The coded trace reconstruction problem asks to construct a code $C\subset \{0,1\}^n$ such that any $x\in C$ is recoverable from independent outputs ("traces") of $x$ from a binary deletion channel (BDC). We present binary codes of rate $1-\varepsilon$ that are efficiently recoverable from ${\exp(O_q(\log^{1/3}(\frac{1}{\varepsilon})))}$ (a constant independent of $n$) traces of a $\operatorname{BDC}_q$ for any constant deletion probability $q\in(0,1)$. We also show that, for rate $1-\varepsilon$ binary codes, $\tilde Ω(\log^{5/2}(1/\varepsilon))$ traces are required. The results follow from a pair of black-box reductions that show that average-case trace reconstruction is essentially equivalent to coded trace reconstruction. We also show that there exist codes of rate $1-\varepsilon$ over an $O_{\varepsilon}(1)$-sized alphabet that are recoverable from $O(\log(1/\varepsilon))$ traces, and that this is tight.
The problem of reconstructing strings from their substring spectra has a long history and in its most simple incarnation asks for determining under which conditions the spectrum uniquely determines the string. We study the problem of coded string reconstruction from multiset substring spectra, where the strings are restricted to lie in some codebook. In particular, we consider binary codebooks that allow for unique string reconstruction and propose a new method, termed repeat replacement, to create the codebook. Our contributions include algorithmic solutions for repeat replacement and constru…
In this paper, we investigate the problem of recovering source information from an incomplete set of network coded data. We first study the theoretical performance of such systems under maximum a posteriori (MAP) decoding and derive the upper bound on the probability of decoding error as a function of the system parameters. We also establish the sufficient conditions on the number of network coded symbols required to achieve decoding error probability below a certain level. We then propose a low complexity iterative decoding algorithm based on message passing for decoding the network coded dat…
Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of \emph{coded trace reconstruction}, the design and analysis of high-rate efficiently encodable codes that can be efficiently decoded with high probability from few reads (also called \emph{traces}) corrupted by edit errors. Codes used in current portable DNA-based storage systems with nanopore sequencers are largely based on heuristics, and have no provable robustness or performance guarantees even for an error model with i.i.d.\ deletions and constant deletion probability.…
The University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）；University of California, Merced（加利福尼亚大学默塞德分校）
This paper describes a coded aperture and keyed exposure approach to compressive video measurement which admits a small physical platform, high photon efficiency, high temporal resolution, and fast reconstruction algorithms. The proposed projections satisfy the Restricted Isometry Property (RIP), and hence compressed sensing theory provides theoretical guarantees on the video reconstruction quality. Moreover, the projections can be easily implemented using existing optical elements such as spatial light modulators (SLMs). We extend these coded mask designs to novel dual-scale masks (DSMs) whic…
We consider a compressive hyperspectral imaging reconstruction problem, where three-dimensional spatio-spectral information about a scene is sensed by a coded aperture snapshot spectral imager (CASSI). The approximate message passing (AMP) framework is utilized to reconstruct hyperspectral images from CASSI measurements, and an adaptive Wiener filter is employed as a three-dimensional image denoiser within AMP. We call our algorithm "AMP-3D-Wiener." The simulation results show that AMP-3D-Wiener outperforms existing widely-used algorithms such as gradient projection for sparse reconstruction (…
Duke University（杜克大学）；Fitzpatrick Institute for Photonics（菲茨帕特里克光子学研究所）
We use mechanical translation of a coded aperture for code division multiple access compression of video. We present experimental results for reconstruction at 148 frames per coded snapshot.
The problem of string reconstruction from substring information has found many applications due to its relevance in DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry readouts. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide matching upper and lower bounds on the asymptotic rate of the underlying codebooks. Under certain mild constraints …
Most DNA sequencing technologies are based on the shotgun paradigm: many short reads are obtained from random unknown locations in the DNA sequence. A fundamental question, studied in arXiv:1203.6233, is what read length and coverage depth (i.e., the total number of reads) are needed to guarantee reliable sequence reconstruction. Motivated by DNA-based storage, we study the coded version of this problem;i.e., the scenario where the DNA molecule being sequenced is a codeword from a predefined codebook. Our main result is an exact characterization of the capacity of the resulting shotgun sequenc…
Motivated by DNA-based storage applications, we study the problem of reconstructing a coded sequence from multiple traces. We consider the model where the traces are outputs of independent deletion channels, where each channel deletes each bit of the input codeword \(\mathbf{x} \in \{0,1\}^n\) independently with probability \(p\). We focus on the regime where the deletion probability \(p \to 0\) when \(n\to \infty\). Our main contribution is designing a novel code for trace reconstruction that allows reconstructing a coded sequence efficiently from a constant number of traces. We provide theor…
We study coded distributed matrix multiplication from an approximate recovery viewpoint. We consider a system of $P$ computation nodes where each node stores $1/m$ of each multiplicand via linear encoding. Our main result shows that the matrix product can be recovered with $ε$ relative error from any $m$ of the $P$ nodes for any $ε> 0$. We obtain this result through a careful specialization of MatDot codes -- a class of matrix multiplication codes previously developed in the context of exact recovery ($ε=0$). Since prior results showed that MatDot codes achieve the best exact recovery threshol…
University of Washington（华盛顿大学）；Princeton University（普林斯顿大学）；University of California, San Diego（加州大学圣迭戈分校）
In the usual trace reconstruction problem, the goal is to exactly reconstruct an unknown string of length $n$ after it passes through a deletion channel many times independently, producing a set of traces (i.e., random subsequences of the string). We consider the relaxed problem of approximate reconstruction. Here, the goal is to output a string that is close to the original one in edit distance while using much fewer traces than is needed for exact reconstruction. We present several algorithms that can approximately reconstruct strings that belong to certain classes, where the estimate is wit…
Advances of information-theoretic understanding of sparse sampling of continuous uncoded signals at sampling rates exceeding the Landau rate were reported in recent works. This work examines sparse sampling of coded signals at sub-Landau sampling rates. It is shown that with coded signals the Landau condition may be relaxed and the sampling rate required for signal reconstruction and for support detection can be lower than the effective bandwidth. Equivalently, the number of measurements in the corresponding sparse sensing problem can be smaller than the support size. Tight bounds on informati…
The trace reconstruction problem studies the number of noisy samples needed to recover an unknown string $\boldsymbol{x}\in\{0,1\}^n$ with high probability, where the samples are independently obtained by passing $\boldsymbol{x}$ through a random deletion channel with deletion probability $q$. The problem is receiving significant attention recently due to its applications in DNA sequencing and DNA storage. Yet, there is still an exponential gap between upper and lower bounds for the trace reconstruction problem. In this paper we study the trace reconstruction problem when $\boldsymbol{x}$ is c…
University of California San Diego（加利福尼亚大学圣地亚哥分校）；Qualcomm Institute（高通研究所）
The problem of reconstructing a string from its error-prone copies, the trace reconstruction problem, was introduced by Vladimir Levenshtein two decades ago. While there has been considerable theoretical work on trace reconstruction, practical solutions have only recently started to emerge in the context of two rapidly developing research areas: immunogenomics and DNA data storage. In immunogenomics, traces correspond to mutated copies of genes, with mutations generated naturally by the adaptive immune system. In DNA data storage, traces correspond to noisy copies of DNA molecules that encode …
In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string $x$ given random "traces" of $x$ where each trace is generated by deleting each coordinate of $x$ independently with probability $p<1$. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and gene…
Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that $\exp(O(n^{1/3}))$ traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon. We introduce oblivious synchroniz…
The deletion channel takes as input a bit string $\mathbf{x} \in \{0,1\}^n$, and deletes each bit independently with probability $q$, yielding a shorter string. The trace reconstruction problem is to recover an unknown string $\mathbf{x}$ from many independent outputs (called "traces") of the deletion channel applied to $\mathbf{x}$. We show that if $\mathbf{x}$ is drawn uniformly at random and $q < 1/2$, then $e^{O(\log^{1/2} n)}$ traces suffice to reconstruct $\mathbf{x}$ with high probability. The previous best bound, established in 2008 by Holenstein-Mitzenmacher-Panigrahy-Wieder, uses $n^…
We consider a large-scale matrix multiplication problem where the computation is carried out using a distributed system with a master node and multiple worker nodes, where each worker can store parts of the input matrices. We propose a computation strategy that leverages ideas from coding theory to design intermediate computations at the worker nodes, in order to efficiently deal with straggling workers. The proposed strategy, named as \emph{polynomial codes}, achieves the optimum recovery threshold, defined as the minimum number of workers that the master needs to wait for in order to compute…
In the trace reconstruction problem, an unknown bit string ${\bf x}\in\{0,1 \}^n$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string $\widetilde{\bf x}$. How many i.i.d.\ samples of $\widetilde{\bf x}$ are needed to reconstruct $\bf x$ with high probability? We prove that there exist ${\bf x},{\bf y} \in\{0,1 \}^n$ such that at least $c\, n^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\bf x}$ and ${\bf y}$ for some absolute constant $c$, improving the previous lower bound of $c\,n$. Fur…
The insertion-deletion channel takes as input a bit string ${\bf x}\in\{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called "traces") of the insertion-deletion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of …
The {\em insertion-deletion channel} takes as input a binary string $x \in\{0, 1\}^n$, and outputs a string $\widetilde{x}$ where some of the bits have been deleted and others inserted independently at random. In the {\em trace reconstruction problem}, one is given many outputs (called {\em traces}) of the insertion-deletion channel on the same input message $x$, and asked to recover the input message. Nazarov and Peres (STOC 2017), and De, O'Donnell and Servedio (STOC 2017) showed that any string $x$ can be reconstructed from $\exp(O(n^{1/3}))$ traces. Holden, Pemantle, Peres and Zhai (COLT 2…
In the trace reconstruction problem, one seeks to reconstruct a binary string $s$ from a collection of traces, each of which is obtained by passing $s$ through a deletion channel. It is known that $\exp(\tilde O(n^{1/5}))$ traces suffice to reconstruct any length-$n$ string with high probability. We consider a variant of the trace reconstruction problem where the goal is to recover a "density map" that indicates the locations of each length-$k$ substring throughout $s$. We show that $ε^{-2}\cdot \text{poly}(n)$ traces suffice to recover the density map with error at most $ε$. As a result, when…
In the trace reconstruction problem, an unknown bit string $x \in \{0,1\}^n$ is observed through the deletion channel, which deletes each bit of $x$ with some constant probability $q$, yielding a contracted string $\widetilde{x}$. How many independent copies of $\widetilde{x}$ are needed to reconstruct $x$ with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was $\exp(\widetilde{O}(n^{1/2}))$. We improve this bound to $\exp(O(n^{1/3}))$ using statistics of individual bits in the output and show that this bound is sharp …
In the trace reconstruction problem an unknown string ${\bf x}=(x_0,\dots,x_{n-1})\in\{0,1,...,m-1\}^n$ is observed through the deletion channel, which deletes each $x_k$ with a certain probability, yielding a contracted string $\widetilde{\bf X}$. Earlier works have proved that if each $x_k$ is deleted with the same probability $q\in[0,1)$, then $\exp(O(n^{1/3}))$ independent copies of the contracted string $\widetilde{\bf X}$ suffice to reconstruct $\bf x$ with high probability. We extend this upper bound to the setting where the deletion probabilities vary, assuming certain regularity condi…
This work studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and non-binary sequences from synchronization (insertion/deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We i…
A new video coding method based on compressive sampling is proposed. In this method, a video is coded using compressive measurements on video cubes. Video reconstruction is performed by minimization of total variation (TV) of the pixelwise DCT coefficients along the temporal direction. A new reconstruction algorithm is developed from TVAL3, an efficient TV minimization algorithm based on the alternating minimization and augmented Lagrangian methods. Video coding with this method is inherently scalable, and has applications in mobile broadcast.
