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Given a family $\varphi= (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $\rho$, we consider the mean value $$ M_{k, \rho} (\varphi, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_{\varphi}(\mathbf{x}, \mathbf{y}; N) \right|^\rho d\mathbf{x} $$ of exponential sums $$ S_{\varphi}( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 \pi i \left(\sum_{j=1}^k x_j \varphi_j(n)+ \sum_{j=1}^{d-k}y_j\varphi_{k+j}(n)\right)\right), $$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of polynomials $\varphi_i(T) = T^i$, $i =1, \ldots, d$ 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 $\sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_{\varphi}( \mathbf{x}, \mathbf{y}; N) \right|$ for almost all $\mathbf{x} \in [0,1]^k$, we obtain nontrivial bounds on $M_{k, \rho} (\varphi, N)$., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1903.07330

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., Comment: To appear in IROS 2019

Let $\alpha, \beta \in (0,1)$ such that at least one of them is irrational. We take a random walk on the real line such that the choice of $\alpha$ and $\beta$ has equal probability $1/2$. We prove that almost surely the $\alpha\beta$-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., Comment: 17 pages

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., Comment: Accepted to IEEE Robotics and Automation Letters (RAL)

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.

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., Comment: 11 pages, accepted in the Web Conference (WWW'2019)

Dynamical models estimate and predict the temporal evolution of physical systems. State Space Models (SSMs) in particular represent the system dynamics with many desirable properties, such as being able to model uncertainty in both the model and measurements, and optimal (in the Bayesian sense) recursive formulations e.g. the Kalman Filter. However, they require significant domain knowledge to derive the parametric form and considerable hand-tuning to correctly set all the parameters. Data driven techniques e.g. Recurrent Neural Networks have emerged as compelling alternatives to SSMs with wide success across a number of challenging tasks, in part due to their ability to extract relevant features from rich inputs. They however lack interpretability and robustness to unseen conditions. In this work, we present DynaNet, a hybrid deep learning and time-varying state-space model which can be trained end-to-end. Our neural Kalman dynamical model allows us to exploit the relative merits of each approach. We demonstrate state-of-the-art estimation and prediction on a number of physically challenging tasks, including visual odometry, sensor fusion for visual-inertial navigation and pendulum control. In addition we show how DynaNet can indicate failures through investigation of properties such as the rate of innovation (Kalman Gain)., Comment: Accepted by IEEE Transactions on Neural Networks and Learning Systems (TNNLS)

We prove that the set of $(x_1, \ldots, x_d)\in [0,1)^d$, such that $$ \underline{\lim}_{N\to \infty}\left| \sum_{n=1}^N\exp(2 \pi i (x_1n+\ldots + x_dn^d)) \right| =0, $$ contains a dense $\mathcal{G}_\delta$ set in $[0,1)^d$ and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums $$ \sum_{n=1}^N\exp(2\pi i x n^d), \qquad x \in [0,1). $$, Comment: 22 pages

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 $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| \ge N^{\alpha} $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets., Comment: 10 pages

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., Comment: 37 pages, To appear in Int. Math. Res. Notices

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., Comment: Accepted by CVPR 2019

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for $c=\frac{(1-\sigma)(2\sigma-1)}{6\sigma+4}$. This improves the bound of Guth, Katz, and Zahl for large $\sigma$., Comment: 12 pages, to appear in Moscow Journal of Combinatorics and Number Theory

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 $$ \left| \sum_{n=1}^{N}\exp\left(2\pi i \left(x_1 n+\ldots +x_d n^{d}\right)\right) \right| \leqslant N^{1/2+o(1)}, \qquad N\to \infty. $$ 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., Comment: 44 pages, 2 figures

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., Comment: A short version is accepted by the NIPS 2018 workshop: Infer2Control

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., Comment: NIPS 2018 workshop on Modeling the Physical World: Perception, Learning, and Control. A complete version will be released soon

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

In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if $A\subseteq \mathbb{F}_p$ satisfies $|A|\le p^{64/117}$ then $$ \max\{|A\pm A|, |AA|\} \gtrsim |A|^{39/32}. $$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^+(P)$ to a point-plane incidence bound of Rudnev rather than a point line incidence bound of Stevens and de Zeeuw as done by Shakan and Shkredov., Comment: 16 pages

The study of substructures in random objects has a long history, beginning with Erd\H{o}s and R\'enyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set., Comment: 23 pages. This version addresses referees' comments

Inertial sensors play a pivotal role in indoor localization, which in turn lays the foundation for pervasive personal applications. However, low-cost inertial sensors, as commonly found in smartphones, are plagued by bias and noise, which leads to unbounded growth in error when accelerations are double integrated to obtain displacement. Small errors in state estimation propagate to make odometry virtually unusable in a matter of seconds. We propose to break the cycle of continuous integration, and instead segment inertial data into independent windows. The challenge becomes estimating the latent states of each window, such as velocity and orientation, as these are not directly observable from sensor data. We demonstrate how to formulate this as an optimization problem, and show how deep recurrent neural networks can yield highly accurate trajectories, outperforming state-of-the-art shallow techniques, on a wide range of tests and attachments. In particular, we demonstrate that IONet can generalize to estimate odometry for non-periodic motion, such as a shopping trolley or baby-stroller, an extremely challenging task for existing techniques., Comment: To appear in AAAI18 (Oral)

We study restriction problem in vector spaces over finite fields. We obtain finite field analogue of Mockenhaupt-Mitsis-Bak-Seenger restriction theorem, and we show that the range of the exponentials is sharp., Comment: 8 pages, no figures

We study the restricted families of projections in vector spaces over finite fields. We show that there are families of random subspaces which admit a Marstrand-Mattila type projection theorem., Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1706.03456

We study the restricted families of orthogonal projections in $\mathbb{R}^3$. We show that there are families of random subspaces which admit a Marstrand- Mattila type projection theorem., Comment: 7 pages, remove finite fields version of restricted families of projection

We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional projections via projections of random sets. In the end we study the projections of sets which have the (discrete) Fourier decay., Comment: 14 pages, no figures, some typos were fixed, last version, to appear in Ann. Acad. Sci. Fenn. Math

We prove that almost all random subsets of a finite vector space are weak Salem sets (small Fourier coefficient), which extends a result of Hayes to a different probability model., Comment: 5 pages

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets., Comment: 38 pages, 5 figures

Let $E$ be a subset of a doubling metric space $(X,d)$. We prove that for any $s\in [0, \dim_{A}E]$, where $\dim_{A}$ denotes the Assouad dimension, there exists a subset $F$ of $E$ such that $\dim_{A}F=s$. We also show that the same statement holds for the lower dimension $\dim_L$., Comment: 13 pages

We prove that a typical compact set does not contain any similar copy of a given pattern. We also prove that a typical compact set of $[0,1]^{d} (d\geq 2)$ intersects any $(d-1)$-dimensional plane in at most $d$ points. We study the "hitting probabilities" of compact sets in the sense of Baire category. In the end we study the arithmetic properties of typical compact sets in $[0,1]$ and the "hitting probabilities" of continuous functions., Comment: 12 pages

We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less than $\tfrac12-\varepsilon$, or the lower porosity is larger than $\varepsilon$. Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton-Watson process., Comment: 29 pages, 8 figures

We study subsets of $\R^{d}$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of $\R^{d}$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also prove that a self-affine set that satisfies $OSCH$ (open set condition with holes) is thin for isotropic doubling measures. For doubling measures, we prove that Bara\'nski carpets are thin for doubling measures., Comment: 16pages, 4 figures