arXiv subjects

Eric Price

Publications and source records attributed to Eric Price.

At least 73 records · Page 4Linked to original sources

arXiv2018

Deep Neural Network-based Cooperative Visual Tracking through Multiple Micro Aerial Vehicles

Multi-camera full-body pose capture of humans and animals in outdoor environments is a highly challenging problem. Our approach to it involves a team of cooperating micro aerial vehicles (MAVs) with on-board cameras only. The key enabling-aspect of our approach is the on-board person detection and tracking method. Recent state-of-the-art methods based on deep neural networks (DNN) are highly promising in this context. However, real time DNNs are severely constrained in input data dimensions, in contrast to available camera resolutions. Therefore, DNNs often fail at objects with small scale or far away from the camera, which are typical characteristics of a scenario with aerial robots. Thus, the core problem addressed in this paper is how to achieve on-board, real-time, continuous and accurate vision-based detections using DNNs for visual person tracking through MAVs. Our solution leverages cooperation among multiple MAVs. First, each MAV fuses its own detections with those obtained by other MAVs to perform cooperative visual tracking. This allows for predicting future poses of the tracked person, which are used to selectively process only the relevant regions of future images, even at high resolutions. Consequently, using our DNN-based detector we are able to continuously track even distant humans with high accuracy and speed. We demonstrate the efficiency of our approach through real robot experiments involving two aerial robots tracking a person, while maintaining an active perception-driven formation. Our solution runs fully on-board our MAV's CPU and GPU, with no remote processing. ROS-based source code is provided for the benefit of the community.

cs.RO↗

arXiv2017

Robust polynomial regression up to the information theoretic limit

We consider the problem of robust polynomial regression, where one receives samples $(x_i, y_i)$ that are usually within $σ$ of a polynomial $y = p(x)$, but have a $ρ$ chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate $p$ only when $ρ< \frac{1}{\log d}$. We give an algorithm that works for the entire feasible range of $ρ< 1/2$, while simultaneously improving other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a $1.09$ approximation is impossible even with infinitely many samples.

cs.DS↗

arXiv2017

Fast Regression with an $\ell_\infty$ Guarantee

Sketching has emerged as a powerful technique for speeding up problems in numerical linear algebra, such as regression. In the overconstrained regression problem, one is given an $n \times d$ matrix $A$, with $n \gg d$, as well as an $n \times 1$ vector $b$, and one wants to find a vector $\hat{x}$ so as to minimize the residual error $\|Ax-b\|_2$. Using the sketch and solve paradigm, one first computes $S \cdot A$ and $S \cdot b$ for a randomly chosen matrix $S$, then outputs $x' = (SA)^{\dagger} Sb$ so as to minimize $\|SAx' - Sb\|_2$. The sketch-and-solve paradigm gives a bound on $\|x'-x^*\|_2$ when $A$ is well-conditioned. Our main result is that, when $S$ is the subsampled randomized Fourier/Hadamard transform, the error $x' - x^*$ behaves as if it lies in a "random" direction within this bound: for any fixed direction $a\in \mathbb{R}^d$, we have with $1 - d^{-c}$ probability that \[ \langle a, x'-x^*\rangle \lesssim \frac{\|a\|_2\|x'-x^*\|_2}{d^{\frac{1}{2}-γ}}, \quad (1) \] where $c, γ> 0$ are arbitrary constants. This implies $\|x'-x^*\|_{\infty}$ is a factor $d^{\frac{1}{2}-γ}$ smaller than $\|x'-x^*\|_2$. It also gives a better bound on the generalization of $x'$ to new examples: if rows of $A$ correspond to examples and columns to features, then our result gives a better bound for the error introduced by sketch-and-solve when classifying fresh examples. We show that not all oblivious subspace embeddings $S$ satisfy these properties. In particular, we give counterexamples showing that matrices based on Count-Sketch or leverage score sampling do not satisfy these properties. We also provide lower bounds, both on how small $\|x'-x^*\|_2$ can be, and for our new guarantee (1), showing that the subsampled randomized Fourier/Hadamard transform is nearly optimal.

cs.DS↗

arXiv2017

Compressed Sensing using Generative Models

The goal of compressed sensing is to estimate a vector from an underdetermined system of noisy linear measurements, by making use of prior knowledge on the structure of vectors in the relevant domain. For almost all results in this literature, the structure is represented by sparsity in a well-chosen basis. We show how to achieve guarantees similar to standard compressed sensing but without employing sparsity at all. Instead, we suppose that vectors lie near the range of a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. Our main theorem is that, if $G$ is $L$-Lipschitz, then roughly $O(k \log L)$ random Gaussian measurements suffice for an $\ell_2/\ell_2$ recovery guarantee. We demonstrate our results using generative models from published variational autoencoder and generative adversarial networks. Our method can use $5$-$10$x fewer measurements than Lasso for the same accuracy.

stat.ML↗

arXiv2016

Collision-based Testers are Optimal for Uniformity and Closeness

We study the fundamental problems of (i) uniformity testing of a discrete distribution, and (ii) closeness testing between two discrete distributions with bounded $\ell_2$-norm. These problems have been extensively studied in distribution testing and sample-optimal estimators are known for them~\cite{Paninski:08, CDVV14, VV14, DKN:15}. In this work, we show that the original collision-based testers proposed for these problems ~\cite{GRdist:00, BFR+:00} are sample-optimal, up to constant factors. Previous analyses showed sample complexity upper bounds for these testers that are optimal as a function of the domain size $n$, but suboptimal by polynomial factors in the error parameter $ε$. Our main contribution is a new tight analysis establishing that these collision-based testers are information-theoretically optimal, up to constant factors, both in the dependence on $n$ and in the dependence on $ε$.

cs.DS↗

arXiv2016

Extensions and Limitations of the Neural GPU

The Neural GPU is a recent model that can learn algorithms such as multi-digit binary addition and binary multiplication in a way that generalizes to inputs of arbitrary length. We show that there are two simple ways of improving the performance of the Neural GPU: by carefully designing a curriculum, and by increasing model size. The latter requires a memory efficient implementation, as a naive implementation of the Neural GPU is memory intensive. We find that these techniques increase the set of algorithmic problems that can be solved by the Neural GPU: we have been able to learn to perform all the arithmetic operations (and generalize to arbitrarily long numbers) when the arguments are given in the decimal representation (which, surprisingly, has not been possible before). We have also been able to train the Neural GPU to evaluate long arithmetic expressions with multiple operands that require respecting the precedence order of the operands, although these have succeeded only in their binary representation, and not with perfect accuracy. In addition, we gain insight into the Neural GPU by investigating its failure modes. We find that Neural GPUs that correctly generalize to arbitrarily long numbers still fail to compute the correct answer on highly-symmetric, atypical inputs: for example, a Neural GPU that achieves near-perfect generalization on decimal multiplication of up to 100-digit long numbers can fail on $000000\dots002 \times 000000\dots002$ while succeeding at $2 \times 2$. These failure modes are reminiscent of adversarial examples.

cs.NE↗

arXiv2016

Equality of Opportunity in Supervised Learning

We propose a criterion for discrimination against a specified sensitive attribute in supervised learning, where the goal is to predict some target based on available features. Assuming data about the predictor, target, and membership in the protected group are available, we show how to optimally adjust any learned predictor so as to remove discrimination according to our definition. Our framework also improves incentives by shifting the cost of poor classification from disadvantaged groups to the decision maker, who can respond by improving the classification accuracy. In line with other studies, our notion is oblivious: it depends only on the joint statistics of the predictor, the target and the protected attribute, but not on interpretation of individualfeatures. We study the inherent limits of defining and identifying biases based on such oblivious measures, outlining what can and cannot be inferred from different oblivious tests. We illustrate our notion using a case study of FICO credit scores.

cs.LG↗

arXiv2016

Fourier-sparse interpolation without a frequency gap

We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval $[0, T]$ and the frequencies can be "off-grid". Previous methods for this problem required the gap between frequencies to be above 1/T, the threshold required to robustly identify individual frequencies. We show the frequency gap is not necessary to estimate the signal as a whole: for arbitrary $k$-Fourier-sparse signals under $\ell_2$ bounded noise, we show how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in $k$ and logarithmic in the bandwidth and signal-to-noise ratio. As a special case, we get an algorithm to interpolate degree $d$ polynomials from noisy measurements, using $O(d)$ samples and increasing the noise by a constant factor in $\ell_2$.

cs.DS↗

arXiv2016

A Robust Sparse Fourier Transform in the Continuous Setting

In recent years, a number of works have studied methods for computing the Fourier transform in sublinear time if the output is sparse. Most of these have focused on the discrete setting, even though in many applications the input signal is continuous and naive discretization significantly worsens the sparsity level. We present an algorithm for robustly computing sparse Fourier transforms in the continuous setting. Let $x(t) = x^*(t) + g(t)$, where $x^*$ has a $k$-sparse Fourier transform and $g$ is an arbitrary noise term. Given sample access to $x(t)$ for some duration $T$, we show how to find a $k$-Fourier-sparse reconstruction $x'(t)$ with $$\frac{1}{T}\int_0^T |x'(t) - x(t) |^2 \mathrm{d} t \lesssim \frac{1}{T}\int_0^T | g(t)|^2 \mathrm{d}t.$$ The sample complexity is linear in $k$ and logarithmic in the signal-to-noise ratio and the frequency resolution. Previous results with similar sample complexities could not tolerate an infinitesimal amount of i.i.d. Gaussian noise, and even algorithms with higher sample complexities increased the noise by a polynomial factor. We also give new results for how precisely the individual frequencies of $x^*$ can be recovered.

cs.DS↗

arXiv2015

Tight bounds for learning a mixture of two gaussians

We consider the problem of identifying the parameters of an unknown mixture of two arbitrary $d$-dimensional gaussians from a sequence of independent random samples. Our main results are upper and lower bounds giving a computationally efficient moment-based estimator with an optimal convergence rate, thus resolving a problem introduced by Pearson (1894). Denoting by $σ^2$ the variance of the unknown mixture, we prove that $Θ(σ^{12})$ samples are necessary and sufficient to estimate each parameter up to constant additive error when $d=1.$ Our upper bound extends to arbitrary dimension $d>1$ up to a (provably necessary) logarithmic loss in $d$ using a novel---yet simple---dimensionality reduction technique. We further identify several interesting special cases where the sample complexity is notably smaller than our optimal worst-case bound. For instance, if the means of the two components are separated by $Ω(σ)$ the sample complexity reduces to $O(σ^2)$ and this is again optimal. Our results also apply to learning each component of the mixture up to small error in total variation distance, where our algorithm gives strong improvements in sample complexity over previous work. We also extend our lower bound to mixtures of $k$ Gaussians, showing that $Ω(σ^{6k-2})$ samples are necessary to estimate each parameter up to constant additive error.

cs.LG↗

arXiv2015

Nearly-optimal bounds for sparse recovery in generic norms, with applications to $k$-median sketching

We initiate the study of trade-offs between sparsity and the number of measurements in sparse recovery schemes for generic norms. Specifically, for a norm $\|\cdot\|$, sparsity parameter $k$, approximation factor $K>0$, and probability of failure $P>0$, we ask: what is the minimal value of $m$ so that there is a distribution over $m \times n$ matrices $A$ with the property that for any $x$, given $Ax$, we can recover a $k$-sparse approximation to $x$ in the given norm with probability at least $1-P$? We give a partial answer to this problem, by showing that for norms that admit efficient linear sketches, the optimal number of measurements $m$ is closely related to the doubling dimension of the metric induced by the norm $\|\cdot\|$ on the set of all $k$-sparse vectors. By applying our result to specific norms, we cast known measurement bounds in our general framework (for the $\ell_p$ norms, $p \in [1,2]$) as well as provide new, measurement-efficient schemes (for the Earth-Mover Distance norm). The latter result directly implies more succinct linear sketches for the well-studied planar $k$-median clustering problem. Finally, our lower bound for the doubling dimension of the EMD norm enables us to address the open question of [Frahling-Sohler, STOC'05] about the space complexity of clustering problems in the dynamic streaming model.

cs.DS↗

arXiv2015

The Noisy Power Method: A Meta Algorithm with Applications

We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications including streaming PCA and privacy-preserving singular vector computation.

cs.DS↗

arXiv2014

Optimal Lower Bound for Itemset Frequency Indicator Sketches

Given a database, a common problem is to find the pairs or $k$-tuples of items that frequently co-occur. One specific problem is to create a small space "sketch" of the data that records which $k$-tuples appear in more than an $ε$ fraction of rows of the database. We improve the lower bound of Liberty, Mitzenmacher, and Thaler [LMT14], showing that $Ω(\frac{1}εd \log (εd))$ bits are necessary even in the case of $k=2$. This matches the sampling upper bound for all $ε\geq 1/d^{.99}$, and (in the case of $k=2$) another trivial upper bound for $ε= 1/d$.

cs.DS↗

arXiv2013

Improved Concentration Bounds for Count-Sketch

We present a refined analysis of the classic Count-Sketch streaming heavy hitters algorithm [CCF02]. Count-Sketch uses O(k log n) linear measurements of a vector x in R^n to give an estimate x' of x. The standard analysis shows that this estimate x' satisfies ||x'-x||_infty^2 < ||x_tail||_2^2 / k, where x_tail is the vector containing all but the largest k coordinates of x. Our main result is that most of the coordinates of x' have substantially less error than this upper bound; namely, for any c < O(log n), we show that each coordinate i satisfies (x'_i - x_i)^2 < (c/log n) ||x_tail||_2^2/k with probability 1-2^{-Omega(c)}, as long as the hash functions are fully independent. This subsumes the previous bound and is optimal for all c. Using these improved point estimates, we prove a stronger concentration result for set estimates by first analyzing the covariance matrix and then using a median-of-median-of-medians argument to bootstrap the failure probability bounds. These results also give improved results for l_2 recovery of exactly k-sparse estimates x^* when x is drawn from a distribution with suitable decay, such as a power law or lognormal. We complement our results with simulations of Count-Sketch on a power law distribution. The empirical evidence indicates that our theoretical bounds give a precise characterization of the algorithm's performance: the asymptotics are correct and the associated constants are small. Our proof shows that any symmetric random variable with finite variance and positive Fourier transform concentrates around 0 at least as well as a Gaussian. This result, which may be of independent interest, gives good concentration even when the noise does not converge to a Gaussian.

cs.DS↗

arXiv2013

Sample-Optimal Average-Case Sparse Fourier Transform in Two Dimensions

We present the first sample-optimal sublinear time algorithms for the sparse Discrete Fourier Transform over a two-dimensional sqrt{n} x sqrt{n} grid. Our algorithms are analyzed for /average case/ signals. For signals whose spectrum is exactly sparse, our algorithms use O(k) samples and run in O(k log k) time, where k is the expected sparsity of the signal. For signals whose spectrum is approximately sparse, our algorithm uses O(k log n) samples and runs in O(k log^2 n) time; the latter algorithm works for k=Theta(sqrt{n}). The number of samples used by our algorithms matches the known lower bounds for the respective signal models. By a known reduction, our algorithms give similar results for the one-dimensional sparse Discrete Fourier Transform when n is a power of a small composite number (e.g., n = 6^t).

cs.DS↗

arXiv2012

New constructions of RIP matrices with fast multiplication and fewer rows

In compressed sensing, the "restricted isometry property" (RIP) is a sufficient condition for the efficient reconstruction of a nearly k-sparse vector x in C^d from m linear measurements Phi x. It is desirable for m to be small, and for Phi to support fast matrix-vector multiplication. In this work, we give a randomized construction of RIP matrices Phi in C^{m x d}, preserving the L_2 norms of all k-sparse vectors with distortion 1+eps, where the matrix-vector multiply Phi x can be computed in nearly linear time. The number of rows m is on the order of eps^{-2}klog dlog^2(klog d). Previous analyses of constructions of RIP matrices supporting fast matrix-vector multiplies, such as the sampled discrete Fourier matrix, required m to be larger by roughly a log k factor. Supporting fast matrix-vector multiplication is useful for iterative recovery algorithms which repeatedly multiply by Phi or Phi^*. Furthermore, our construction, together with a connection between RIP matrices and the Johnson-Lindenstrauss lemma in [Krahmer-Ward, SIAM. J. Math. Anal. 2011], implies fast Johnson-Lindenstrauss embeddings with asymptotically fewer rows than previously known. Our approach is a simple twist on previous constructions. Rather than choosing the rows for the embedding matrix to be rows sampled from some larger structured matrix (such as the discrete Fourier transform or a random circulant matrix), we instead choose each row of the embedding matrix to be a linear combination of a small number of rows of the original matrix, with random sign flips as coefficients. The main tool in our analysis is a recent bound for the supremum of certain types of Rademacher chaos processes in [Krahmer-Mendelson-Rauhut, arXiv:1207.0235].

cs.DS↗

arXiv2012

Lower Bounds for Adaptive Sparse Recovery

We give lower bounds for the problem of stable sparse recovery from /adaptive/ linear measurements. In this problem, one would like to estimate a vector $x \in \R^n$ from $m$ linear measurements $A_1x,..., A_mx$. One may choose each vector $A_i$ based on $A_1x,..., A_{i-1}x$, and must output $x*$ satisfying |x* - x|_p \leq (1 + ε) \min_{k\text{-sparse} x'} |x - x'|_p with probability at least $1-δ>2/3$, for some $p \in \{1,2\}$. For $p=2$, it was recently shown that this is possible with $m = O(\frac{1}εk \log \log (n/k))$, while nonadaptively it requires $Θ(\frac{1}εk \log (n/k))$. It is also known that even adaptively, it takes $m = Ω(k/ε)$ for $p = 2$. For $p = 1$, there is a non-adaptive upper bound of $\tilde{O}(\frac{1}{\sqrtε} k\log n)$. We show: * For $p=2$, $m = Ω(\log \log n)$. This is tight for $k = O(1)$ and constant $ε$, and shows that the $\log \log n$ dependence is correct. * If the measurement vectors are chosen in $R$ "rounds", then $m = Ω(R \log^{1/R} n)$. For constant $ε$, this matches the previously known upper bound up to an O(1) factor in $R$. * For $p=1$, $m = Ω(k/(\sqrtε \cdot \log k/ε))$. This shows that adaptivity cannot improve more than logarithmic factors, providing the analog of the $m = Ω(k/ε)$ bound for $p = 2$.

cs.DS↗

arXiv2012

K-Median Clustering, Model-Based Compressive Sensing, and Sparse Recovery for Earth Mover Distance

We initiate the study of sparse recovery problems under the Earth-Mover Distance (EMD). Specifically, we design a distribution over m x n matrices A such that for any x, given Ax, we can recover a k-sparse approximation to x under the EMD distance. One construction yields m = O(k log(n/k)) and a 1 + epsilon approximation factor, which matches the best achievable bound for other error measures, such as the L_1 norm. Our algorithms are obtained by exploiting novel connections to other problems and areas, such as streaming algorithms for k-median clustering and model-based compressive sensing. We also provide novel algorithms and results for the latter problems.

cs.DS↗