Predicting the value of a function f at a new point given its values at old points is an ubiquitous scientific endeavor, somewhat less developed when f produces several values depending on one another, e.g. when it outputs a probability vector. Considering the points as fixed (not random) entities and focusing on the worst-case, this article uncovers a prediction procedure that is optimal relatively to some model-set information about the vector-valued function f . When the model sets are convex, this procedure turns out to be an affine map constructed by solving a convex optimization program. The theoretical result is specified in the two practical frameworks of (reproducing kernel) Hilbert spaces and of spaces of continuous functions.

We study the matrix completion problem when the observation pattern is deterministic and possibly non-uniform. We propose a simple and efficient debiased projection scheme for recovery from noisy observations and analyze the error under a suitable weighted metric. We introduce a simple function of the weight matrix and the sampling pattern that governs the accuracy of the recovered matrix. We derive theoretical guarantees that upper bound the recovery error and nearly matching lower bounds that showcase optimality in several regimes. Our numerical experiments demonstrate the computational efficiency and accuracy of our approach, and show that debiasing is essential when using non-uniform sampling patterns.

When analyzing communities of microorganisms from their sequenced DNA, an important task is taxonomic profiling: enumerating the presence and relative abundance of all organisms, or merely of all taxa, contained in the sample. This task can be tackled via compressive-sensing-based approaches, which favor communities featuring the fewest organisms among those consistent with the observed DNA data. Despite their successes, these parsimonious approaches sometimes conflict with biological realism by overlooking organism similarities. Here, we leverage a recently developed notion of biological diversity that simultaneously accounts for organism similarities and retains the optimization strategy underlying compressive-sensing-based approaches. We demonstrate that minimizing biological diversity still produces sparse taxonomic profiles and we experimentally validate superiority to existing compressive-sensing-based approaches. Despite showing that the objective function is almost never convex and often concave, generally yielding NP-hard problems, we exhibit ways of representing organism similarities for which minimizing diversity can be performed via a sequence of linear programs guaranteed to decrease diversity. Better yet, when biological similarity is quantified by k-mer co-occurrence (a popular notion in bioinformatics), minimizing diversity actually reduces to one linear program that can utilize multiple k-mer sizes to enhance performance. In proof-of-concept experiments, we verify that the latter procedure can lead to significant gains when taxonomically profiling a metagenomic sample, both in terms of reconstruction accuracy and computational performance.

A matrix whose entries are independent subexponential random variables is not likely to satisfy the classical restricted isometry property in the optimal regime of parameters. However, it is known that uniform sparse recovery is still possible with high probability in the optimal regime if ones uses l1-minimization as a recovery algorithm. We show in this letter that such a statement remains valid if one uses a new variation of iterative hard thresholding as a recovery algorithm. The argument is based on a modified restricted isometry property featuring the l1-norm as the inner norm.

Binary measurements arise naturally in a variety of statistics and engineering applications. They may be inherent to the problem-for example, in determining the relationship between genetics and the presence or absence of a disease-or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can greatly improve the ability to reconstruct a signal from binary measurements. This is exemplified by one-bit compressed sensing, which takes the compressed sensing model but assumes that only the sign of each measurement is retained. It has recently been shown that the number of one-bit measurements required for signal estimation mirrors that of unquantized compressed sensing. Indeed, s-sparse signals in Rn can be estimated (up to normalization) from Ω(slog (n/s)) one-bit measurements. Nevertheless, controlling the precise accuracy of the error estimate remains an open challenge. In this paper, we focus on optimizing the decay of the error as a function of the oversampling factor λ := m/(s log(n/s)), where m is the number of measurements. It is known that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by Ω(λ-1). Without adjusting the measurement procedure, reducing this polynomial error decay rate is impossible. However, we show that an adaptive choice of the thresholds used for quantization can lower the error rate to e-Ω(λ). This improves upon guarantees for other methods of adaptive thresholding, such as sigma- delta quantization. We develop a general recursive strategy to achieve this exponential decay and two specific polynomial-time algorithms, which fall into this framework, one based on convex programming and one on hard thresholding. Our work bridges the one-bit compressed sensing model, in which the engineer controls the measurement procedure, to sigma-delta and successive approximation quantization. Moreover, the principle is extendable to signal reconstruction problems in a variety of binary statistical models as well as statistical estimation problems like logistic regression.

This letter demonstrates that sparse recovery can be achieved by an L1-minimization ersatz easily implemented using a conventional nonnegative least squares algorithm. A connection with orthogonal matching pursuit is also highlighted. The preliminary results call for more investigations on the potential of the method and on its relations to classical sparse recovery algorithms.

MOTIVATION: Many metagenomic studies compare hundreds to thousands of environmental and health-related samples by extracting and sequencing their 16S rRNA amplicons and measuring their similarity using beta-diversity metrics. However, one of the first steps--to classify the operational taxonomic units within the sample--can be a computationally time-consuming task because most methods rely on computing the taxonomic assignment of each individual read out of tens to hundreds of thousands of reads. RESULTS: We introduce Quikr: a QUadratic, K-mer-based, Iterative, Reconstruction method, which computes a vector of taxonomic assignments and their proportions in the sample using an optimization technique motivated from the mathematical theory of compressive sensing. On both simulated and actual biological data, we demonstrate that Quikr typically has less error and is typically orders of magnitude faster than the most commonly used taxonomic assignment technique (the Ribosomal Database Project's Naïve Bayesian Classifier). Furthermore, the technique is shown to be unaffected by the presence of chimeras, thereby allowing for the circumvention of the time-intensive step of chimera filtering. AVAILABILITY: The Quikr computational package (in MATLAB, Octave, Python and C) for the Linux and Mac platforms is available at http://sourceforge.net/projects/quikr/.

We provide sharp lower and upper bounds for the Gelfand widths of $\ell_p$-balls in the $N$-dimensional $\ell_q^N$-space for $0<p\leq 1$ and $p