12,470 research outputs found
Minimax rates of convergence for nonparametric location-scale models
This paper studies minimax rates of convergence for nonparametric
location-scale models, which include mean, quantile and expectile regression
settings. Under Hellinger differentiability on the error distribution and other
mild conditions, we show that the minimax rate of convergence for estimating
the regression function under the squared loss is determined by the
metric entropy of the nonparametric function class. Different error
distributions, including asymmetric Laplace distribution, asymmetric connected
double truncated gamma distribution, connected normal-Laplace distribution,
Cauchy distribution and asymmetric normal distribution are studied as examples.
Applications on low order interaction models and multiple index models are also
given
Development of Flame Retardant and Antibacterial Dual Functionalised Flexible Polyurethane Foam
Flexible Polyurethane foam (PUF), with its unique properties, such as lightweight and softness, has been utilised extensively. Nevertheless, owing to the intrinsic high flammability and low ignition temperature, PUF-associated fire risks are always a concern. During PUFâs combustion, excessive heat and toxic gases can be generated, threatening the health and life of human beings and causing huge property loss. Consequently, improving the flame retardancy of the PUF is of importance. Later, the global COVID-19 pandemic broke out in 2019, leading to the publicâs increased awareness of maintaining good hygiene conditions. Since PUF products are frequently in contact with humans daily, rendering the PUF with bacterial-killing properties should also be addressed.
This dissertation delivers studies on introducing flame retardancy to the PUF via a surface engineering method named the layer-by-layer (LbL) assembly. Due to the consequent COVID-19 situation, this thesis expands the investigations to endow the PUF with antibacterial performances. Preliminary research on fabricating a newly emerged two-dimensional material called MXene (Ti3C2) and chitosan (CH) as flame retardants (FRs) to impart fire safety performances to the PUF was conducted. With only 6.9 wt.% mass added to the PUF, unprecedented fire resistance and smoke suppression properties were received. It was revealed that the FR mechanism was ascribed to the hybrid coatingâs excellent barrier and carbonisation effects. Further investigations on improving the PUFsâ biodegradability identified synergistic effects between the MXene with the CH and phytic acid, demonstrating the great potential for reducing the toxicity and improving the eco-friendliness of the PUFs. Additionally, this thesis analysed the FR and antibacterial dual-functionalised PUFs. The synthesised MXene, CH, and silver ion hybridised coating endows the foam with exceptional bactericidal properties with decreases of 99.7 % in gram-negative bacteria and 88.9 % in gram-positive bacteria compared with the unmodified counterpart. Excellent flame retardancy possessed by the dual-functionalised PUFs was discovered. The compatibility of the two functional coatings was evaluated and confirmed. The results manifest the great potential for eradicating the fire risks of PUFs and providing traditional PUF products with antibacterial properties, further expanding PUFâs applications
Technology for Low Resolution Space Based RSO Detection and Characterisation
Space Situational Awareness (SSA) refers to all activities to detect, identify and track objects in Earth orbit. SSA is critical to all current and future space activities and protect space assets by providing access control, conjunction warnings, and monitoring status of active satellites. Currently SSA methods and infrastructure are not sufficient to account for the proliferations of space debris. In response to the need for better SSA there has been many different areas of research looking to improve SSA most of the requiring dedicated ground or space-based infrastructure. In this thesis, a novel approach for the characterisation of RSOâs (Resident Space Objects) from passive low-resolution space-based sensors is presented with all the background work performed to enable this novel method. Low resolution space-based sensors are common on current satellites, with many of these sensors being in space using them passively to detect RSOâs can greatly augment SSA with out expensive infrastructure or long lead times. One of the largest hurtles to overcome with research in the area has to do with the lack of publicly available labelled data to test and confirm results with. To overcome this hurtle a simulation software, ORBITALS, was created. To verify and validate the ORBITALS simulator it was compared with the Fast Auroral Imager images, which is one of the only publicly available low-resolution space-based images found with auxiliary data. During the development of the ORBITALS simulator it was found that the generation of these simulated images are computationally intensive when propagating the entire space catalog. To overcome this an upgrade of the currently used propagation method, Specialised General Perturbation Method 4th order (SGP4), was performed to allow the algorithm to run in parallel reducing the computational time required to propagate entire catalogs of RSOâs. From the results it was found that the standard facet model with a particle swarm optimisation performed the best estimating an RSOâs attitude with a 0.66 degree RMSE accuracy across a sequence, and ~1% MAPE accuracy for the optical properties. This accomplished this thesis goal of demonstrating the feasibility of low-resolution passive RSO characterisation from space-based platforms in a simulated environment
A stochastic optimization approach to train non-linear neural networks with regularization of higher-order total variation
While highly expressive parametric models including deep neural networks have
an advantage to model complicated concepts, training such highly non-linear
models is known to yield a high risk of notorious overfitting. To address this
issue, this study considers a th order total variation (-TV)
regularization, which is defined as the squared integral of the th order
derivative of the parametric models to be trained; penalizing the -TV is
expected to yield a smoother function, which is expected to avoid overfitting.
While the -TV terms applied to general parametric models are computationally
intractable due to the integration, this study provides a stochastic
optimization algorithm, that can efficiently train general models with the
-TV regularization without conducting explicit numerical integration. The
proposed approach can be applied to the training of even deep neural networks
whose structure is arbitrary, as it can be implemented by only a simple
stochastic gradient descent algorithm and automatic differentiation. Our
numerical experiments demonstrate that the neural networks trained with the
-TV terms are more ``resilient'' than those with the conventional parameter
regularization. The proposed algorithm also can be extended to the
physics-informed training of neural networks (PINNs).Comment: 13 pages, 24 figures, in preparation for submission; comments are
welcome
Gaussian Control Barrier Functions : A Gaussian Process based Approach to Safety for Robots
In recent years, the need for safety of autonomous and intelligent robots has increased. Today, as robots are being increasingly deployed in closer proximity to humans, there is an exigency for safety since human lives may be at risk, e.g., self-driving vehicles or surgical robots. The objective of this thesis is to present a safety framework for dynamical systems that leverages tools from control theory and machine learning. More formally, the thesis presents a data-driven framework for designing safety function candidates which ensure properties of forward invariance. The potential benefits of the results presented in this thesis are expected to help applications such as safe exploration, collision avoidance problems, manipulation tasks, and planning, to name some.
We utilize Gaussian processes (GP) to place a prior on the desired safety function candidate, which is to be utilized as a control barrier function (CBF). The resultant formulation is called Gaussian CBFs and they reside in a reproducing kernel Hilbert space. A key concept behind Gaussian CBFs is the incorporation of both safety belief as well as safety uncertainty, which former barrier function formulations did not consider. This is achieved by using robust posterior estimates from a GP where the posterior mean and variance serve as surrogates for the safety belief and uncertainty respectively. We synthesize safe controllers by framing a convex optimization problem where the kernel-based representation of GPs allows computing the derivatives in closed-form analytically.
Finally, in addition to the theoretical and algorithmic frameworks in this thesis, we rigorously test our methods in hardware on a quadrotor platform. The platform used is a Crazyflie 2.1 which is a versatile palm-sized quadrotor. We provide our insights and detailed discussions on the hardware implementations which will be useful for large-scale deployment of the techniques presented in this dissertation.Ph.D
Singularity Formation in the High-Dimensional Euler Equations and Sampling of High-Dimensional Distributions by Deep Generative Networks
High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ââż, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality.
In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is Cα Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α < α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor Δ as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength Δ < Δ*, and this upper bound Δ* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if Δ < 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up.
In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.</p
Actually Sparse Variational Gaussian Processes
Gaussian processes (GPs) are typically criticised for their unfavourable
scaling in both computational and memory requirements. For large datasets,
sparse GPs reduce these demands by conditioning on a small set of inducing
variables designed to summarise the data. In practice however, for large
datasets requiring many inducing variables, such as low-lengthscale spatial
data, even sparse GPs can become computationally expensive, limited by the
number of inducing variables one can use. In this work, we propose a new class
of inter-domain variational GP, constructed by projecting a GP onto a set of
compactly supported B-spline basis functions. The key benefit of our approach
is that the compact support of the B-spline basis functions admits the use of
sparse linear algebra to significantly speed up matrix operations and
drastically reduce the memory footprint. This allows us to very efficiently
model fast-varying spatial phenomena with tens of thousands of inducing
variables, where previous approaches failed.Comment: 14 pages, 5 figures, published in AISTATS 202
Predicting distributional profiles of physical activity in the NHANES database using a Partially Linear Single-Index Fr\'echet Regression model
Object-oriented data analysis is a fascinating and developing field in modern
statistical science with the potential to make significant and valuable
contributions to biomedical applications. This statistical framework allows for
the formalization of new methods to analyze complex data objects that capture
more information than traditional clinical biomarkers. The paper applies the
object-oriented framework to analyzing and predicting physical activity
measured by accelerometers. As opposed to traditional summary metrics, we
utilize a recently proposed representation of physical activity data as a
distributional object, providing a more sophisticated and complete profile of
individual energetic expenditure in all ranges of monitoring intensity. For the
purpose of predicting these distributional objects, we propose a novel hybrid
Frechet regression model and apply it to US population accelerometer data from
NHANES 2011-2014. The semi-parametric character of the new model allows us to
introduce non-linear effects for essential variables, such as age, that are
known from a biological point of view to have nuanced effects on physical
activity. At the same time, the inclusion of a global for linear term retains
the advantage of interpretability for other variables, particularly categorical
covariates such as ethnicity and sex. The results obtained in our analysis are
helpful from a public health perspective and may lead to new strategies for
optimizing physical activity interventions in specific American subpopulations
Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are
We consider the problem of optimizing the discriminator in generative
adversarial networks (GANs) subject to higher-order gradient regularization. We
show analytically, via the least-squares (LSGAN) and Wasserstein (WGAN) GAN
variants, that the discriminator optimization problem is one of interpolation
in -dimensions. The optimal discriminator, derived using variational
Calculus, turns out to be the solution to a partial differential equation
involving the iterated Laplacian or the polyharmonic operator. The solution is
implementable in closed-form via polyharmonic radial basis function (RBF)
interpolation. In view of the polyharmonic connection, we refer to the
corresponding GANs as Poly-LSGAN and Poly-WGAN. Through experimental validation
on multivariate Gaussians, we show that implementing the optimal RBF
discriminator in closed-form, with penalty orders , results in superior performance, compared to training GAN with
arbitrarily chosen discriminator architectures. We employ the Poly-WGAN
discriminator to model the latent space distribution of the data with
encoder-decoder-based GAN flavors such as Wasserstein autoencoders
Monotone Cubic B-Splines
We present a method for fitting monotone curves using cubic B-splines with a
monotonicity constraint on the coefficients. We explore different ways of
enforcing this constraint and analyze their theoretical and empirical
properties. We propose two algorithms for solving the spline fitting problem:
one that uses standard optimization techniques and one that trains a
Multi-Layer Perceptrons (MLP) generator to approximate the solutions under
various settings and perturbations. The generator approach can speed up the
fitting process when we need to solve the problem repeatedly, such as when
constructing confidence bands using bootstrap. We evaluate our method against
several existing methods, some of which do not use the monotonicity constraint,
on some monotone curves with varying noise levels. We demonstrate that our
method outperforms the other methods, especially in high-noise scenarios. We
also apply our method to analyze the polarization-hole phenomenon during star
formation in astrophysics. The source code is accessible at
\texttt{\url{https://github.com/szcf-weiya/MonotoneSplines.jl}}
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