Convergence of Adapted Empirical Measures on Rd\mathbb{R}^{d}

We consider empirical measures of $\R^{d}$-valued stochastic process in finite discrete-time. We show that the adapted empirical measure introduced in the recent work \cite{backhoff2022estimating} by Backhoff et al. in compact spaces can be defined analogously on $\R^{d}$, and that it converges almost surely to the underlying measure under the adapted Wasserstein distance. Moreover, we quantitatively analyze the convergence of the adapted Wasserstein \add{distance} between those two measures. We establish convergence rates of the expected error as well as the deviation error under different moment conditions. \add{Under suitable integrability and kernel assumptions, we recover the optimal convergence rates of both expected error and deviation error.} Furthermore, we propose a modification of the adapted empirical measure with \add{projection} on a non-uniform grid, which obtains the same convergence rate but under weaker assumptions

Instance-Dependent Generalization Bounds via Optimal Transport

Existing generalization bounds fail to explain crucial factors that drive generalization of modern neural networks. Since such bounds often hold uniformly over all parameters, they suffer from over-parametrization, and fail to account for the strong inductive bias of initialization and stochastic gradient descent. As an alternative, we propose a novel optimal transport interpretation of the generalization problem. This allows us to derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the earned prediction function in the data space. Therefore, our bounds are agnostic to the parametrization of the model and work well when the number of training samples is much smaller than the number of parameters. With small modifications, our approach yields accelerated rates for data on low-dimensional manifolds, and guarantees under distribution shifts. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.Comment: 50 pages, 7 figure

Visible-telecom broadband optical isolator based on dynamic modulation in thin-film lithium niobate

Optical isolators are an essential component of photonic systems. Current integrated optical isolators have limited bandwidths due to stringent phase-matching conditions, resonant structures, or material absorption. Here, we demonstrate an ultra-broadband integrated optical isolator in thin-film lithium niobate photonics. We use dynamic standing-wave modulation in a tandem configuration to break Lorentz reciprocity and achieve isolation. We measure an isolation ratio of 15 dB and insertion loss below 0.5 dB for a design wavelength of 1550 nm. In addition, we experimentally show that this isolator can simultaneously operate at visible and telecom wavelengths with comparable performance. Isolation bandwidths ~100 nm can be achieved simultaneously at both visible and telecom wavelengths. Our device's large bandwidth, high flexibility, and real-time tunability can enable novel non-reciprocal functionality on integrated photonic platforms

Improving Linewidth and Extinction Ratio Performances of Lithium Niobate Ring Modulator Using Ring-pair Structure

Electro-optic modulators lie at the heart of complex integration and high density electro-optic systems. One of the representative electro-optic modulators is thin film lithium niobate based microring modulator which has demonstrated advantages of compact footprint, low optical loss and high modulation efficiency. However, the linewidth and extinction ratio of ring modulators are fundamentally limited by the ring losses and coupling, respectively. To this end, we propose a novel type of electro-optic modulators with ring-pair structure on thin film lithium niobate platform, which brings substantially improvement of linewidth and extinction ratio. The ring-pair modulator exhibits a larger linewidth up to 22 GHz, 1.74-time larger than that of single ring resonator with same design parameters. Moreover, the experimental results also reveal that the added-up extinction ratio of ring-pair resonator goes beyond 30 dB, much larger than that in an individual ring resonator. These advantages of ring-pair modulator pave a new way for the application of compact ring-based modulators with large working wavelength window and high extinction ratio, to be exploited in quantum optics, programmable nanophotonics and optical sensors, etc.Comment: 14 page

Study on manipulating photonic environment of emitters with scale dependent optical cavities

Photon emitters placed into an optical cavity will experience a surrounding photonic environment change, which is essential to push nanophotonic devices into the practical realm, including photonic switches, quantum networking and nano-lasers. Spontaneous emission of emitters plays a critical role in determining the performance of many photonic devices. Both dielectric microcavity and plasmonic nanocavity provide a platform to control the decay channels of integrated emitters, and their coupling strength, g, yields: , where N is the involved number of excitons and V is the mode volume of a cavity. The coupling strength can be enhanced by scaling down the cavity volume. Although it difficult to boost this interaction in individual single photon emitters, approaching the quantum limit of coherent interactions between individual quantum emitters and cavity resonators is crucial for fundamental quantum mechanics as well as practical applications. In this thesis, we investigated the light-matter interaction by coupling emitters with photonic and plasmonic optical cavities. We scaled down both cavity volume from photonic crystals to gap plasmonic nanocavities and the size of emitters from film, monolayer to localized single photon sources. Photonic emitters experienced a photonic environment change from broadband confinement in photonic crystals to extreme field confinement in gap plasmonic nanocavities. By scaling down the size of the cavity volume from micrometers to nanometers, we successfully manipulate light emission with high extraction in photonic crystals at weak coupling regime and strong light-matter interaction in plasmonic nanocavities. Finally, we also demonstrated single photon emitters in laser irradiated hexagonal boron nitride monolayers and the modification of spontaneous emission of quantum emitters using plasmonic resonators.Doctor of Philosoph

Deep Partial Hedging

Using techniques from deep learning, we show that neural networks can be trained successfully to replicate the modified payoff functions that were first derived in the context of partial hedging by Föllmer and Leukert. Not only does this approach better accommodate the realistic setting of hedging in discrete time, it also allows for the inclusion of transaction costs as well as general market dynamics. It needs to be noted that, without further modifications, the approach works only if the risk aversion is beyond a certain level

Deep Partial Hedging

Using techniques from deep learning, we show that neural networks can be trained successfully to replicate the modified payoff functions that were first derived in the context of partial hedging by Follmer and Leukert. Not only does this approach better accommodate the realistic setting of hedging in discrete time, it also allows for the inclusion of transaction costs as well as general market dynamics. It needs to be noted that, without further modifications, the approach works only if the risk aversion is beyond a certain level

Long-term fermented organic fertilizer application reduce urea nitrogen-15 loss from plastic shed agricultural soils

Continuous application of fermented organic fertilizer can improve soil quality, while the performance of nitrogen (N) in the improved soils is rarely investigated. To investigate the fate of applied N in the soils with organic management history, the 15NH2CO15NH2 (15N abundance of 19.6 %) was employed as the exogenous N source to conduct an experiment in the Chinese cabbage and tomato rotation system under plastic shed condition. The cultivated soils have received 15-year of effective microorganism (EM) fermented organic fertilizer (EM-OF), N, P, K inorganic fertilizer (NPK-IF) and no fertilizer (NoF). The 15N use by cabbage and tomato, the soil 15N forms, as well as the 15N distribution were observed. Results showed that the 15N use efficiency of cabbage in the EM-OF, NPK-IF and NoF soils were 55.1 %, 37.3 % and 26.6 % respectively, showing significant (p ≤ 0.05) differences. The succeeding crop tomato could take up the soil residual 15N, and the highest 15N reuse efficiency was 7.1 % that detected in the NoF soil. The total 15N loss (6.0 %) from the rotation system was the lowest in the EM-OF soil, compared to that in the NPK-IF and NoF soils. It was concluded that the long-term fermented organic fertilizer applied soils can reduce urea 15N loss from plastic shed agriculture, mainly through improving the in-season crop 15N use efficiency

Directing Cherenkov photons with spatial nonlocality

Cherenkov radiation in natural transparent materials is generally forward-propagating, owing to the positive group index of radiation modes. While negative-index metamaterials enable reversed Cherenkov radiation, the forward photon emission from a swift charged particle is prohibited. In this work, we theoretically investigate emission behaviours of a swift charged particle in the nanometallic layered structure. Our results show that Cherenkov photons are significantly enhanced by longitudinal plasmon modes resulting from the spatial nonlocality in metamaterials. More importantly, longitudinal Cherenkov photons can be directed either forward or backward, stringently depending on the particle velocity. The enhanced flexibility to route Cherenkov photons holds promise for many practical applications of Cherenkov radiation, such as novel free-electron radiation sources and new types of Cherenkov detectors