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In STOC'95 [ADMSS'95] Arya et al. showed that any set of $n$ points in $\mathbb R^d$ admits a $(1+\epsilon)$-spanner with hop-diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges). They also gave a general upper bound tradeoff of hop-diameter at most $k$ and $O(n \alpha_k(n))$ edges, for any $k \ge 2$. The function $\alpha_k$ is the inverse of a certain Ackermann-style function at the $\lfloor k/2 \rfloor$th level of the primitive recursive hierarchy, where $\alpha_0(n) = \lceil n/2 \rceil$, $\alpha_1(n) = \left\lceil \sqrt{n} \right\rceil$, $\alpha_2(n) = \lceil \log{n} \rceil$, $\alpha_3(n) = \lceil \log\log{n} \rceil$, $\alpha_4(n) = \log^* n$, $\alpha_5(n) = \lfloor \frac{1}{2} \log^*n \rfloor$, \ldots. Roughly speaking, for $k \ge 2$ the function $\alpha_{k}$ is close to $\lfloor \frac{k-2}{2} \rfloor$-iterated log-star function, i.e., $\log$ with $\lfloor \frac{k-2}{2} \rfloor$ stars. Also, $\alpha_{2\alpha(n)+4}(n) \le 4$, where $\alpha(n)$ is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases $k = 2$ and $k = 3$. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of $k$. In this paper we prove a tight lower bound for any constant $k$: For any fixed $\epsilon > 0$, any $(1+\epsilon)$-spanner for the uniform line metric with hop-diameter at most $k$ must have at least $\Omega(n \alpha_k(n))$ edges.

Illegal vehicle parking is a common urban problem faced by major cities in the world, as it incurs traffic jams, which lead to air pollution and traffic accidents. The government highly relies on active human efforts to detect illegal parking events. However, such an approach is extremely ineffective to cover a large city since the police have to patrol over the entire city roads. The massive and high-quality sharing bike trajectories from Mobike offer us a unique opportunity to design a ubiquitous illegal parking detection approach, as most of the illegal parking events happen at curbsides and have significant impact on the bike users. The detection result can guide the patrol schedule, i.e. send the patrol policemen to the region with higher illegal parking risks, and further improve the patrol efficiency. Inspired by this idea, three main components are employed in the proposed framework: 1)~{\em trajectory pre-processing}, which filters outlier GPS points, performs map-matching, and builds trajectory indexes; 2)~{\em illegal parking detection}, which models the normal trajectories, extracts features from the evaluation trajectories, and utilizes a distribution test-based method to discover the illegal parking events; and 3)~{\em patrol scheduling}, which leverages the detection result as reference context, and models the scheduling task as a multi-agent reinforcement learning problem to guide the patrol police. Finally, extensive experiments are presented to validate the effectiveness of illegal parking detection, as well as the improvement of patrol efficiency.

We propose a simple and efficient clustering method for high-dimensional data with a large number of clusters. Our algorithm achieves high-performance by evaluating distances of datapoints with a subset of the cluster centres. Our contribution is substantially more efficient than k-means as it does not require an all to all comparison of data points and clusters. We show that the optimal solutions of our approximation are the same as in the exact solution. However, our approach is considerably more efficient at extracting these clusters compared to the state-of-the-art. We compare our approximation with the exact k-means and alternative approximation approaches on a series of standardised clustering tasks. For the evaluation, we consider the algorithmic complexity, including number of operations to convergence, and the stability of the results.

We study a multivariate version of trend filtering, called Kronecker trend filtering or KTF, for the case in which the design points form a lattice in $d$ dimensions. KTF is a natural extension of univariate trend filtering (Steidl et al., 2006; Kim et al., 2009; Tibshirani, 2014), and is defined by minimizing a penalized least squares problem whose penalty term sums the absolute (higher-order) differences of the parameter to be estimated along each of the coordinate directions. The corresponding penalty operator can be written in terms of Kronecker products of univariate trend filtering penalty operators, hence the name Kronecker trend filtering. Equivalently, one can view KTF in terms of an $\ell_1$-penalized basis regression problem where the basis functions are tensor products of falling factorial functions, a piecewise polynomial (discrete spline) basis that underlies univariate trend filtering. This paper is a unification and extension of the results in Sadhanala et al. (2016, 2017). We develop a complete set of theoretical results that describe the behavior of $k^{\mathrm{th}}$ order Kronecker trend filtering in $d$ dimensions, for every $k \geq 0$ and $d \geq 1$. This reveals a number of interesting phenomena, including the dominance of KTF over linear smoothers in estimating heterogeneously smooth functions, and a phase transition at $d=2(k+1)$, a boundary past which (on the high dimension-to-smoothness side) linear smoothers fail to be consistent entirely. We also leverage recent results on discrete splines from Tibshirani (2020), in particular, discrete spline interpolation results that enable us to extend the KTF estimate to any off-lattice location in constant-time (independent of the size of the lattice $n$).

In Chen and Zhou 2021, they consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other is bounded by $(ts)^{H-1}$ with $H\in (\frac12,\,1)$, up to a constant factor. In this paper, we investigate the same problem but with the assumption of $H\in (0,\,\frac12)$. It is well known that there is a significant difference between the Hilbert space associated with the fractional Gaussian processes in the case of $H\in (\frac12, 1)$ and that of $H\in (0, \frac12)$. The starting point of this paper is a new relationship between the inner product of $\mathfrak{H}$ associated with the Gaussian process $(G_t)_{t\ge 0}$ and that of the Hilbert space $\mathfrak{H}_1$ associated with the fractional Brownian motion $(B^{H}_t)_{t\ge 0}$. Then we prove the strong consistency with $H\in (0, \frac12)$, and the asymptotic normality and the Berry-Ess\'{e}en bounds with $H\in (0,\frac38)$ for both the least squares estimator and the moment estimator of the drift parameter constructed from the continuous observations. A good many inequality estimates are involved in and we also make use of the estimation of the inner product based on the results of $\mathfrak{H}_1$ in Hu, Nualart and Zhou 2019.

For $m, d \in {\mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c \ge 0$ and $C$ such that for all $d$ and all $m \ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies \[c \,dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)} \le {\mathbb E} D^*(P) \le C\, dm^{\frac{d-1}{2}} \sqrt{1 + \log(\tfrac md)}.\] This discrepancy is thus smaller by a factor of $\Theta\big(\sqrt{\frac{1+\log(m/d)}{m/d}}\,\big)$ than the one of a uniformly distributed random point set of $m^d$ points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity (2016)). It also removes the asymptotic requirement that $m$ is sufficiently large compared to $d$.

We prove the validity of a non-overlapping block bootstrap for the empirical distance covariance under the assumption of strictly stationary and absolutely regular sample data. From this, we develop a test for independence of two strictly stationary and absolutely regular processes. In proving our results, we derive explicit bounds on the expected Wasserstein distance between an empirical measure and its limit for strictly stationary and strongly mixing sample sequences.

We consider large-scale Markov decision processes with an unknown cost function and address the problem of learning a policy from a finite set of expert demonstrations. We assume that the learner is not allowed to interact with the expert and has no access to reinforcement signal of any kind. Existing inverse reinforcement learning methods come with strong theoretical guarantees, but are computationally expensive, while state-of-the-art policy optimization algorithms achieve significant empirical success, but are hampered by limited theoretical understanding. To bridge the gap between theory and practice, we introduce a novel bilinear saddle-point framework using Lagrangian duality. The proposed primal-dual viewpoint allows us to develop a model-free provably efficient algorithm through the lens of stochastic convex optimization. The method enjoys the advantages of simplicity of implementation, low memory requirements, and computational and sample complexities independent of the number of states. We further present an equivalent no-regret online-learning interpretation.

We initiate a systematic study on $\mathit{dynamic}$ $\mathit{influence}$ $\mathit{maximization}$ (DIM). In the DIM problem, one maintains a seed set $S$ of at most $k$ nodes in a dynamically involving social network, with the goal of maximizing the expected influence spread while minimizing the amortized updating cost. We consider two evolution models. In the $\mathit{incremental}$ model, the social network gets enlarged over time and one only introduces new users and establishes new social links, we design an algorithm that achieves $(1-1/e-\epsilon)$-approximation to the optimal solution and has $k \cdot\mathsf{poly}(\log n, \epsilon^{-1})$ amortized running time, which matches the state-of-art offline algorithm with only poly-logarithmic overhead. In the $\mathit{fully}$ $\mathit{dynamic}$ model, users join in and leave, influence propagation gets strengthened or weakened in real time, we prove that under the Strong Exponential Time Hypothesis (SETH), no algorithm can achieve $2^{-(\log n)^{1-o(1)}}$-approximation unless the amortized running time is $n^{1-o(1)}$. On the technical side, we exploit novel adaptive sampling approaches that reduce DIM to the dynamic MAX-k coverage problem, and design an efficient $(1-1/e-\epsilon)$-approximation algorithm for it. Our lower bound leverages the recent developed distributed PCP framework.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.