arXiv subjects

Eric Price

Publications and source records attributed to Eric Price.

At least 55 records · Page 3Linked to original sources

arXiv2021

Separations and Equivalences between Turnstile Streaming and Linear Sketching

A longstanding observation, which was partially proven in \cite{LNW14,AHLW16}, is that any turnstile streaming algorithm can be implemented as a linear sketch (the reverse is trivially true). We study the relationship between turnstile streaming and linear sketching algorithms in more detail, giving both new separations and new equivalences between the two models. It was shown in \cite{LNW14} that, if a turnstile algorithm works for arbitrarily long streams with arbitrarily large coordinates at intermediate stages of the stream, then the turnstile algorithm is equivalent to a linear sketch. We show separations of the opposite form: if either the stream length or the maximum value of the stream are substantially restricted, there exist problems where linear sketching is exponentially harder than turnstile streaming. A further limitation of the \cite{LNW14} equivalence is that the turnstile sketching algorithm is neither explicit nor uniform, but requires an exponentially long advice string. We show how to remove this limitation for deterministic streaming algorithms: we give an explicit small-space algorithm that takes the streaming algorithm and computes an equivalent module.

cs.DS↗

arXiv2020

Simulation and Control of Deformable Autonomous Airships in Turbulent Wind

Abstract. Fixed wing and multirotor UAVs are common in the field of robotics. Solutions for simulation and control of these vehicles are ubiquitous. This is not the case for airships, a simulation of which needs to address unique properties, i) dynamic deformation in response to aerodynamic and control forces, ii) high susceptibility to wind and turbulence at low airspeed, iii) high variability in airship designs regarding placement, direction and vectoring of thrusters and control surfaces. We present a flexible framework for modeling, simulation and control of airships, based on the Robot operating system (ROS), simulation environment (Gazebo) and commercial off the shelf (COTS) electronics, both of which are open source. Based on simulated wind and deformation, we predict substantial effects on controllability, verified in real world flight experiments. All our code is shared as open source, for the benefit of the community and to facilitate lighter-than-air vehicle (LTAV) research. https://github.com/robot-perception-group/airship_simulation

cs.RO↗

arXiv2020

Compressed Sensing with Deep Image Prior and Learned Regularization

We propose a novel method for compressed sensing recovery using untrained deep generative models. Our method is based on the recently proposed Deep Image Prior (DIP), wherein the convolutional weights of the network are optimized to match the observed measurements. We show that this approach can be applied to solve any differentiable linear inverse problem, outperforming previous unlearned methods. Unlike various learned approaches based on generative models, our method does not require pre-training over large datasets. We further introduce a novel learned regularization technique, which incorporates prior information on the network weights. This reduces reconstruction error, especially for noisy measurements. Finally, we prove that, using the DIP optimization approach, moderately overparameterized single-layer networks can perfectly fit any signal despite the non-convex nature of the fitting problem. This theoretical result provides justification for early stopping.

stat.ML↗

arXiv2020

Optimal Testing of Discrete Distributions with High Probability

We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property $\mathcal{P}$, and parameters $0< ε, δ<1$, we want to distinguish {\em with probability at least $1-δ$} whether these distributions satisfy $\mathcal{P}$ or are $ε$-far from $\mathcal{P}$ in total variation distance. Most prior work in distribution testing studied the constant confidence case (corresponding to $δ= Ω(1)$), and provided sample-optimal testers for a range of properties. While one can always boost the confidence probability of any such tester by black-box amplification, this generic boosting method typically leads to sub-optimal sample bounds. Here we study the following broad question: For a given property $\mathcal{P}$, can we {\em characterize} the sample complexity of testing $\mathcal{P}$ as a function of all relevant problem parameters, including the error probability $δ$? Prior to this work, uniformity testing was the only statistical task whose sample complexity had been characterized in this setting. As our main results, we provide the first algorithms for closeness and independence testing that are sample-optimal, within constant factors, as a function of all relevant parameters. We also show matching information-theoretic lower bounds on the sample complexity of these problems. Our techniques naturally extend to give optimal testers for related problems. To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions and testing closeness with unequal sized samples.

cs.DS↗

arXiv2020

A Fast Binary Splitting Approach to Non-Adaptive Group Testing

In this paper, we consider the problem of noiseless non-adaptive group testing under the for-each recovery guarantee, also known as probabilistic group testing. In the case of $n$ items and $k$ defectives, we provide an algorithm attaining high-probability recovery with $O(k \log n)$ scaling in both the number of tests and runtime, improving on the best known $O(k^2 \log k \cdot \log n)$ runtime previously available for any algorithm that only uses $O(k \log n)$ tests. Our algorithm bears resemblance to Hwang's adaptive generalized binary splitting algorithm (Hwang, 1972); we recursively work with groups of items of geometrically vanishing sizes, while maintaining a list of "possibly defective" groups and circumventing the need for adaptivity. While the most basic form of our algorithm requires $Ω(n)$ storage, we also provide a low-storage variant based on hashing, with similar recovery guarantees.

cs.IT↗

arXiv2019

Lower Bounds for Compressed Sensing with Generative Models

The goal of compressed sensing is to learn a structured signal $x$ from a limited number of noisy linear measurements $y \approx Ax$. In traditional compressed sensing, "structure" is represented by sparsity in some known basis. Inspired by the success of deep learning in modeling images, recent work starting with~\cite{BJPD17} has instead considered structure to come from a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. We present two results establishing the difficulty of this latter task, showing that existing bounds are tight. First, we provide a lower bound matching the~\cite{BJPD17} upper bound for compressed sensing from $L$-Lipschitz generative models $G$. In particular, there exists such a function that requires roughly $Ω(k \log L)$ linear measurements for sparse recovery to be possible. This holds even for the more relaxed goal of \emph{nonuniform} recovery. Second, we show that generative models generalize sparsity as a representation of structure. In particular, we construct a ReLU-based neural network $G: \mathbb{R}^{2k} \to \mathbb{R}^n$ with $O(1)$ layers and $O(kn)$ activations per layer, such that the range of $G$ contains all $k$-sparse vectors.

cs.DS↗

arXiv2019

Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering

We study high-dimensional sparse estimation tasks in a robust setting where a constant fraction of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust sparse mean estimation and robust sparse PCA. We give the first practically viable robust estimators for these problems. In more detail, our algorithms are sample and computationally efficient and achieve near-optimal robustness guarantees. In contrast to prior provable algorithms which relied on the ellipsoid method, our algorithms use spectral techniques to iteratively remove outliers from the dataset. Our experimental evaluation on synthetic data shows that our algorithms are scalable and significantly outperform a range of previous approaches, nearly matching the best error rate without corruptions.

cs.DS↗

arXiv2019

Active Perception based Formation Control for Multiple Aerial Vehicles

Autonomous motion capture (mocap) systems for outdoor scenarios involving flying or mobile cameras rely on i) a robotic front-end to track and follow a human subject in real-time while he/she performs physical activities, and ii) an algorithmic back-end that estimates full body human pose and shape from the saved videos. In this paper we present a novel front-end for our aerial mocap system that consists of multiple micro aerial vehicles (MAVs) with only on-board cameras and computation. In previous work, we presented an approach for cooperative detection and tracking (CDT) of a subject using multiple MAVs. However, it did not ensure optimal view-point configurations of the MAVs to minimize the uncertainty in the person's cooperatively tracked 3D position estimate. In this article we introduce an active approach for CDT. In contrast to cooperatively tracking only the 3D positions of the person, the MAVs can now actively compute optimal local motion plans, resulting in optimal view-point configurations, which minimize the uncertainty in the tracked estimate. We achieve this by decoupling the goal of active tracking as a convex quadratic objective and non-convex constraints corresponding to angular configurations of the MAVs w.r.t. the person. We derive it using Gaussian observation model assumptions within the CDT algorithm. We also show how we embed all the non-convex constraints, including those for dynamic and static obstacle avoidance, as external control inputs in the MPC dynamics. Multiple real robot experiments and comparisons involving 3 MAVs in several challenging scenarios are presented (video link : https://youtu.be/1qWW2zWvRhA). Extensive simulation results demonstrate the scalability and robustness of our approach. ROS-based source code is also provided.

cs.RO↗

arXiv2019

Estimating the Frequency of a Clustered Signal

We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function $g(t)$ with Fourier spectrum in a narrow range $[f_0 - Δ, f_0 + Δ]$, how accurately is it possible to identify $f_0$? We present generic conditions on $g$ that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for $k$-Fourier-sparse signals that imply recovery of $f_0$ to within $Δ+ \tilde{O}(k^3)$ from samples on $[-1, 1]$. This improves upon the best previous bound of $O\big( Δ+ \tilde{O}(k^5) \big)^{1.5}$. We also show that no algorithm can do better than $Δ+ \tilde{O}(k^2)$. In the process we provide a new $\tilde{O}(k^3)$ bound on the ratio between the maximum and average value of continuous $k$-Fourier-sparse signals, which has independent application.

cs.DS↗

arXiv2019

A Hybrid Sampling Scheme for Triangle Counting

We study the problem of estimating the number of triangles in a graph stream. No streaming algorithm can get sublinear space on all graphs, so methods in this area bound the space in terms of parameters of the input graph such as the maximum number of triangles sharing a single edge. We give a sampling algorithm that is additionally parameterized by the maximum number of triangles sharing a single vertex. Our bound matches the best known turnstile results in all graphs, and gets better performance on simple graphs like $G(n, p)$ or a set of independent triangles. We complement the upper bound with a lower bound showing that no sampling algorithm can do better on those graphs by more than a log factor. In particular, any insertion stream algorithm must use $\sqrt{T}$ space when all the triangles share a common vertex, and any sampling algorithm must take $T^\frac{1}{3}$ samples when all the triangles are independent. We add another lower bound, also matching our algorithm's performance, which applies to all graph classes. This lower bound covers "triangle-dependent" sampling algorithms, a subclass that includes our algorithm and all previous sampling algorithms for the problem. Finally, we show how to generalize our algorithm to count arbitrary subgraphs of constant size.

cs.DS↗

arXiv2019

Binary Embedding: Fundamental Limits and Fast Algorithm

Binary embedding is a nonlinear dimension reduction methodology where high dimensional data are embedded into the Hamming cube while preserving the structure of the original space. Specifically, for an arbitrary $N$ distinct points in $\mathbb{S}^{p-1}$, our goal is to encode each point using $m$-dimensional binary strings such that we can reconstruct their geodesic distance up to $δ$ uniform distortion. Existing binary embedding algorithms either lack theoretical guarantees or suffer from running time $O\big(mp\big)$. We make three contributions: (1) we establish a lower bound that shows any binary embedding oblivious to the set of points requires $m = Ω(\frac{1}{δ^2}\log{N})$ bits and a similar lower bound for non-oblivious embeddings into Hamming distance; (2) [DELETED, see comment]; (3) we also provide an analytic result about embedding a general set of points $K \subseteq \mathbb{S}^{p-1}$ with even infinite size. Our theoretical findings are supported through experiments on both synthetic and real data sets.

cs.DS↗

arXiv2019

Optimal Identity Testing with High Probability

We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution $p$ over $n$ elements, an explicitly given distribution $q$, and parameters $0< ε, δ< 1$, we wish to distinguish, {\em with probability at least $1-δ$}, whether the distributions are identical versus $\varepsilon$-far in total variation distance. Most prior work focused on the case that $δ= Ω(1)$, for which the sample complexity of identity testing is known to be $Θ(\sqrt{n}/ε^2)$. Given such an algorithm, one can achieve arbitrarily small values of $δ$ via black-box amplification, which multiplies the required number of samples by $Θ(\log(1/δ))$. We show that black-box amplification is suboptimal for any $δ= o(1)$, and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is \[ Θ\left( \frac{1}{ε^2}\left(\sqrt{n \log(1/δ)} + \log(1/δ) \right)\right) \] for any $n, \varepsilon$, and $δ$. For the special case of uniformity testing, where the given distribution is the uniform distribution $U_n$ over the domain, our new tester is surprisingly simple: to test whether $p = U_n$ versus $d_{\mathrm TV}(p, U_n) \geq \varepsilon$, we simply threshold $d_{\mathrm TV}(\widehat{p}, U_n)$, where $\widehat{p}$ is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant $δ$ case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of $\varepsilon$ and $δ$.

cs.DS↗

arXiv2019

Compressed Sensing with Adversarial Sparse Noise via L1 Regression

We present a simple and effective algorithm for the problem of \emph{sparse robust linear regression}. In this problem, one would like to estimate a sparse vector $w^* \in \mathbb{R}^n$ from linear measurements corrupted by sparse noise that can arbitrarily change an adversarially chosen $η$ fraction of measured responses $y$, as well as introduce bounded norm noise to the responses. For Gaussian measurements, we show that a simple algorithm based on L1 regression can successfully estimate $w^*$ for any $η< η_0 \approx 0.239$, and that this threshold is tight for the algorithm. The number of measurements required by the algorithm is $O(k \log \frac{n}{k})$ for $k$-sparse estimation, which is within constant factors of the number needed without any sparse noise. Of the three properties we show---the ability to estimate sparse, as well as dense, $w^*$; the tolerance of a large constant fraction of outliers; and tolerance of adversarial rather than distributional (e.g., Gaussian) dense noise---to the best of our knowledge, no previous result achieved more than two.

cs.DS↗

arXiv2018

Adversarial Examples from Cryptographic Pseudo-Random Generators

In our recent work (Bubeck, Price, Razenshteyn, arXiv:1805.10204) we argued that adversarial examples in machine learning might be due to an inherent computational hardness of the problem. More precisely, we constructed a binary classification task for which (i) a robust classifier exists; yet no non-trivial accuracy can be obtained with an efficient algorithm in (ii) the statistical query model. In the present paper we significantly strengthen both (i) and (ii): we now construct a task which admits (i') a maximally robust classifier (that is it can tolerate perturbations of size comparable to the size of the examples themselves); and moreover we prove computational hardness of learning this task under (ii') a standard cryptographic assumption.

cs.LG↗

arXiv2018

The Sketching Complexity of Graph and Hypergraph Counting

Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e.g., estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems. In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph $H$ in a bounded degree graph $G$ presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph $H$. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of $H$. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs $H$, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity.

cs.DS↗

arXiv2018

Batch Sparse Recovery, or How to Leverage the Average Sparsity

We introduce a \emph{batch} version of sparse recovery, where the goal is to report a sequence of vectors $A_1',\ldots,A_m' \in \mathbb{R}^n$ that estimate unknown signals $A_1,\ldots,A_m \in \mathbb{R}^n$ using a few linear measurements, each involving exactly one signal vector, under an assumption of \emph{average sparsity}. More precisely, we want to have \newline $(1) \;\;\; \sum_{j \in [m]}{\|A_j- A_j'\|_p^p} \le C \cdot \min \Big\{ \sum_{j \in [m]}{\|A_j - A_j^*\|_p^p} \Big\}$ for predetermined constants $C \ge 1$ and $p$, where the minimum is over all $A_1^*,\ldots,A_m^*\in\mathbb{R}^n$ that are $k$-sparse on average. We assume $k$ is given as input, and ask for the minimal number of measurements required to satisfy $(1)$. The special case $m=1$ is known as stable sparse recovery and has been studied extensively. We resolve the question for $p =1$ up to polylogarithmic factors, by presenting a randomized adaptive scheme that performs $\tilde{O}(km)$ measurements and with high probability has output satisfying $(1)$, for arbitrarily small $C > 1$. Finally, we show that adaptivity is necessary for every non-trivial scheme.

cs.DS↗

arXiv2018

Adversarial examples from computational constraints

Why are classifiers in high dimension vulnerable to "adversarial" perturbations? We show that it is likely not due to information theoretic limitations, but rather it could be due to computational constraints. First we prove that, for a broad set of classification tasks, the mere existence of a robust classifier implies that it can be found by a possibly exponential-time algorithm with relatively few training examples. Then we give a particular classification task where learning a robust classifier is computationally intractable. More precisely we construct a binary classification task in high dimensional space which is (i) information theoretically easy to learn robustly for large perturbations, (ii) efficiently learnable (non-robustly) by a simple linear separator, (iii) yet is not efficiently robustly learnable, even for small perturbations, by any algorithm in the statistical query (SQ) model. This example gives an exponential separation between classical learning and robust learning in the statistical query model. It suggests that adversarial examples may be an unavoidable byproduct of computational limitations of learning algorithms.

stat.ML↗

arXiv2018

Stochastic Multi-armed Bandits in Constant Space

We consider the stochastic bandit problem in the sublinear space setting, where one cannot record the win-loss record for all $K$ arms. We give an algorithm using $O(1)$ words of space with regret \[ \sum_{i=1}^{K}\frac{1}{Δ_i}\log \frac{Δ_i}Δ\log T \] where $Δ_i$ is the gap between the best arm and arm $i$ and $Δ$ is the gap between the best and the second-best arms. If the rewards are bounded away from $0$ and $1$, this is within an $O(\log 1/Δ)$ factor of the optimum regret possible without space constraints.

cs.DS↗