Reinhard Heckel

Robust Chemical Preservation of Digital Information on DNA in Silica with Error‐Correcting Codes

Information, such as text printed on paper or images projected onto microfilm, can survive for over 500 years. However, the storage of digital information for time frames exceeding 50 years is challenging. Here we show that digital information can be stored on DNA and recovered without errors for considerably longer time frames. To allow for the perfect recovery of the information, we encapsulate the DNA in an inorganic matrix, and employ error-correcting codes to correct storage-related errors. Specifically, we translated 83 kB of information to 4991 DNA segments, each 158 nucleotides long, which were encapsulated in silica. Accelerated aging experiments were performed to measure DNA decay kinetics, which show that data can be archived on DNA for millennia under a wide range of conditions. The original information could be recovered error free, even after treating the DNA in silica at 70 °C for one week. This is thermally equivalent to storing information on DNA in central Europe for 2000 years.

Noisy subspace clustering via thresholding

We consider the problem of clustering noisy high-dimensional data points into a union of low-dimensional subspaces and a set of outliers. The number of subspaces, their dimensions, and their orientations are unknown. A probabilistic performance analysis of the thresholding-based subspace clustering (TSC) algorithm introduced recently in [1] shows that TSC succeeds in the noisy case, even when the subspaces intersect. Our results reveal an explicit tradeoff between the allowed noise level and the affinity of the subspaces. We furthermore find that the simple outlier detection scheme introduced in [1] provably succeeds in the noisy case.

Robust Subspace Clustering via Thresholding

The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity subspace clustering algorithm, which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points. In other words, the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance. A statistical performance analysis shows that the algorithm exhibits robustness to additive noise and succeeds even when the subspaces intersect. Specifically, our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level. We furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be (up to a log-factor) linear in the ambient dimension. We also propose a simple scheme that provably detects outliers, and we present numerical results on real and synthetic data.

Identification of Sparse Linear Operators

We consider the problem of identifying a linear deterministic operator from its response to a given probing signal. For a large class of linear operators, we show that stable identifiability is possible if the total support area of the operators spreading function satisfies Δ ≤ 1/2. This result holds for an arbitrary (possibly fragmented) support region of the spreading function, does not impose limitations on the total extent of the support region, and, most importantly, does not require the support region to be known prior to identification. Furthermore, we prove that stable identifiability of almost all operators is possible if Δ <; 1. This result is surprising as it says that there is no penalty for not knowing the support region of the spreading function prior to identification. Algorithms that provably recover all operators with Δ ≤ 1/2, and almost all operators with Δ <; 1 are presented.

Subspace clustering via thresholding and spectral clustering

We consider the problem of clustering a set of high-dimensional data points into sets of low-dimensional linear subspaces. The number of subspaces, their dimensions, and their orientations are unknown. We propose a simple and low-complexity clustering algorithm based on thresholding the correlations between the data points followed by spectral clustering. A probabilistic performance analysis shows that this algorithm succeeds even when the subspaces intersect, and when the dimensions of the subspaces scale (up to a log-factor) linearly in the ambient dimension. Moreover, we prove that the algorithm also succeeds for data points that are subject to erasures with the number of erasures scaling (up to a log-factor) linearly in the ambient dimension. Finally, we propose a simple scheme that provably detects outliers.

Joint sparsity with different measurement matrices

We consider a generalization of the multiple measurement vector (MMV) problem, where the measurement matrices are allowed to differ across measurements. This problem arises naturally when multiple measurements are taken over time, e.g., and the measurement modality (matrix) is time-varying. We derive probabilistic recovery guarantees showing that - under certain (mild) conditions on the measurement matrices - l2/l1-norm minimization and a variant of orthogonal matching pursuit fail with a probability that decays exponentially in the number of measurements. This allows us to conclude that, perhaps surprisingly, recovery performance does not suffer from the individual measurements being taken through different measurement matrices. What is more, recovery performance typically benefits (significantly) from diversity in the measurement matrices; we specify conditions under which such improvements are obtained. These results continue to hold when the measurements are subject to (bounded) noise.

Subspace clustering of dimensionality-reduced data

Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, assumed unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from “undersampling” due to complexity and speed constraints on the acquisition device. More pertinently, even if one has access to the high-dimensional data set it is often desirable to first project the data points into a lower-dimensional space and to perform the clustering task there; this reduces storage requirements and computational cost. The purpose of this paper is to quantify the impact of dimensionality-reduction through random projection on the performance of the sparse subspace clustering (SSC) and the thresholding based subspace clustering (TSC) algorithms. We find that for both algorithms dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation. The mathematical engine behind our theorems is a result quantifying how the affinities between subspaces change under random dimensionality reducing projections.

Compressive nonparametric graphical model selection for time series

We propose a method for inferring the conditional independence graph (CIG) of a high-dimensional discrete-time Gaussian vector random process from finite-length observations. Our approach does not rely on a parametric model (such as, e.g., an autoregressive model) for the vector random process; rather, it only assumes certain spectral smoothness properties. The proposed inference scheme is compressive in that it works for sample sizes that are (much) smaller than the number of scalar process components. We provide analytical conditions for our method to correctly identify the CIG with high probability.

Detecting and Number Counting of Single Engineered Nanoparticles by Digital Particle Polymerase Chain Reaction

The concentrations of nanoparticles present in colloidal dispersions are usually measured and given in mass concentration (e.g. mg/mL), and number concentrations can only be obtained by making assumptions about nanoparticle size and morphology. Additionally traditional nanoparticle concentration measures are not very sensitive, and only the presence/absence of millions/billions of particles occurring together can be obtained. Here, we describe a method, which not only intrinsically results in number concentrations, but is also sensitive enough to count individual nanoparticles, one by one. To make this possible, the sensitivity of the polymerase chain reaction (PCR) was combined with a binary (=0/1, yes/no) measurement arrangement, binomial statistics and DNA comprising monodisperse silica nanoparticles. With this method, individual tagged particles in the range of 60-250 nm could be detected and counted in drinking water in absolute number, utilizing a standard qPCR device within 1.5 h of measurement time. For comparison, the method was validated with single particle inductively coupled plasma mass spectrometry (sp-ICPMS).

Neighborhood selection for thresholding-based subspace clustering

Subspace clustering refers to the problem of clustering high-dimensional data points into a union of low-dimensional linear subspaces, where the number of subspaces, their dimensions and orientations are all unknown. In this paper, we propose a variation of the recently introduced thresholding-based subspace clustering (TSC) algorithm, which applies spectral clustering to an adjacency matrix constructed from the nearest neighbors of each data point with respect to the spherical distance measure. The new element resides in an individual and data-driven choice of the number of nearest neighbors. Previous performance results for TSC, as well as for other subspace clustering algorithms based on spectral clustering, come in terms of an intermediate performance measure, which does not address the clustering error directly. Our main analytical contribution is a performance analysis of the modified TSC algorithm (as well as the original TSC algorithm) in terms of the clustering error directly.