the information reservoir must be larger than d=2, corresponding to the dimensions where a simple random walker is transient. Second, for integer dimensions d>4 the unbiased random walk optimizes work extraction corresponding to the dimensions where a simple random walker is strongly transient.

Morphological tuning of photo-booster g-C3N4 with higher surface area and better charge transfers for enhanced power conversion efficiency of quantum dot sensitized solar cells

Photo-booster effect of g-C3N4 Nanotubes (g-C3N4 NTs) on the photovoltaic properties are investigated using the composites having ZnO Nanorods (ZnO NRs) with different composition ratios, sensitized by CdS quantum dots. Enhanced performance is attributed to the cumulative effects of this composite i.e., (i) a significant decrement of fluorescence intensity in steady state PL, (ii) faster electron lifetime and good electron injection rate from dynamic PL, and (iii) sufficient loading of sensitizer at photoanodic scaffold for better harvesting of solar energy. An increase of 32% in power conversion efficiency (PCE, η) is observed in case of g-C3N4 NTs based device as compare to g-C3N4 Nanoflakes (g-C3N4 NFs). Increased PCE value is mainly due to (i) efficient separation and transportation of the photogenerated charge carriers along a one dimensional (1D) path, resulting in shorter lifetime of charge carriers and (ii) high surface area which provides more active sites for loading of sensitizer particles, results in an improvement of light harvesting ability. Further, electrochemical impedance spectroscopic (EIS) analyses showed an efficient interfacial charge transfer by reducing the recombination processes i.e., the back transferring of photoexcited electron at electrode/electrolyte interface.

Moment Magnitudes of Local/Regional Events from 1D Coda Calibrations in the Broader Middle East Region

Abstract Reliable moment magnitude estimates for seismic events in the Middle East region can be difficult to obtain due to the uneven distribution of stations, the complex tectonic structure, and regions of high attenuation. In this study, we take advantage of the many new broadband seismic stations that have become available through improved national networks and numerous temporary deployments. We make coda envelope‐amplitude measurements for 2247 events recorded by 68 stations over 13 narrow frequency bands ranging between 0.03 and 8 Hz. The absolute scaling of these spectra was calculated based on independent waveform modeling solutions of the moment magnitudes for a subset of these events to avoid circularity. Using our 1D path calibrations, we determined coda‐based magnitudes for a majority of the events. We obtain fairly good agreement with waveform‐modeled seismic moments for the larger events ( M w >4.5) at low frequencies ( 0.7 Hz) because of unaccounted 2D path effects, as well as mixing of both Sn coda and Lg coda, which have different attenuation behavior. This scatter leads to increased variance in the magnitudes estimated for smaller events in which low‐frequency amplitudes are below the noise levels and the higher frequencies are the only signals available. We quantify the expected variance in coda envelope amplitudes as a function of frequency using interstation scatter as our metric. The net results of this study provide thousands of new 1D coda magnitude estimates for events in the broad region, as well as the necessary initial starting model for use in a new related 2D coda study (Pasyanos et al. , 2016). Online Material: Table of site terms and moment magnitudes.

Highly anisotropic P3HT films with enhanced thermoelectric performance via organic small molecule epitaxy

Conducting polymers are potential candidates for thermoelectric (TE) applications owing to their low thermal conductivity, non-toxicity and low cost. However, the coil conformation and random aggregation of polymer chains usually degrade electrical transport properties, thus deteriorating TE performance. In this work, we fabricated poly(3-hexylthiophene) (P3HT) films with highly oriented morphology using 1,3,5-trichlorobenzene (TCB), an organic small-molecule, as a template for polymer epitaxy under a temperature gradient crystallization process. The resulting P3HT molecules, which were confirmed to be highly anisotropic by a combination of scanning electron microscopy, atomic force microscopy, polarizing microscope, polarized Raman spectroscopy, and two-dimensional-grazing incidence X-ray diffraction (GIXRD) analysis, not only markedly reduced the conjugated defects along the polymer backbone, but also effectively increased the degree of electron delocalization. These combined phenomena produced an efficient, 1D path for carrier movement and therefore resulted in enhanced carrier mobility in the TCB-treated P3HT films. The maximum values of the electrical conductivity and Seebeck coefficient were 320 S cm−1 and 269 μV K−1, respectively. Consequently, the maximum TE power factor and ZT value at 365 K reached 62.4 μW mK−2 and 0.1, respectively, in the parallel direction of the TCB-treated P3HT film. To the best of our knowledge, these are the highest values reported for pure P3HT TE materials. The method of using organic small-molecule epitaxy to generate highly anisotropic polymer films is expected to be valid for many conducting polymers. A process for stretching twisted polymer chains into ordered fibers can benefit production of low-cost materials that harvest heat losses. Most thermoelectric energy converters are made from inorganic semiconductors. Conductive polymers, however, are an attractive alternative because of their simple manufacturing requirements and intrinsic thermal insulating properties. Lidong Chen from the Shanghai Institute of Ceramics and co-workers have developed a way to reduce the defects that often plague plastic thermoelectrics with a dissolvable organic template. The team co-crystallized the conductive polymer poly(3-hexylthiophene) with small, trichlorobenzene-based molecules that solidify into needle-like structures under a temperature gradient. The aromatic stacking forces within the trichlorobenzene crystals lock the polymer chains into elongated positions that persist after template removal. Experiments on the resulting poly(3-hexylthiophene) filaments revealed near-record thermoelectric conversion efficiencies for this material. Herein, we fabricated pure poly(3-hexylthiophene) (P3HT) film with highly oriented morphology using a small-molecule 1,3,5-trichlorobenzene (TCB) as the template for polymer epitaxy with a temperature gradient crystallization process. The strong π–π conjugated interactions as well as the very close matching between the repeat distance of the thiophene units in P3HT and the repeat distance of TCB molecules have successfully induced the epitaxy process, allowing the P3HT polymer chains to lock onto the lattice of TCB, forming highly ordered P3HT chains in the direction parallel to the fiber axis. As a result, the electrical conductivity in the direction parallel to the fiber axis was significantly improved. The maximum thermoelectric power factor and ZT value at 365 K reached 62.4 μW mK-2 and 0.1 in the parallel direction of the TCB-treated P3HT film.

One-Dimensional Migration of Olfactory Ensheathing Cells on Synthetic Materials: Experimental and Numerical Characterization

Olfactory ensheathing cells (OECs) are of great interest for regenerative purposes since they are believed to aid axonal growth. With the view set on the strategies to achieve reconnection between neuronal structures, it is of great importance to characterize the behaviour of these cells on long thread-like structures that may efficiently guide cell spread in a targeted way. Here, rat OECs were studied on polycaprolactone (PCL) long monofilaments, on long bars and on discs. PCL turns out to be an excellent substrate for OECs. The cells cover long distances along the monofilaments and colonize completely these structures. With the help of a one-dimensional (1D) analytical model, a migration coefficient, a net proliferation rate constant and the fraction of all cells which undergo migration were obtained. The separate effect of the three phenomena summarized by these parameters on the colonization patterns of the 1D path was qualitatively discussed. Other features of interest were also determined, such as the speed of the advance front of colonization and the order of the kinetics of net cell proliferation. Characterizing migration by means of these quantities may be useful for comparing and predicting features of the colonization process (such as times, patterns, advance fronts and proportion of motile cells) of different cell-substrate combinations.

Emergent magnetic monopoles, disorder, and avalanches in artificial kagome spin ice (invited)

We study artificial spin ice with isolated elongated nanoscale islands arranged in a kagome lattice and solely interacting via long range dipolar fields. The artificial kagome spin ice displays a phenomenology similar to the microscopic pyrochlore system, where excitations at sub-Kelvin temperatures consist of emergent monopole quasiparticles that are connected via a solenoidal flux line, a classical and observable version of the Dirac string. We show that magnetization reversal in kagome spin ice is fundamentally different from the nucleation and extensive domain growth scenario expected for a generic 2D system. Here, the magnetization reverses in a strictly 1D fashion: After nucleation, a monopole-antimonopole dissociates along a 1D path, leaving a (Dirac) string of islands with reversed magnetization in its wake. Since the 2D artificial spin ice spontaneously decays into a 1D subsystem, magnetization reversal in kagome spin ice provides an example of dimensional reduction via frustration.

Seam carving for media retargeting

Traditional image resizing techniques are oblivious to the content of the image when changing its width or height. In contrast, media (i.e., image and video) retargeting take s content into account. For example, one would like to change the aspect ratio of a video without making human figures look too fat or too skinny, or change the size of an image by automatically removing "unnecessary" portions while keeping the "important" features intact. We propose a simple operator; we term seam carving to support image and video retargeting. A seam is an optimal 1D path of pixels in an image, or a 2D manifold in a video cube, going from top to bottom, or left to right. Optimality is defined by minimizing an energy function that assigns costs to pixels. We show that computing a seam reduces to a dynamic programming problem for images and a graph min-cut search for video. We demonstrate that several image and video operations, such as aspect ratio correction, size change, and object removal, can be recast as a successive operation of the seam carving operator.

2D Coda and Direct-Wave Attenuation Tomography in Northern Italy

Abstract A 1D coda method was proposed by Mayeda et al. (2003) in order to obtain stable seismic source moment-rate spectra using narrowband coda envelope measurements. That study took advantage of the averaging nature of coda waves to derive stable amplitude measurements taking into account all propagation, site, and S -to-coda transfer function effects. Recently, this methodology was applied to microearthquake data sets from three subregions of northern Italy (i.e., western Alps, northern Apennines, and eastern Alps). Because the study regions were small, ranging between local-to-near-regional distances, the simple 1D path assumptions used in the coda method worked very well. The lateral complexity of this region would suggest, however, that a 2D path correction might provide even better results if the data sets were combined, especially when paths traverse larger distances and complicated regions. The structural heterogeneity of northern Italy makes the region ideal to test the extent to which coda variance can be reduced further by using a 2D Q tomography technique. The approach we use has been developed by Phillips et al. (2005) and is an extension of previous amplitude ratio techniques to remove source effects from the inversion. The method requires some assumptions, such as isotropic source radiation, which is generally true for coda waves. Our results are compared against direct S -wave inversions for 1/ Q and results from both share very similar attenuation features that coincide with known geologic structures. We compare our results with those derived from direct waves as well as some recent results from northern California obtained by Mayeda et al. (2005) that tested the same tomographic methodology applied in this study to invert for 1/ Q . We find that 2D coda path corrections for this region significantly improve upon the 1D corrections, in contrast to California where only a marginal improvement was observed. We attribute this difference to stronger lateral variations in Q for northern Italy relative to California.

Determination of the multi-scattered solar radiation from a leaf canopy for use in climate models

The terrestrial component of climate models requires computationally efficient algorithms for determining the multi-scattered radiation contributing to its heating from solar radiation. Much of the vegetated land cover has strong 3D controls on its radiation. The scattering from a 3D object of isotropic scatters is formulated abstractly and an approach to solution is described in the context of a spherical object. A Laplace integral representation of the 3D integral equation for radiative transfer is discretized. Such discretization provides the solution in terms of solutions to 3D Helmholtz equations, a single such equation in the lowest order approach. A Green’s function approximate solution along the paths of entering and exiting radiation is integrated over such radiation for the paths assumed to coincide except for direction. The resulting approximate description of multi-scattered radiation corresponds to replacing the 3D scattering paths with a 1D path with attenuation amplified by a diffusivity factor. This description combines with previously derived analytic solutions for single scattered radiation to provide an efficient representation of the bidirectional scattering from the 3D object, intended for use in climate models and remote sensing.

BRST treatment of zero modes for the worldline formalism in curved space

One-loop quantities in QFT can be computed in an efficient way using the worldline formalism. The latter rests on the ability of calculating 1D path integrals on the circle. In this paper we give a systematic discussion for treating zero modes on the circle of 1D path integrals for both bosonic and supersymmetric nonlinear sigma models, following an approach originally introduced by Friedan. We use BRST techniques and place a special emphasis on the issue of reparametrization invariance. Various examples are extensively analyzed to verify and test the general set-up. In particular, we explicitly check that the chiral anomaly, which can be obtained by the semiclassical approximation of a supersymmetric 1D path integral, does not receive higher order worldline contributions, as implied by supersymmetry.