arXiv subjects

Siu-Wing Cheng

Publications and source records attributed to Siu-Wing Cheng.

At least 19 recordsLinked to original sources

arXiv2026

Sparse Signal Recovery from Random Measurements

Given the compressed sensing measurements of an unknown vector $z \in \mathbb{R}^n$ using random matrices, we present a simple method to determine $z$ without solving any optimization problem or linear system. Our method uses $\Theta(\log n)$ random sensing matrices in $\mathbb{R}^{k \times n}$ and runs in $O(kn\log n)$ time, where $k = \Theta(s\log n)$ and $s$ is the number of nonzero coordinates in $z$. We adapt our method to determine the support set of $z$ and experimentally compare with some optimization-based methods on binary signals.

cs.IT↗

arXiv2025

A Dynamic Working Set Method for Compressed Sensing

We propose a dynamic working set method (DWS) for the problem $\min_{\mathtt{x} \in \mathbb{R}^n} \frac{1}{2}\|\mathtt{Ax}-\mathtt{b}\|^2 + \eta\|\mathtt{x}\|_1$ that arises from compressed sensing. DWS manages the working set while iteratively calling a regression solver to generate progressively better solutions. Our experiments show that DWS is more efficient than other state-of-the-art software in the context of compressed sensing. Scale space such that $\|b\|=1$. Let $s$ be the number of non-zeros in the unknown signal. We prove that for any given $\varepsilon > 0$, DWS reaches a solution with an additive error $\varepsilon/\eta^2$ such that each call of the solver uses only $O(\frac{1}{\varepsilon}s\log s \log\frac{1}{\varepsilon})$ variables, and each intermediate solution has $O(\frac{1}{\varepsilon}s\log s\log\frac{1}{\varepsilon})$ non-zero coordinates.

cs.DS↗

arXiv2025

Simplification of Trajectory Streams

While there are software systems that simplify trajectory streams on the fly, few curve simplification algorithms with quality guarantees fit the streaming requirements. We present streaming algorithms for two such problems under the Fr\'{e}chet distance $d_F$ in $\mathbb{R}^d$ for some constant $d \geq 2$. Consider a polygonal curve $\tau$ in $\mathbb{R}^d$ in a stream. We present a streaming algorithm that, for any $\varepsilon\in (0,1)$ and $\delta > 0$, produces a curve $\sigma$ such that $d_F(\sigma,\tau[v_1,v_i])\le (1+\varepsilon)\delta$ and $|\sigma|\le 2\,\mathrm{opt}-2$, where $\tau[v_1,v_i]$ is the prefix in the stream so far, and $\mathrm{opt} = \min\{|\sigma'|: d_F(\sigma',\tau[v_1,v_i])\le \delta\}$. Let $\alpha = 2(d-1){\lfloor d/2 \rfloor}^2 + d$. The working storage is $O(\varepsilon^{-\alpha})$. Each vertex is processed in $O(\varepsilon^{-\alpha}\log\frac{1}{\varepsilon})$ time for $d \in \{2,3\}$ and $O(\varepsilon^{-\alpha})$ time for $d \geq 4$ . Thus, the whole $\tau$ can be simplified in $O(\varepsilon^{-\alpha}|\tau|\log\frac{1}{\varepsilon})$ time. Ignoring polynomial factors in $1/\varepsilon$, this running time is a factor $|\tau|$ faster than the best static algorithm that offers the same guarantees. We present another streaming algorithm that, for any integer $k \geq 2$ and any $\varepsilon \in (0,\frac{1}{17})$, maintains a curve $\sigma$ such that $|\sigma| \leq 2k-2$ and $d_F(\sigma,\tau[v_1,v_i])\le (1+\varepsilon) \cdot \min\{d_F(\sigma',\tau[v_1,v_i]): |\sigma'| \leq k\}$, where $\tau[v_1,v_i]$ is the prefix in the stream so far. The working storage is $O((k\varepsilon^{-1}+\varepsilon^{-(\alpha+1)})\log \frac{1}{\varepsilon})$. Each vertex is processed in $O(k\varepsilon^{-(\alpha+1)}\log^2\frac{1}{\varepsilon})$ time for $d \in \{2,3\}$ and $O(k\varepsilon^{-(\alpha+1)}\log\frac{1}{\varepsilon})$ time for $d \geq 4$.

cs.CG↗

arXiv2025

Constant Approximation of Fr\'echet Distance in Strongly Subquadratic Time

Let $\tau$ and $\sigma$ be two polygonal curves in $\mathbb{R}^d$ for any fixed $d$. Suppose that $\tau$ and $\sigma$ have $n$ and $m$ vertices, respectively, and $m\le n$. While conditional lower bounds prevent approximating the Fr\'echet distance between $\tau$ and $\sigma$ within a factor of 3 in strongly subquadratic time, the current best approximation algorithm attains a ratio of $n^c$ in strongly subquadratic time, for some constant $c\in(0,1)$. We present a randomized algorithm with running time $O(nm^{0.99}\log(n/\varepsilon))$ that approximates the Fr\'echet distance within a factor of $7+\varepsilon$, with a success probability at least $1-1/n^6$. We also adapt our techniques to develop a randomized algorithm that approximates the \emph{discrete} Fr\'echet distance within a factor of $7+\varepsilon$ in strongly subquadratic time. They are the first algorithms to approximate the Fr\'echet distance and the discrete Fr\'echet distance within constant factors in strongly subquadratic time.

cs.CG↗

arXiv2024

Fr\'echet Distance in Subquadratic Time

Let $m$ and $n$ be the numbers of vertices of two polygonal curves in $\mathbb{R}^d$ for any fixed $d$ such that $m \leq n$. Since it was known in 1995 how to compute the Fr\'{e}chet distance of these two curves in $O(mn\log (mn))$ time, it has been an open problem whether the running time can be reduced to $o(n^2)$ when $m = \Omega(n)$. In the mean time, several well-known quadratic time barriers in computational geometry have been overcome: 3SUM, some 3SUM-hard problems, and the computation of some distances between two polygonal curves, including the discrete Fr\'{e}chet distance, the dynamic time warping distance, and the geometric edit distance. It is curious that the quadratic time barrier for Fr\'{e}chet distance still stands. We present an algorithm to compute the Fr\'echet distance in $O(mn(\log\log n)^{2+\mu}\log n/\log^{1+\mu} m)$ expected time for some constant $\mu \in (0,1)$. It is the first algorithm that returns the Fr\'{e}chet distance in $o(mn)$ time when $m = \Omega(n^{\varepsilon})$ for any fixed $\varepsilon \in (0,1]$.

cs.CG↗

arXiv2024

A Fast Method for Lasso and Logistic Lasso

We propose a fast method for solving compressed sensing, Lasso regression, and Logistic Lasso regression problems that iteratively runs an appropriate solver using an active set approach. We design a strategy to update the active set that achieves a large speedup over a single call of several solvers, including gradient projection for sparse reconstruction (GPSR), lassoglm of Matlab, and glmnet. For compressed sensing, the hybrid of our method and GPSR is 31.41 times faster than GPSR on average for Gaussian ensembles and 25.64 faster on average for binary ensembles. For Lasso regression, the hybrid of our method and GPSR achieves a 30.67-fold average speedup in our experiments. In our experiments on Logistic Lasso regression, the hybrid of our method and lassoglm gives an 11.95-fold average speedup, and the hybrid of our method and glmnet gives a 1.40-fold average speedup.

cs.LG↗

arXiv2023

Geometric Matching and Bottleneck Problems

Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. A (many-to-many) matching is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$ and the $a_{pr}$'s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing $\sum_{(p,r,a_{pr}) \in \mathcal{A}} a_{pr}$. Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of $n$ red points $P$ and a set of $n$ blue points $Q$ that minimizes the length of the longest edge. For the $L_\infty$-metric, we can do this in time $O(n^{1+\varepsilon})$ in any fixed dimension, for the $L_2$-metric in the plane in time $O(n^{4/3 + \varepsilon})$, for any $\varepsilon > 0$.

cs.CG↗

arXiv2023

Solving Fr\'echet Distance Problems by Algebraic Geometric Methods

We study several polygonal curve problems under the Fr\'{e}chet distance via algebraic geometric methods. Let $\mathbb{X}_m^d$ and $\mathbb{X}_k^d$ be the spaces of all polygonal curves of $m$ and $k$ vertices in $\mathbb{R}^d$, respectively. We assume that $k \leq m$. Let $\mathcal{R}^d_{k,m}$ be the set of ranges in $\mathbb{X}_m^d$ for all possible metric balls of polygonal curves in $\mathbb{X}_k^d$ under the Fr\'{e}chet distance. We prove a nearly optimal bound of $O(dk\log (km))$ on the VC dimension of the range space $(\mathbb{X}_m^d,\mathcal{R}_{k,m}^d)$, improving on the previous $O(d^2k^2\log(dkm))$ upper bound and approaching the current $\Omega(dk\log k)$ lower bound. Our upper bound also holds for the weak Fr\'{e}chet distance. We also obtain exact solutions that are hitherto unknown for curve simplification, range searching, nearest neighbor search, and distance oracle.

cs.CG↗

arXiv2023

Solving Proximity Problems for Polygonal Shapes via Polynomials

We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by distance measures, such as the Hausdorff distance, the Fr\'echet distance and the dynamic time warping distance. These range spaces appear in various applications like classification, simplification, range searching, density estimation and clustering when the instances are trajectories, time series or polygons. We show for the Fr\'echet distance of polygonal curves and the Hausdorff distance of polygonal curves and planar polygonal regions that the VC-dimension is upper-bounded by $O(dk\log(km))$, where $k$ is the complexity of the center of a ball, $m$ is the complexity of each polygonal curve or region in the ground set, and $d$ is the ambient dimension. For $d \geq 4$ this bound is tight in each of the parameters $d,k$ and $m$ separately. For a variant of dynamic time warping distance of polygonal curves using squared distances, our analysis directly yields an upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$. We also obtain exact solutions that are hitherto unknown for simplification, range searching, nearest neighbor search, and distance oracle.

cs.CG↗

arXiv2023

Approximate Nearest Neighbor for Polygonal Curves under Fr\'echet Distance

We propose $\kappa$-approximate nearest neighbor (ANN) data structures for $n$ polygonal curves under the Fr\'{e}chet distance in $\mathbb{R}^d$, where $\kappa \in \{1+\varepsilon,3+\varepsilon\}$ and $d \geq 2$. We assume that every input curve has at most $m$ vertices, every query curve has at most $k$ vertices, $k \ll m$, and $k$ is given for preprocessing. The query times are $\tilde{O}(k(mn)^{0.5+\varepsilon}/\varepsilon^d+ k(d/\varepsilon)^{O(dk)})$ for $(1+\varepsilon)$-ANN and $\tilde{O}(k(mn)^{0.5+\varepsilon}/\varepsilon^d)$ for $(3+\varepsilon)$-ANN. The space and expected preprocessing time are $\tilde{O}(k(mnd^d/\varepsilon^d)^{O(k+1/\varepsilon^2)})$ in both cases. In two and three dimensions, we improve the query times to $O(1/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$ for $(1+\varepsilon)$-ANN and $\tilde{O}(k)$ for $(3+\varepsilon)$-ANN. The space and expected preprocessing time improve to $O(mn/\varepsilon)^{O(k)} \cdot \tilde{O}(k)$ in both cases. For ease of presentation, we treat factors in our bounds that depend purely on $d$ as~$O(1)$. The hidden polylog factors in the big-$\tilde{O}$ notation have powers dependent on $d$.

cs.CG↗

arXiv2022

Curve Simplification and Clustering under Fr\'echet Distance

We present new approximation results on curve simplification and clustering under Fr\'echet distance. Let $T = \{\tau_i : i \in [n] \}$ be polygonal curves in $R^d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $\delta_i > 0$ for $i \in [n]$, find a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i$ for $i \in [n]$. We present an algorithm that returns a null output or a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i + \epsilon\delta_{\max}$ for $i \in [n]$, where $\delta_{\max} = \max_{i \in [n]} \delta_i$. If the output is null, there is no curve of at most $l$ vertices within a Fr\'echet distance of $\delta_i$ from $\tau_i$ for $i \in [n]$. The running time is $\tilde{O}\bigl(n^{O(l)} m^{O(l^2)} (dl/\epsilon)^{O(dl)}\bigr)$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $\tau$ to another curve $\sigma$, where the vertices of $\sigma$ can be anywhere in $R^d$, so that $d_F(\sigma,\tau) \le (1+\epsilon)\delta$ and $|\sigma| \le (1+\alpha) \min\{|c| : d_F(c,\tau) \le \delta\}$ for any given $\delta > 0$ and any fixed $\alpha, \epsilon \in (0,1)$. The running time is $\tilde{O}\bigl(m^{O(1/\alpha)} (d/(\alpha\epsilon))^{O(d/\alpha)}\bigr)$. By combining our technique with some previous results in the literature, we obtain an approximation algorithm for $(k,l)$-median clustering. Given $T$, it computes a set $\Sigma$ of $k$ curves, each of $l$ vertices, such that $\sum_{i \in [n]} \min_{\sigma \in \Sigma} d_F(\sigma,\tau_i)$ is within a factor $1+\epsilon$ of the optimum with probability at least $1-\mu$ for any given $\mu, \epsilon \in (0,1)$. The running time is $\tilde{O}\bigl(n m^{O(kl^2)} \mu^{-O(kl)} (dkl/\epsilon)^{O((dkl/\epsilon)\log(1/\mu))}\bigr)$.

cs.CG↗

arXiv2022

On Non-Negative Quadratic Programming in Geometric Optimization

We present experimental and theoretical results on a method that applies a numerical solver iteratively to solve several non-negative quadratic programming problems in geometric optimization. The method gains efficiency by exploiting the potential sparsity of the intermediate solutions. We implemented the method to call quadprog of MATLAB iteratively. In comparison with a single call of quadprog, we obtain a 10-fold speedup on two proximity graph problems in $\mathbb{R}^d$ on some public data sets, a 10-fold speedup on the minimum enclosing ball problem on random points in a unit cube in $\mathbb{R}^d$, and a 5-fold speedup on the polytope distance problem on random points from a cube in $\mathbb{R}^d$ when the input size is significantly larger than the dimension; we also obtain a 2-fold or more speedup on deblurring some gray-scale space and thermal images via non-negative least square. We compare with two minimum enclosing ball software by G\"{a}rtner and Fischer et al.; for 1000 nearly cospherical points or random points in a unit cube, the iterative method overtakes the software by G\"{a}rtner at 20 dimensions and the software by Fischer et al. at 170 dimensions. In the image deblurring experiments, the iterative method compares favorably with other software that can solve non-negative least square, including FISTA with backtracking, SBB, FNNLS, and lsqnonneg of MATLAB. We analyze theoretically the number of iterations taken by the iterative scheme to reduce the gap between the current solution value and the optimum by a factor $e$. Under certain assumptions, we prove a bound proportional to the square root of the number of variables.

cs.CG↗

arXiv2021

Multistage Online Maxmin Allocation of Indivisible Entities

We consider an online allocation problem that involves a set $P$ of $n$ players and a set $E$ of $m$ indivisible entities over discrete time steps $1,2,\ldots,\tau$. At each time step $t \in [1,\tau]$, for every entity $e \in E$, there is a restriction list $L_t(e)$ that prescribes the subset of players to whom $e$ can be assigned and a non-negative value $v_t(e,p)$ of $e$ to every player $p \in P$. The sets $P$ and $E$ are fixed beforehand. The sets $L_t(\cdot)$ and values $v_t(\cdot,\cdot)$ are given in an online fashion. An allocation is a distribution of $E$ among $P$, and we are interested in the minimum total value of the entities received by a player according to the allocation. In the static case, it is NP-hard to find an optimal allocation the maximizes this minimum value. On the other hand, $\rho$-approximation algorithms have been developed for certain values of $\rho \in (0,1]$. We propose a $w$-lookahead algorithm for the multistage online maxmin allocation problem for any fixed $w \geq 1$ in which the restriction lists and values of entities may change between time steps, and there is a fixed stability reward for an entity to be assigned to the same player from one time step to the next. The objective is to maximize the sum of the minimum values and stability rewards over the time steps $1, 2, \ldots, \tau$. Our algorithm achieves a competitive ratio of $(1-c)\rho$, where $c$ is the positive root of the equation $wc^2 = \rho (w+1)(1-c)$. When $w = 1$, it is greater than $\frac{\rho}{4\rho+2} + \frac{\rho}{10}$, which improves upon the previous ratio of $\frac{\rho}{4\rho+2 - 2^{1-\tau}(2\rho+1)}$ obtained for the case of 1-lookahead.

cs.DS↗

arXiv2021

Self-Improving Voronoi Construction for a Hidden Mixture of Product Distributions

We propose a self-improving algorithm for computing Voronoi diagrams under a given convex distance function with constant description complexity. The $n$ input points are drawn from a hidden mixture of product distributions; we are only given an upper bound $m = o(\sqrt{n})$ on the number of distributions in the mixture, and the property that for each distribution, an input instance is drawn from it with a probability of $\Omega(1/n)$. For any $\varepsilon \in (0,1)$, after spending $O\bigl(mn\log^{O(1)} (mn) + m^{\varepsilon} n^{1+\varepsilon}\log(mn)\bigr)$ time in a training phase, our algorithm achieves an $O\bigl(\frac{1}{\varepsilon}n\log m + \frac{1}{\varepsilon}n2^{O(\log^* n)} + \frac{1}{\varepsilon}H\bigr)$ expected running time with probability at least $1 - O(1/n)$, where $H$ is the entropy of the distribution of the Voronoi diagram output. The expectation is taken over the input distribution and the randomized decisions of the algorithm. For the Euclidean metric, the expected running time improves to $O\bigl(\frac{1}{\varepsilon}n\log m + \frac{1}{\varepsilon}H\bigr)$.

cs.CG↗

arXiv2020

Dynamic Distribution-Sensitive Point Location

We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in $\mathbb{R}^2$, and we are given an oracle that can return in $O(1)$ time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision $\cal S$ with $n$ vertices such that each query is answered in $O(\mathrm{OPT})$ expected time, where OPT is the minimum expected time of the best linear decision tree for point location in $\cal S$. The space and construction time are $O(n\log^2 n)$. An update of $\cal S$ as a mixed sequence of $k$ edge insertions and deletions takes $O(k\log^5 n)$ amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of $n$ sites can be performed in $O(n\log^5 n)$ expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.

cs.CG↗

arXiv2020

A Generalization of Self-Improving Algorithms

Ailon et al. [SICOMP'11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances $x_1,\cdots,x_n$ follow some unknown \emph{product distribution}. That is, $x_i$ comes from a fixed unknown distribution $\mathsf{D}_i$, and the $x_i$'s are drawn independently. After spending $O(n^{1+\varepsilon})$ time in a learning phase, the subsequent expected running time is $O((n+ H)/\varepsilon)$, where $H \in \{H_\mathrm{S},H_\mathrm{DT}\}$, and $H_\mathrm{S}$ and $H_\mathrm{DT}$ are the entropies of the distributions of the sorting and DT output, respectively. In this paper, we allow dependence among the $x_i$'s under the \emph{group product distribution}. There is a hidden partition of $[1,n]$ into groups; the $x_i$'s in the $k$-th group are fixed unknown functions of the same hidden variable $u_k$; and the $u_k$'s are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map $u_k$ to $x_i$'s are well-behaved. After an $O(\mathrm{poly}(n))$-time training phase, we achieve $O(n + H_\mathrm{S})$ and $O(n\alpha(n) + H_\mathrm{DT})$ expected running times for sorting and DT, respectively, where $\alpha(\cdot)$ is the inverse Ackermann function.

cs.CG↗

arXiv2019

Extensions of Self-Improving Sorters

Ailon et al. (SICOMP 2011) proposed a self-improving sorter that tunes its performance to an unknown input distribution in a training phase. The input numbers $x_1,x_2,\ldots,x_n$ come from a product distribution, that is, each $x_i$ is drawn independently from an arbitrary distribution ${\cal D}_i$. We study two relaxations of this requirement. The first extension models hidden classes in the input. We consider the case that numbers in the same class are governed by linear functions of the same hidden random parameter. The second extension considers a hidden mixture of product distributions.

cs.DS↗

arXiv2019

Restricted Max-Min Allocation: Approximation and Integrality Gap

Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most $4$. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a $(6 + \delta)$-approximation algorithm where $\delta$ can be any positive constant, and there is still a gap of roughly $2$. In this paper, we narrow the gap significantly by proposing a $(4+\delta)$-approximation algorithm where $\delta$ can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is $\mathit{poly}(m,n)\cdot n^{\mathit{poly}(\frac{1}{\delta})}$ where $n$ is the number of players and $m$ is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to $3 + \frac{21}{26} \approx 3.808$.

cs.DS↗