arXiv subjects

Jian Ding

Publications and source records attributed to Jian Ding.

At least 127 records · Page 7Linked to original sources

arXiv2018

Learning RoI Transformer for Detecting Oriented Objects in Aerial Images

Object detection in aerial images is an active yet challenging task in computer vision because of the birdview perspective, the highly complex backgrounds, and the variant appearances of objects. Especially when detecting densely packed objects in aerial images, methods relying on horizontal proposals for common object detection often introduce mismatches between the Region of Interests (RoIs) and objects. This leads to the common misalignment between the final object classification confidence and localization accuracy. Although rotated anchors have been used to tackle this problem, the design of them always multiplies the number of anchors and dramatically increases the computational complexity. In this paper, we propose a RoI Transformer to address these problems. More precisely, to improve the quality of region proposals, we first designed a Rotated RoI (RRoI) learner to transform a Horizontal Region of Interest (HRoI) into a Rotated Region of Interest (RRoI). Based on the RRoIs, we then proposed a Rotated Position Sensitive RoI Align (RPS-RoI-Align) module to extract rotation-invariant features from them for boosting subsequent classification and regression. Our RoI Transformer is with light weight and can be easily embedded into detectors for oriented object detection. A simple implementation of the RoI Transformer has achieved state-of-the-art performances on two common and challenging aerial datasets, i.e., DOTA and HRSC2016, with a neglectable reduction to detection speed. Our RoI Transformer exceeds the deformable Position Sensitive RoI pooling when oriented bounding-box annotations are available. Extensive experiments have also validated the flexibility and effectiveness of our RoI Transformer. The results demonstrate that it can be easily integrated with other detector architectures and significantly improve the performances.

cs.CV↗

arXiv2018

Localization for random walks among random obstacles in a single Euclidean ball

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that for environments with probability tending to one as $n\to \infty$ there exists a unique discrete Euclidean ball of volume $d \log_{1/p} n$ asymptotically such that the following holds: conditioned on survival up to time $n$ we have that at any time $t \in [o(n),n]$ with probability tending to one the simple random walk is in this ball. This work relies on and substantially improves a previous result of the authors on localization in a region of volume poly-logarithmic in $n$ for the same problem.

math.PR↗

arXiv2018

Capacity lower bound for the Ising perceptron

We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube $\{-1,+1\}^N$ intersected by $M$ random half-spaces. The perceptron's capacity is $α_N \equiv M_N/N$ for the largest integer $M_N$ such that the intersection in nonempty. It is conjectured by Krauth and Mézard (1989) that the (random) ratio $α_N$ converges in probability to an explicit constant $α_\star \doteq 0.83$. Kim and Roche (1998) proved the existence of a positive constant $γ$ such that $γ\le α_N \le 1-γ$ with high probability; see also Talagrand (1999). In this paper we show that the Krauth--Mézard conjecture $α_\star$ is a lower bound with positive probability, under the condition that an explicit univariate function $S(λ)$ is maximized at $λ=0$. Our proof is an application of the second moment method to a certain slice of perceptron configurations, as selected by the so-called TAP (Thouless, Anderson, and Palmer, 1977) or AMP (approximate message passing) iteration, whose scaling limit has been characterized by Bayati and Montanari (2011) and Bolthausen (2012). For verifying the condition on $S(λ)$ we outline one approach, which is implemented in the current version using (nonrigorous) numerical integration packages. In a future version of this paper we intend to complete the verification by implementing a rigorous numerical method.

math.PR↗

arXiv2018

Liouville first-passage percolation: subsequential scaling limits at high temperature

Let $\{Y_{\mathfrak{B}}(x)\,:\,x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{γY_{\mathfrak{B}}(x)}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov--Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

math.PR↗

arXiv2018

Cut-off for lamplighter chains on tori: dimension interpolation and phase transition

Given a finite, connected graph $G$, the lamplighter chain on $G$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $G^\diamond={\mathbf Z}_2 \wr G$. The mixing time of the lamplighter chain on the torus ${\mathbf Z}_n^d$ is known to have a cutoff at a time asymptotic to the cover time of ${\mathbf Z}_n^d$ if $d=2$, and to half the cover time if $d \ge 3$. We show that the mixing time of the lamplighter chain on $G_n(a)={\mathbf Z}_n^2 \times {\mathbf Z}_{a \log n}$ has a cutoff at $ψ(a)$ times the cover time of $G_n(a)$ as $n \to \infty$, where $ψ$ is an explicit weakly decreasing map from $(0,\infty)$ onto $[1/2,1)$. In particular, as $a > 0$ varies, the threshold continuously interpolates between the known thresholds for ${\mathbf Z}_n^2$ and ${\mathbf Z}_n^3$. Perhaps surprisingly, we find a phase transition (non-smoothness of $ψ$) at the point $a_*=πr_3 (1+\sqrt{2})$, where high dimensional behavior ($ψ(a)=1/2$ for all $a \ge a_*$) commences. Here $r_3$ is the effective resistance from $0$ to $\infty$ in ${\mathbf Z}^3$.

math.PR↗

arXiv2018

Upper bounds on Liouville first passage percolation and Watabiki's prediction

Given a planar continuum Gaussian free field $h^{\mathcal U}$ in a domain $\mathcal U$ with Dirichlet boundary condition and any $δ>0$, we let $\{h_δ^{\mathcal U}(v): v\in \mathcal U\}$ be a real-valued smooth Gaussian process where $h_δ^{\mathcal U}(v)$ is the average of $h^{\mathcal U}$ along a circle of radius $δ$ with center $v$. For $γ> 0$, we study the Liouville first passage percolation (in scale $δ$), i.e., the shortest path distance in $\mathcal U$ where the weight of each path $P$ is given by $\int_P \mathrm{e}^{γh_δ^{\mathcal U}(z)} |dz|$. We show that the distance between two typical points is $O(δ^{c^* γ^{4/3}/\log γ^{-1}})$ for all sufficiently small but fixed $γ>0$ and some constant $c^* > 0$. In addition, we obtain similar upper bounds on the Liouville first passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance which roughly speaking is the minimal number of Euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict with some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.

math.PR↗

arXiv2018

Poly-logarithmic localization for random walks among random obstacles

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following \emph{path localization} holds for environments with probability tending to 1 as $n\to \infty$: conditioned on survival up to time $n$ we have that ever since $o(n)$ steps the simple random walk is localized in a region of volume poly-logarithmic in $n$ with probability tending to 1. The previous best result of this type went back to Sznitman (1996) on Brownian motion among Poisson obstacles, where a localization (only for the end point) in a region of volume $t^{o(1)}$ was derived conditioned on the survival of Brownian motion up to time $t$.

math.PR↗

arXiv2018

Heat kernel for Liouville Brownian motion and Liouville graph distance

We show the existence of the scaling exponent $χ\in (0,4[(1+γ^2/4)- \sqrt{1+γ^4/16}]/γ^2]$ of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $γ<2$ on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel satisfies, for any fixed $u,v\in \mathbb V^o$, the short time estimates $$ \lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^γ(u,v)| }{|\log t|}=\fracχ{2-χ}, \ \mbox{\rm a.s.} $$

math.PR↗

arXiv2018

Hidden Hamiltonian Cycle Recovery via Linear Programming

We introduce the problem of hidden Hamiltonian cycle recovery, where there is an unknown Hamiltonian cycle in an $n$-vertex complete graph that needs to be inferred from noisy edge measurements. The measurements are independent and distributed according to $\calP_n$ for edges in the cycle and $\calQ_n$ otherwise. This formulation is motivated by a problem in genome assembly, where the goal is to order a set of contigs (genome subsequences) according to their positions on the genome using long-range linking measurements between the contigs. Computing the maximum likelihood estimate in this model reduces to a Traveling Salesman Problem (TSP). Despite the NP-hardness of TSP, we show that a simple linear programming (LP) relaxation, namely the fractional $2$-factor (F2F) LP, recovers the hidden Hamiltonian cycle with high probability as $n \to \infty$ provided that $α_n - \log n \to \infty$, where $α_n \triangleq -2 \log \int \sqrt{d P_n d Q_n}$ is the Rényi divergence of order $\frac{1}{2}$. This condition is information-theoretically optimal in the sense that, under mild distributional assumptions, $α_n \geq (1+o(1)) \log n$ is necessary for any algorithm to succeed regardless of the computational cost. Departing from the usual proof techniques based on dual witness construction, the analysis relies on the combinatorial characterization (in particular, the half-integrality) of the extreme points of the F2F polytope. Represented as bicolored multi-graphs, these extreme points are further decomposed into simpler "blossom-type" structures for the large deviation analysis and counting arguments. Evaluation of the algorithm on real data shows improvements over existing approaches.

cs.DM↗

arXiv2018

Chemical distances for percolation of planar Gaussian free fields and critical random walk loop soups

We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall.

math.PR↗

arXiv2017

First passage percolation on the exponential of two-dimensional branching random walk

We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{γ\mathcal R^N_z}$ for $γ>0$. We show that for $γ>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - γ^2/10})$.

math.PR↗

arXiv2017

Continuous monitoring of $\ell_p$ norms in data streams

In insertion-only streaming, one sees a sequence of indices $a_1, a_2, \ldots, a_m\in [n]$. The stream defines a sequence of $m$ frequency vectors $x^{(1)},\ldots,x^{(m)}\in\mathbb{R}^n$ with $(x^{(t)})_i = |\{j : j\in[t], a_j = i\}|$. That is, $x^{(t)}$ is the frequency vector after seeing the first $t$ items in the stream. Much work in the streaming literature focuses on estimating some function $f(x^{(m)})$. Many applications though require obtaining estimates at time $t$ of $f(x^{(t)})$, for every $t\in[m]$. Naively this guarantee is obtained by devising an algorithm with failure probability $\ll 1/m$, then performing a union bound over all stream updates to guarantee that all $m$ estimates are simultaneously accurate with good probability. When $f(x)$ is some $\ell_p$ norm of $x$, recent works have shown that this union bound is wasteful and better space complexity is possible for the continuous monitoring problem, with the strongest known results being for $p=2$ [HTY14, BCIW16, BCINWW17]. In this work, we improve the state of the art for all $0<p<2$, which we obtain via a novel analysis of Indyk's $p$-stable sketch [Indyk06].

cs.DS↗

arXiv2017

Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields

We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with \emph{Liouville quantum gravity} (LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that the FPP distance between two typically sampled vertices (according to the LQG measure) is $N^{1+ O(ε)}$, where $N$ is the side length of the box and $ε$ can be made arbitrarily small if we tune a certain parameter in our construction. That is, the exponents can be arbitrarily close to $1$. Combined with a recent work of the first author and Goswami on an upper bound for this exponent when the underlying field is a GFF, our result implies that such exponent is \emph{not} universal among the family of log-correlated Gaussian fields.

math.PR↗

arXiv2017

Three favorite sites occurs infinitely often for one-dimensional simple random walk

For a one-dimensional simple random walk $(S_t)$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Tóth (2001), and disproves a conjecture of Erdös and Révész (1984) and of Tóth (2001).

math.PR↗

arXiv2017

On the Liouville heat kernel for k-coarse MBRW and nonuniversality

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon>0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, $$ \exp \left( - t^{ - \frac 1 { 1 + \frac 1 2 γ^2 } - \varepsilon } \right) \le p_t^γ(x, y) \le \exp \left( - t^{- \frac 1 { 1 + \frac 1 2 γ^2 } + \varepsilon } \right) , $$ for $γ<1/2$. In particular, these are different from predictions, due to Watabiki, concerning the Liouville heat kernel for the two dimensional Gaussian free field.

math.PR↗

arXiv2016

When multiplicative noise stymies control

We consider the stabilization of an unstable discrete-time linear system that is observed over a channel corrupted by continuous multiplicative noise. Our main result shows that if the system growth is large enough, then the system cannot be stabilized in a second-moment sense. This is done by showing that the probability that the state magnitude remains bounded must go to zero with time. Our proof technique recursively bounds the conditional density of the system state (instead of focusing on the second moment) to bound the progress the controller can make. This sidesteps the difficulty encountered in using the standard data-rate theorem style approach; that approach does not work because the mutual information per round between the system state and the observation is potentially unbounded. It was known that a system with multiplicative observation noise can be stabilized using a simple memoryless linear strategy if the system growth is suitably bounded. In this paper, we show that while memory cannot improve the performance of a linear scheme, a simple non-linear scheme that uses one-step memory can do better than the best linear scheme.

eess.SY↗

arXiv2016

Convergence in law of the maximum of nonlattice branching random walk

Let $η^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $η_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $η^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni. Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest.

math.PR↗

arXiv2016

New quantum codes from dual-containing cyclic codes over finite rings

Let $R=\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}+\cdots+u^{k}\mathbb{F}_{2^{m}}$ , where $\mathbb{F}_{2^{m}}$ is a finite field with $2^{m}$ elements, $m$ is a positive integer, $u$ is an indeterminate with $u^{k+1}=0.$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over $R$. A new Gray map over $R$ is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over $R$ is given. A new family of $2^{m}$-ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$ Furthermore, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over $R.$

cs.IT↗