arXiv subjects

Eric Price

Publications and source records attributed to Eric Price.

At least 37 records · Page 2Linked to original sources

arXiv2022

Fast Splitting Algorithms for Sparsity-Constrained and Noisy Group Testing

In group testing, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether at least one defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, communication protocols, and many more. In this paper, we study (i) a sparsity-constrained version of the problem, in which the testing procedure is subjected to one of the following two constraints: items are finitely divisible and thus may participate in at most $γ$ tests; or tests are size-constrained to pool no more than $ρ$ items per test; and (ii) a noisy version of the problem, where each test outcome is independently flipped with some constant probability. Under each of these settings, considering the for-each recovery guarantee with asymptotically vanishing error probability, we introduce a fast splitting algorithm and establish its near-optimality not only in terms of the number of tests, but also in terms of the decoding time. While the most basic formulations of our algorithms require $Ω(n)$ storage for each algorithm, we also provide low-storage variants based on hashing, with similar recovery guarantees.

cs.IT↗

arXiv2022

Factorial Lower Bounds for (Almost) Random Order Streams

In this paper we introduce and study the \textsc{StreamingCycles} problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph $G$, comprising $n/\ell$ disjoint length-$\ell$ cycles on $n$ vertices, are partitioned randomly among $n$ players. Every edge is annotated with an independent uniformly random bit, and the players' task is to output the parity of some cycle in $G$ after one round of sequential communication. Our main result is an $\ell^{Ω(\ell)}$ lower bound on the communication complexity of \textsc{StreamingCycles}, which is tight up to constant factors in $\ell$. Applications of our lower bound for \textsc{StreamingCycles} include an essentially tight lower bound for component collection in (almost) random order graph streams, making progress towards a conjecture of Peng and Sohler [SODA'18] and the first exponential space lower bounds for random walk generation.

cs.DS↗

arXiv2022

Finite-Sample Maximum Likelihood Estimation of Location

We consider 1-dimensional location estimation, where we estimate a parameter $λ$ from $n$ samples $λ+ η_i$, with each $η_i$ drawn i.i.d. from a known distribution $f$. For fixed $f$ the maximum-likelihood estimate (MLE) is well-known to be optimal in the limit as $n \to \infty$: it is asymptotically normal with variance matching the Cramér-Rao lower bound of $\frac{1}{n\mathcal{I}}$, where $\mathcal{I}$ is the Fisher information of $f$. However, this bound does not hold for finite $n$, or when $f$ varies with $n$. We show for arbitrary $f$ and $n$ that one can recover a similar theory based on the Fisher information of a smoothed version of $f$, where the smoothing radius decays with $n$.

math.ST↗

arXiv2022

Hardness and Algorithms for Robust and Sparse Optimization

We explore algorithms and limitations for sparse optimization problems such as sparse linear regression and robust linear regression. The goal of the sparse linear regression problem is to identify a small number of key features, while the goal of the robust linear regression problem is to identify a small number of erroneous measurements. Specifically, the sparse linear regression problem seeks a $k$-sparse vector $x\in\mathbb{R}^d$ to minimize $\|Ax-b\|_2$, given an input matrix $A\in\mathbb{R}^{n\times d}$ and a target vector $b\in\mathbb{R}^n$, while the robust linear regression problem seeks a set $S$ that ignores at most $k$ rows and a vector $x$ to minimize $\|(Ax-b)_S\|_2$. We first show bicriteria, NP-hardness of approximation for robust regression building on the work of [OWZ15] which implies a similar result for sparse regression. We further show fine-grained hardness of robust regression through a reduction from the minimum-weight $k$-clique conjecture. On the positive side, we give an algorithm for robust regression that achieves arbitrarily accurate additive error and uses runtime that closely matches the lower bound from the fine-grained hardness result, as well as an algorithm for sparse regression with similar runtime. Both our upper and lower bounds rely on a general reduction from robust linear regression to sparse regression that we introduce. Our algorithms, inspired by the 3SUM problem, use approximate nearest neighbor data structures and may be of independent interest for solving sparse optimization problems. For instance, we demonstrate that our techniques can also be used for the well-studied sparse PCA problem.

cs.LG↗

arXiv2022

Sharp Constants in Uniformity Testing via the Huber Statistic

Uniformity testing is one of the most well-studied problems in property testing, with many known test statistics, including ones based on counting collisions, singletons, and the empirical TV distance. It is known that the optimal sample complexity to distinguish the uniform distribution on $m$ elements from any $ε$-far distribution with $1-δ$ probability is $n = Θ\left(\frac{\sqrt{m \log (1/δ)}}{ε^2} + \frac{\log (1/δ)}{ε^2}\right)$, which is achieved by the empirical TV tester. Yet in simulation, these theoretical analyses are misleading: in many cases, they do not correctly rank order the performance of existing testers, even in an asymptotic regime of all parameters tending to $0$ or $\infty$. We explain this discrepancy by studying the \emph{constant factors} required by the algorithms. We show that the collisions tester achieves a sharp maximal constant in the number of standard deviations of separation between uniform and non-uniform inputs. We then introduce a new tester based on the Huber loss, and show that it not only matches this separation, but also has tails corresponding to a Gaussian with this separation. This leads to a sample complexity of $(1 + o(1))\frac{\sqrt{m \log (1/δ)}}{ε^2}$ in the regime where this term is dominant, unlike all other existing testers.

stat.ML↗

arXiv2022

Linear Bandit Algorithms with Sublinear Time Complexity

We propose two linear bandits algorithms with per-step complexity sublinear in the number of arms $K$. The algorithms are designed for applications where the arm set is extremely large and slowly changing. Our key realization is that choosing an arm reduces to a maximum inner product search (MIPS) problem, which can be solved approximately without breaking regret guarantees. Existing approximate MIPS solvers run in sublinear time. We extend those solvers and present theoretical guarantees for online learning problems, where adaptivity (i.e., a later step depends on the feedback in previous steps) becomes a unique challenge. We then explicitly characterize the tradeoff between the per-step complexity and regret. For sufficiently large $K$, our algorithms have sublinear per-step complexity and $\tilde O(\sqrt{T})$ regret. Empirically, we evaluate our proposed algorithms in a synthetic environment and a real-world online movie recommendation problem. Our proposed algorithms can deliver a more than 72 times speedup compared to the linear time baselines while retaining similar regret.

cs.LG↗

arXiv2022

Deep Residual Reinforcement Learning based Autonomous Blimp Control

Blimps are well suited to perform long-duration aerial tasks as they are energy efficient, relatively silent and safe. To address the blimp navigation and control task, in previous work we developed a hardware and software-in-the-loop framework and a PID-based controller for large blimps in the presence of wind disturbance. However, blimps have a deformable structure and their dynamics are inherently non-linear and time-delayed, making PID controllers difficult to tune. Thus, often resulting in large tracking errors. Moreover, the buoyancy of a blimp is constantly changing due to variations in ambient temperature and pressure. To address these issues, in this paper we present a learning-based framework based on deep residual reinforcement learning (DRRL), for the blimp control task. Within this framework, we first employ a PID controller to provide baseline performance. Subsequently, the DRRL agent learns to modify the PID decisions by interaction with the environment. We demonstrate in simulation that DRRL agent consistently improves the PID performance. Through rigorous simulation experiments, we show that the agent is robust to changes in wind speed and buoyancy. In real-world experiments, we demonstrate that the agent, trained only in simulation, is sufficiently robust to control an actual blimp in windy conditions. We openly provide the source code of our approach at https://github.com/ robot-perception-group/AutonomousBlimpDRL.

cs.RO↗

arXiv2022

Coresets for Data Discretization and Sine Wave Fitting

In the \emph{monitoring} problem, the input is an unbounded stream $P={p_1,p_2\cdots}$ of integers in $[N]:=\{1,\cdots,N\}$, that are obtained from a sensor (such as GPS or heart beats of a human). The goal (e.g., for anomaly detection) is to approximate the $n$ points received so far in $P$ by a single frequency $\sin$, e.g. $\min_{c\in C}cost(P,c)+λ(c)$, where $cost(P,c)=\sum_{i=1}^n \sin^2(\frac{2π}{N} p_ic)$, $C\subseteq [N]$ is a feasible set of solutions, and $λ$ is a given regularization function. For any approximation error $\varepsilon>0$, we prove that \emph{every} set $P$ of $n$ integers has a weighted subset $S\subseteq P$ (sometimes called core-set) of cardinality $|S|\in O(\log(N)^{O(1)})$ that approximates $cost(P,c)$ (for every $c\in [N]$) up to a multiplicative factor of $1\pm\varepsilon$. Using known coreset techniques, this implies streaming algorithms using only $O((\log(N)\log(n))^{O(1)})$ memory. Our results hold for a large family of functions. Experimental results and open source code are provided.

cs.LG↗

arXiv2022

AirPose: Multi-View Fusion Network for Aerial 3D Human Pose and Shape Estimation

In this letter, we present a novel markerless 3D human motion capture (MoCap) system for unstructured, outdoor environments that uses a team of autonomous unmanned aerial vehicles (UAVs) with on-board RGB cameras and computation. Existing methods are limited by calibrated cameras and off-line processing. Thus, we present the first method (AirPose) to estimate human pose and shape using images captured by multiple extrinsically uncalibrated flying cameras. AirPose itself calibrates the cameras relative to the person instead of relying on any pre-calibration. It uses distributed neural networks running on each UAV that communicate viewpoint-independent information with each other about the person (i.e., their 3D shape and articulated pose). The person's shape and pose are parameterized using the SMPL-X body model, resulting in a compact representation, that minimizes communication between the UAVs. The network is trained using synthetic images of realistic virtual environments, and fine-tuned on a small set of real images. We also introduce an optimization-based post-processing method (AirPose$^{+}$) for offline applications that require higher MoCap quality. We make our method's code and data available for research at https://github.com/robot-perception-group/AirPose. A video describing the approach and results is available at https://youtu.be/xLYe1TNHsfs.

cs.CV↗

arXiv2021

Simulating Random Walks in Random Streams

The random order graph streaming model has received significant attention recently, with problems such as matching size estimation, component counting, and the evaluation of bounded degree constant query testable properties shown to admit surprisingly space efficient algorithms. The main result of this paper is a space efficient single pass random order streaming algorithm for simulating nearly independent random walks that start at uniformly random vertices. We show that the distribution of $k$-step walks from $b$ vertices chosen uniformly at random can be approximated up to error $\varepsilon$ per walk using $(1/\varepsilon)^{O(k)} 2^{O(k^2)}\cdot b$ words of space with a single pass over a randomly ordered stream of edges, solving an open problem of Peng and Sohler [SODA `18]. Applications of our result include the estimation of the average return probability of the $k$-step walk (the trace of the $k^\text{th}$ power of the random walk matrix) as well as the estimation of PageRank. We complement our algorithm with a strong impossibility result for directed graphs.

cs.DS↗

arXiv2021

Robust Compressed Sensing MRI with Deep Generative Priors

The CSGM framework (Bora-Jalal-Price-Dimakis'17) has shown that deep generative priors can be powerful tools for solving inverse problems. However, to date this framework has been empirically successful only on certain datasets (for example, human faces and MNIST digits), and it is known to perform poorly on out-of-distribution samples. In this paper, we present the first successful application of the CSGM framework on clinical MRI data. We train a generative prior on brain scans from the fastMRI dataset, and show that posterior sampling via Langevin dynamics achieves high quality reconstructions. Furthermore, our experiments and theory show that posterior sampling is robust to changes in the ground-truth distribution and measurement process. Our code and models are available at: \url{https://github.com/utcsilab/csgm-mri-langevin}.

cs.LG↗

arXiv2021

Autonomous Blimp Control using Deep Reinforcement Learning

Aerial robot solutions are becoming ubiquitous for an increasing number of tasks. Among the various types of aerial robots, blimps are very well suited to perform long-duration tasks while being energy efficient, relatively silent and safe. To address the blimp navigation and control task, in our recent work, we have developed a software-in-the-loop simulation and a PID-based controller for large blimps in the presence of wind disturbance. However, blimps have a deformable structure and their dynamics are inherently non-linear and time-delayed, often resulting in large trajectory tracking errors. Moreover, the buoyancy of a blimp is constantly changing due to changes in the ambient temperature and pressure. In the present paper, we explore a deep reinforcement learning (DRL) approach to address these issues. We train only in simulation, while keeping conditions as close as possible to the real-world scenario. We derive a compact state representation to reduce the training time and a discrete action space to enforce control smoothness. Our initial results in simulation show a significant potential of DRL in solving the blimp control task and robustness against moderate wind and parameter uncertainty. Extensive experiments are presented to study the robustness of our approach. We also openly provide the source code of our approach.

cs.RO↗

arXiv2021

Optimal Non-Adaptive Probabilistic Group Testing in General Sparsity Regimes

In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which the goal is high-probability recovery of the defective set. We show that in the case of $n$ items among which $k$ are defective, the smallest possible number of tests equals $\min\{ C_{k,n} k \log n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ is a uniformly bounded constant (varying depending on the scaling of $k$ with respect to $n$) with a simple explicit expression. The algorithmic upper bound follows from a minor adaptation of an existing analysis of the Definite Defectives (DD) algorithm, and the algorithm-independent lower bound builds on existing works for the regimes $k \le n^{1-Ω(1)}$ and $k = Θ(n)$. In sufficiently sparse regimes (including $k = o\big( \frac{n}{\log n} \big)$), our main result generalizes that of Coja-Oghlan {\em et al.} (2020) by avoiding the assumption $k \le n^{1-Ω(1)}$, whereas in sufficiently dense regimes (including $k = ω\big( \frac{n}{\log n} \big)$), our main result shows that individual testing is asymptotically optimal for any non-zero target success probability, thus strengthening an existing result of Aldridge (2019) in terms of both the error probability and the assumed scaling of $k$.

cs.IT↗

arXiv2021

Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu

We provide finite sample guarantees for the classical Chow-Liu algorithm (IEEE Trans.~Inform.~Theory, 1968) to learn a tree-structured graphical model of a distribution. For a distribution $P$ on $Σ^n$ and a tree $T$ on $n$ nodes, we say $T$ is an $\varepsilon$-approximate tree for $P$ if there is a $T$-structured distribution $Q$ such that $D(P\;||\;Q)$ is at most $\varepsilon$ more than the best possible tree-structured distribution for $P$. We show that if $P$ itself is tree-structured, then the Chow-Liu algorithm with the plug-in estimator for mutual information with $\widetilde{O}(|Σ|^3 n\varepsilon^{-1})$ i.i.d.~samples outputs an $\varepsilon$-approximate tree for $P$ with constant probability. In contrast, for a general $P$ (which may not be tree-structured), $Ω(n^2\varepsilon^{-2})$ samples are necessary to find an $\varepsilon$-approximate tree. Our upper bound is based on a new conditional independence tester that addresses an open problem posed by Canonne, Diakonikolas, Kane, and Stewart~(STOC, 2018): we prove that for three random variables $X,Y,Z$ each over $Σ$, testing if $I(X; Y \mid Z)$ is $0$ or $\geq \varepsilon$ is possible with $\widetilde{O}(|Σ|^3/\varepsilon)$ samples. Finally, we show that for a specific tree $T$, with $\widetilde{O} (|Σ|^2n\varepsilon^{-1})$ samples from a distribution $P$ over $Σ^n$, one can efficiently learn the closest $T$-structured distribution in KL divergence by applying the add-1 estimator at each node.

cs.DS↗

arXiv2021

Fairness for Image Generation with Uncertain Sensitive Attributes

This work tackles the issue of fairness in the context of generative procedures, such as image super-resolution, which entail different definitions from the standard classification setting. Moreover, while traditional group fairness definitions are typically defined with respect to specified protected groups -- camouflaging the fact that these groupings are artificial and carry historical and political motivations -- we emphasize that there are no ground truth identities. For instance, should South and East Asians be viewed as a single group or separate groups? Should we consider one race as a whole or further split by gender? Choosing which groups are valid and who belongs in them is an impossible dilemma and being "fair" with respect to Asians may require being "unfair" with respect to South Asians. This motivates the introduction of definitions that allow algorithms to be \emph{oblivious} to the relevant groupings. We define several intuitive notions of group fairness and study their incompatibilities and trade-offs. We show that the natural extension of demographic parity is strongly dependent on the grouping, and \emph{impossible} to achieve obliviously. On the other hand, the conceptually new definition we introduce, Conditional Proportional Representation, can be achieved obliviously through Posterior Sampling. Our experiments validate our theoretical results and achieve fair image reconstruction using state-of-the-art generative models.

cs.LG↗

arXiv2021

Instance-Optimal Compressed Sensing via Posterior Sampling

We characterize the measurement complexity of compressed sensing of signals drawn from a known prior distribution, even when the support of the prior is the entire space (rather than, say, sparse vectors). We show for Gaussian measurements and \emph{any} prior distribution on the signal, that the posterior sampling estimator achieves near-optimal recovery guarantees. Moreover, this result is robust to model mismatch, as long as the distribution estimate (e.g., from an invertible generative model) is close to the true distribution in Wasserstein distance. We implement the posterior sampling estimator for deep generative priors using Langevin dynamics, and empirically find that it produces accurate estimates with more diversity than MAP.

cs.LG↗

arXiv2021

A Simple Proof of a New Set Disjointness with Applications to Data Streams

The multiplayer promise set disjointness is one of the most widely used problems from communication complexity in applications. In this problem there are $k$ players with subsets $S^1, \ldots, S^k$, each drawn from $\{1, 2, \ldots, n\}$, and we are promised that either the sets are (1) pairwise disjoint, or (2) there is a unique element $j$ occurring in all the sets, which are otherwise pairwise disjoint. The total communication of solving this problem with constant probability in the blackboard model is $Ω(n/k)$. We observe for most applications, it instead suffices to look at what we call the ``mostly'' set disjointness problem, which changes case (2) to say there is a unique element $j$ occurring in at least half of the sets, and the sets are otherwise disjoint. This change gives us a much simpler proof of an $Ω(n/k)$ randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Using this we show several new results for data streams: \begin{itemize} \item for $\ell_2$-Heavy Hitters, any $O(1)$-pass streaming algorithm in the insertion-only model for detecting if an $\eps$-$\ell_2$-heavy hitter exists requires $\min(\frac{1}{\eps^2}\log \frac{\eps^2n}δ, \frac{1}{\eps}n^{1/2})$ bits of memory, which is optimal up to a $\log n$ factor. For deterministic algorithms and constant $\eps$, this gives an $Ω(n^{1/2})$ lower bound, improving the prior $Ω(\log n)$ lower bound. We also obtain lower bounds for Zipfian distributions. \item for $\ell_p$-Estimation, $p > 2$, we show an $O(1)$-pass $Ω(n^{1-2/p} \log(1/δ))$ bit lower bound for outputting an $O(1)$-approximation with probability $1-δ$, in the insertion-only model. This is optimal, and the best previous lower bound was $Ω(n^{1-2/p} + \log(1/δ))$. \end{itemize}

cs.DS↗

arXiv2021

L1 Regression with Lewis Weights Subsampling

We consider the problem of finding an approximate solution to $\ell_1$ regression while only observing a small number of labels. Given an $n \times d$ unlabeled data matrix $X$, we must choose a small set of $m \ll n$ rows to observe the labels of, then output an estimate $\widehatβ$ whose error on the original problem is within a $1 + \varepsilon$ factor of optimal. We show that sampling from $X$ according to its Lewis weights and outputting the empirical minimizer succeeds with probability $1-δ$ for $m > O(\frac{1}{\varepsilon^2} d \log \frac{d}{\varepsilon δ})$. This is analogous to the performance of sampling according to leverage scores for $\ell_2$ regression, but with exponentially better dependence on $δ$. We also give a corresponding lower bound of $Ω(\frac{d}{\varepsilon^2} + (d + \frac{1}{\varepsilon^2}) \log\frac{1}δ)$.

cs.LG↗