Mary Wootters

Recovering simple signals

The primary goal of compressed sensing and (non-adaptive) combinatorial group testing is to recover a sparse vector x from an underdetermined set of linear equations Φx = y. Both problems entail solving Φx = y given Φ and y but they use different models of arithmetic, different models of randomness models for F, and different guarantees upon the solution x and the class of signals from which x is drawn. In [1], Lipton introduced a model for error correction where the channel is computationally bounded, subject to standard cryptographic assumptions, and produces the error vector x that must be found and then corrected. This has been extended in [2], [3] to create more efficient schemes against polynomial and logspace bounded channels. Inspired by these results in error correction, we view compressed sensing and combinatorial group testing as an adversarial process, where Mallory the adversary produces the vector x to be measured, with limited information about the matrix Φ. We define a number of computationally bounded models for Mallory and show that there are significant gains (in the minimum number of measurements) to be had by relaxing the model from adversarial to computationally or information-theoretically bounded, and not too much (in some cases, nothing at all) is lost by assuming these models over oblivious or statistical models. We also show that differences in adversarial power give rise to different lower bounds for the number of measurements required to defeat such an adversary. By contrast we show that randomized one pass log space streaming Mallory is almost as powerful as a fully adversarial one for group testing while for compressed sensing such an adversary is as weak as an oblivious one.

On the list decodability of random linear codes with large error rates

It is well known that a random q-ary code of rate Ω(ε2) is list decodable up to radius (1 - 1/q - ε) with list sizes on the order of 1/ε2, with probability 1 - o(1). However, until recently, a similar statement about random linear codes has until remained elusive. In a recent paper, Cheraghchi, Guruswami, and Velingker show a connection between list decodability of random linear codes and the Restricted Isometry Property from compressed sensing, and use this connection to prove that a random linear code of rate Ω( ε2 /log3(1/ε)) achieves the list decoding properties above, with constant probability. We improve on their result to show that in fact we may take the rate to be Ω(ε2), which is optimal, and further that the success probability is 1 - o(1), rather than constant. As an added benefit, our proof is relatively simple. Finally, we extend our methods to more general ensembles of linear codes. As an example, we show that randomly punctured Reed-Muller codes have the same list decoding properties as the original codes, even when the rate is improved to a constant.

Repairing multiple failures for scalar MDS codes

In distributed storage, erasure codes (like Reed–Solomon Codes) are often employed to provide reliability. In this setting, it is desirable to be able to repair one or more failed nodes while minimizing the repair bandwidth. In this paper, motivated by Reed-Solomon codes, we study the problem of repairing multiple failed nodes in a scalar MDS code. We extend the framework of (Guruswami and Wootters, 2017) to give a framework for constructing repair schemes for multiple failures in general scalar MDS codes in the centralized repair model. We then specialize our framework to Reed–Solomon codes, and also extend and improve upon recent results of (Dau et al., 2017).

Accurate Decoding of Pooled Sequenced Data Using Compressed Sensing

In order to overcome the limitations imposed by DNA barcoding when multiplexing a large number of samples in the current generation of high-throughput sequencing instruments, we have recently proposed a new protocol that leverages advances in combinatorial pooling design (group testing) [9]. We have also demonstrated how this new protocol would enable de novo selective sequencing and assembly of large, highly-repetitive genomes. Here we address the problem of decoding pooled sequenced data obtained from such a protocol. Our algorithm employs a synergistic combination of ideas from compressed sensing and the decoding of error-correcting codes. Experimental results on synthetic data for the rice genome and real data for the barley genome show that our novel decoding algorithm enables significantly higher quality assemblies than the previous approach.

Linear-Time List Recovery of High-Rate Expander Codes

We show that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms. List recoverable codes have been useful recently in constructing efficiently list-decodable codes, as well as explicit constructions of matrices for compressive sensing and group testing. Previous list recoverable codes with linear-time decoding algorithms have all had rate at most \(1/2\); in contrast, our codes can have rate \(1 - \varepsilon \) for any \(\varepsilon > 0\). We can plug our high-rate codes into a framework of Alon and Luby (1996) and Meir (2014) to obtain linear-time list recoverable codes of arbitrary rates \(R\), which approach the optimal trade-off between the number of non-trivial lists provided and the rate of the code.

Public key locally decodable codes with short keys

This work considers locally decodable codes in the computationally bounded channel model. The computationally bounded channel model, introduced by Lipton in 1994, views the channel as an adversary which is restricted to polynomial-time computation. Assuming the existence of IND-CPA secure public-key encryption, we present a construction of public-key locally decodable codes, with constant codeword expansion, tolerating constant error rate, with locality O(λ), and negligible probability of decoding failure, for security parameter λ. Hemenway and Ostrovsky gave a construction of locally decodable codes in the public-key model with constant codeword expansion and locality O(λ2), but their construction had two major drawbacks. The keys in their scheme were proportional to n, the length of the message, and their schemes were based on the F-hiding assumption. Our keys are of length proportional to the security parameter instead of the message, and our construction relies only on the existence of IND-CPA secure encryption rather than on specific number-theoretic assumptions. Our scheme also decreases the locality from O(λ2) to O(λ). Our construction can be modified to give a generic transformation of any private-key locally decodable code to a public-key locally decodable code based only on the existence of an IND-CPA secure public-key encryption scheme.

Local List Recovery of High-Rate Tensor Codes & Applications

In this work, we give the first construction of high-rate locally list-recoverable codes. List-recovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving} globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be uniquely decoded in near-linear-time, with higher rate than was previously known.Our techniques are actually quite simple, and are inspired by an approach of Gopalan, Guruswami, and Raghavendra (Siam Journal on Computing, 2011) for list-decoding tensor codes. We show that tensor powers of (globally) list-recoverable codes are approximately locally list-recoverable, and that the approximately modifier may be removed by pre-encoding the message with a suitable locally decodable code. Instantiating this with known constructions of high-rate globally list-recoverable codes and high-rate locally decodable codes finishes the construction.

Optimal entanglement-assisted one-shot classical communication

The one-shot success probability of a noisy classical channel for transmitting one classical bit is the optimal probability with which the bit can be successfully sent via a single use of the channel. Prevedel et al. [Phys. Rev. Lett. 106, 110505 (2011)] recently showed that for a specific channel, this quantity can be increased if the parties using the channel share an entangled quantum state. In this paper, we characterize the optimal entanglement-assisted protocols in terms of the radius of a set of operators associated with the channel. This characterization can be used to construct optimal entanglement-assisted protocols for a given classical channel and to prove the limits of such protocols. As an example, we show that the Prevedel et al. protocol is optimal for two-qubit entanglement. We also prove some tight upper bounds on the improvement that can be obtained from quantum and nonsignaling correlations.

Reusable low-error compressive sampling schemes through privacy

A compressive sampling algorithm recovers approximately a nearly sparse vector x from a much smaller “sketch” given by the matrix vector product Φx. Different settings in the literature make different assumptions to meet strong requirements on the accuracy of the recovered signal. Some are robust to noise (that is, the signal may be far from sparse), but the matrix Φ is only guaranteed to work on a single fixed x with high probability-it may not be re-used arbitrarily many times. Others require Φ to work on all x simultaneously, but are much less resilient to noise. In this note, we examine the case of compressive sampling of a RADAR signal. Through a combination of mathematical theory and assumptions appropriate to our scenario, we show how a single matrix Φ can be used repeatedly on multiple input vectors x, and still give the best possible resilience to noise.

Linear-time list recovery of high-rate expander codes

We show that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms. List recoverable codes have been useful recently in constructing efficiently list-decodable codes, as well as explicit constructions of matrices for compressive sensing and group testing. Previous list recoverable codes with linear-time decoding algorithms have all had rate at most 1/2; in contrast, our codes can have rate