arXiv subjects

Changhao Chen

Publications and source records attributed to Changhao Chen.

At least 73 records · Page 4Linked to original sources

arXiv2020

On Large Values of Weyl Sums

A special case of the Menshov--Rademacher theorem implies for almost all polynomials $x_1Z+\ldots +x_d Z^{d} \in {\mathbb R}[Z]$ of degree $d$ for the Weyl sums satisfy the upper bound $$ \left| \sum_{n=1}^{N}\exp\left(2πi \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty. $$ Here we investigate the exceptional sets of coefficients $(x_1, \ldots, x_d)$ with large values of Weyl sums for infinitely many $N$, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of $[0,1]^d$. We also use a different technique to give similar results for sums with just one monomial $xn^d$. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.

math.NT↗

arXiv2020

DeepPCO: End-to-End Point Cloud Odometry through Deep Parallel Neural Network

Odometry is of key importance for localization in the absence of a map. There is considerable work in the area of visual odometry (VO), and recent advances in deep learning have brought novel approaches to VO, which directly learn salient features from raw images. These learning-based approaches have led to more accurate and robust VO systems. However, they have not been well applied to point cloud data yet. In this work, we investigate how to exploit deep learning to estimate point cloud odometry (PCO), which may serve as a critical component in point cloud-based downstream tasks or learning-based systems. Specifically, we propose a novel end-to-end deep parallel neural network called DeepPCO, which can estimate the 6-DOF poses using consecutive point clouds. It consists of two parallel sub-networks to estimate 3-D translation and orientation respectively rather than a single neural network. We validate our approach on KITTI Visual Odometry/SLAM benchmark dataset with different baselines. Experiments demonstrate that the proposed approach achieves good performance in terms of pose accuracy.

cs.CV↗

arXiv2020

Hybrid bounds on two-parametric family Weyl sums along smooth curves

We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the form $(τx+c) ω(n)+xn$, $n=1, 2, \ldots$, with fixed $ω(T) \in \mathbb{Z}[T]$ and $τ\in \mathbb{R}$, which holds for almost all $c\in [0,1)$ and all $x\in [0,1)$. We improve and generalise some recent results of M.~B.~Erdogan and G.~Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials $xn+y ω(n)$ for all $(x,y)\in [0,1)^2$ with $f(x,y)=z$ for a set of $z \in [0,1)$ of full Lebesgue measure, provided that $f$ is some Hölder function.

math.CA↗

arXiv2020

Restricted mean value theorems and metric theory of restricted Weyl sums

We study an apparently new question about the behaviour of Weyl sums on a subset $\mathcal{X}\subseteq [0,1)^d$ with a natural measure $μ$ on $\mathcal{X}$. For certain measure spaces $(\mathcal{X}, μ)$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for $μ$-almost all points of $\mathcal{X}$ the Weyl sums satisfy the square root cancellation law. Moreover we characterise the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $(\mathcal{X}, μ)$. We obtain general results, which we refine for some special spaces $\mathcal{X}$ such as spheres, moment curves and line segments.

math.CA↗

arXiv2020

DeepTIO: A Deep Thermal-Inertial Odometry with Visual Hallucination

Visual odometry shows excellent performance in a wide range of environments. However, in visually-denied scenarios (e.g. heavy smoke or darkness), pose estimates degrade or even fail. Thermal cameras are commonly used for perception and inspection when the environment has low visibility. However, their use in odometry estimation is hampered by the lack of robust visual features. In part, this is as a result of the sensor measuring the ambient temperature profile rather than scene appearance and geometry. To overcome this issue, we propose a Deep Neural Network model for thermal-inertial odometry (DeepTIO) by incorporating a visual hallucination network to provide the thermal network with complementary information. The hallucination network is taught to predict fake visual features from thermal images by using Huber loss. We also employ selective fusion to attentively fuse the features from three different modalities, i.e thermal, hallucination, and inertial features. Extensive experiments are performed in hand-held and mobile robot data in benign and smoke-filled environments, showing the efficacy of the proposed model.

cs.CV↗

arXiv2020

Deep Learning based Pedestrian Inertial Navigation: Methods, Dataset and On-Device Inference

Modern inertial measurements units (IMUs) are small, cheap, energy efficient, and widely employed in smart devices and mobile robots. Exploiting inertial data for accurate and reliable pedestrian navigation supports is a key component for emerging Internet-of-Things applications and services. Recently, there has been a growing interest in applying deep neural networks (DNNs) to motion sensing and location estimation. However, the lack of sufficient labelled data for training and evaluating architecture benchmarks has limited the adoption of DNNs in IMU-based tasks. In this paper, we present and release the Oxford Inertial Odometry Dataset (OxIOD), a first-of-its-kind public dataset for deep learning based inertial navigation research, with fine-grained ground-truth on all sequences. Furthermore, to enable more efficient inference at the edge, we propose a novel lightweight framework to learn and reconstruct pedestrian trajectories from raw IMU data. Extensive experiments show the effectiveness of our dataset and methods in achieving accurate data-driven pedestrian inertial navigation on resource-constrained devices.

cs.RO↗

arXiv2019

AtLoc: Attention Guided Camera Localization

Deep learning has achieved impressive results in camera localization, but current single-image techniques typically suffer from a lack of robustness, leading to large outliers. To some extent, this has been tackled by sequential (multi-images) or geometry constraint approaches, which can learn to reject dynamic objects and illumination conditions to achieve better performance. In this work, we show that attention can be used to force the network to focus on more geometrically robust objects and features, achieving state-of-the-art performance in common benchmark, even if using only a single image as input. Extensive experimental evidence is provided through public indoor and outdoor datasets. Through visualization of the saliency maps, we demonstrate how the network learns to reject dynamic objects, yielding superior global camera pose regression performance. The source code is avaliable at https://github.com/BingCS/AtLoc.

cs.CV↗

arXiv2019

Discretized sum-product for large sets

Let $A\subset [1, 2]$ be a $(δ, σ)$-set with measure $|A|=δ^{1-σ}$ in the sense of Katz and Tao. For $σ\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox δ^{-c}|A|, $$ for $c=\frac{(1-σ)(2σ-1)}{6σ+4}$. This improves the bound of Guth, Katz, and Zahl for large $σ$.

math.CO↗

arXiv2019

On a Hybrid Version of the Vinogradov Mean Value Theorem

Given a family $φ= (φ_1, \ldots, φ_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $ρ$, we consider the mean value $$ M_{k, ρ} (φ, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ(\mathbf{x}, \mathbf{y}; N) \right|^ρd\mathbf{x} $$ of exponential sums $$ S_φ( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 πi \left(\sum_{j=1}^k x_j φ_j(n)+ \sum_{j=1}^{d-k}y_jφ_{k+j}(n)\right)\right), $$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of polynomials $φ_i(T) = T^i$, $i =1, \ldots, d$ and $k=d$ corresponds to the classical Vinaogradov mean value theorem. Here motivated by recent works of Wooley (2015) and the authors (2019) on bounds on $\sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ( \mathbf{x}, \mathbf{y}; N) \right|$ for almost all $\mathbf{x} \in [0,1]^k$, we obtain nontrivial bounds on $M_{k, ρ} (φ, N)$.

math.CA↗

arXiv2019

New bounds of Weyl sums

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \ldots , u_d)$ onto a lower dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innovations, we also introduce a ``self-improving'' approach, which leads an infinite series of monotonically decreasing bound, converging to our final result.

math.CA↗

arXiv2019

On uniform distribution of $αβ$-orbits

Let $α, β\in (0,1)$ such that at least one of them is irrational. We take a random walk on the real line such that the choice of $α$ and $β$ has equal probability $1/2$. We prove that almost surely the $αβ$-orbit is uniformly distributed module one, and the exponential sums along its orbit has the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property of uniform distribution modulo one, is large in the terms of topology and Hausdorff dimension.

math.CA↗

arXiv2019

Autonomous Learning for Face Recognition in the Wild via Ambient Wireless Cues

Facial recognition is a key enabling component for emerging Internet of Things (IoT) services such as smart homes or responsive offices. Through the use of deep neural networks, facial recognition has achieved excellent performance. However, this is only possibly when trained with hundreds of images of each user in different viewing and lighting conditions. Clearly, this level of effort in enrolment and labelling is impossible for wide-spread deployment and adoption. Inspired by the fact that most people carry smart wireless devices with them, e.g. smartphones, we propose to use this wireless identifier as a supervisory label. This allows us to curate a dataset of facial images that are unique to a certain domain e.g. a set of people in a particular office. This custom corpus can then be used to finetune existing pre-trained models e.g. FaceNet. However, due to the vagaries of wireless propagation in buildings, the supervisory labels are noisy and weak.We propose a novel technique, AutoTune, which learns and refines the association between a face and wireless identifier over time, by increasing the inter-cluster separation and minimizing the intra-cluster distance. Through extensive experiments with multiple users on two sites, we demonstrate the ability of AutoTune to design an environment-specific, continually evolving facial recognition system with entirely no user effort.

cs.CV↗

arXiv2019

Small values of Weyl sums

We prove that the set of $(x_1, \ldots, x_d)\in [0,1)^d$, such that $$ \underline{\lim}_{N\to \infty}\left| \sum_{n=1}^N\exp(2 πi (x_1n+\ldots + x_dn^d)) \right| =0, $$ contains a dense $\mathcal{G}_δ$ set in $[0,1)^d$ and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums $$ \sum_{n=1}^N\exp(2πi x n^d), \qquad x \in [0,1). $$

math.NT↗

arXiv2019

Hausdorff dimension of the large values of Weyl sums

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2πi (x_1 n+\ldots +x_d n^{d})) \right| \ge N^α $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

math.CA↗

arXiv2019

Selective Sensor Fusion for Neural Visual-Inertial Odometry

Deep learning approaches for Visual-Inertial Odometry (VIO) have proven successful, but they rarely focus on incorporating robust fusion strategies for dealing with imperfect input sensory data. We propose a novel end-to-end selective sensor fusion framework for monocular VIO, which fuses monocular images and inertial measurements in order to estimate the trajectory whilst improving robustness to real-life issues, such as missing and corrupted data or bad sensor synchronization. In particular, we propose two fusion modalities based on different masking strategies: deterministic soft fusion and stochastic hard fusion, and we compare with previously proposed direct fusion baselines. During testing, the network is able to selectively process the features of the available sensor modalities and produce a trajectory at scale. We present a thorough investigation on the performances on three public autonomous driving, Micro Aerial Vehicle (MAV) and hand-held VIO datasets. The results demonstrate the effectiveness of the fusion strategies, which offer better performances compared to direct fusion, particularly in presence of corrupted data. In addition, we study the interpretability of the fusion networks by visualising the masking layers in different scenarios and with varying data corruption, revealing interesting correlations between the fusion networks and imperfect sensory input data.

cs.CV↗

arXiv2018

Learning with Stochastic Guidance for Navigation

Due to the sparse rewards and high degree of environment variation, reinforcement learning approaches such as Deep Deterministic Policy Gradient (DDPG) are plagued by issues of high variance when applied in complex real world environments. We present a new framework for overcoming these issues by incorporating a stochastic switch, allowing an agent to choose between high and low variance policies. The stochastic switch can be jointly trained with the original DDPG in the same framework. In this paper, we demonstrate the power of the framework in a navigation task, where the robot can dynamically choose to learn through exploration, or to use the output of a heuristic controller as guidance. Instead of starting from completely random moves, the navigation capability of a robot can be quickly bootstrapped by several simple independent controllers. The experimental results show that with the aid of stochastic guidance we are able to effectively and efficiently train DDPG navigation policies and achieve significantly better performance than state-of-the-art baselines models.

cs.RO↗

arXiv2018

Transferring Physical Motion Between Domains for Neural Inertial Tracking

Inertial information processing plays a pivotal role in ego-motion awareness for mobile agents, as inertial measurements are entirely egocentric and not environment dependent. However, they are affected greatly by changes in sensor placement/orientation or motion dynamics, and it is infeasible to collect labelled data from every domain. To overcome the challenges of domain adaptation on long sensory sequences, we propose a novel framework that extracts domain-invariant features of raw sequences from arbitrary domains, and transforms to new domains without any paired data. Through the experiments, we demonstrate that it is able to efficiently and effectively convert the raw sequence from a new unlabelled target domain into an accurate inertial trajectory, benefiting from the physical motion knowledge transferred from the labelled source domain. We also conduct real-world experiments to show our framework can reconstruct physically meaningful trajectories from raw IMU measurements obtained with a standard mobile phone in various attachments.

cs.LG↗

arXiv2018

OxIOD: The Dataset for Deep Inertial Odometry

Advances in micro-electro-mechanical (MEMS) techniques enable inertial measurements units (IMUs) to be small, cheap, energy efficient, and widely used in smartphones, robots, and drones. Exploiting inertial data for accurate and reliable navigation and localization has attracted significant research and industrial interest, as IMU measurements are completely ego-centric and generally environment agnostic. Recent studies have shown that the notorious issue of drift can be significantly alleviated by using deep neural networks (DNNs), e.g. IONet. However, the lack of sufficient labelled data for training and testing various architectures limits the proliferation of adopting DNNs in IMU-based tasks. In this paper, we propose and release the Oxford Inertial Odometry Dataset (OxIOD), a first-of-its-kind data collection for inertial-odometry research, with all sequences having ground-truth labels. Our dataset contains 158 sequences totalling more than 42 km in total distance, much larger than previous inertial datasets. Another notable feature of this dataset lies in its diversity, which can reflect the complex motions of phone-based IMUs in various everyday usage. The measurements were collected with four different attachments (handheld, in the pocket, in the handbag and on the trolley), four motion modes (halting, walking slowly, walking normally, and running), five different users, four types of off-the-shelf consumer phones, and large-scale localization from office buildings. Deep inertial tracking experiments were conducted to show the effectiveness of our dataset in training deep neural network models and evaluate learning-based and model-based algorithms. The OxIOD Dataset is available at: http://deepio.cs.ox.ac.uk

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