Modern cryptographic practice rests on the use of one-way functions, which are easy to evaluate but difficult to invert. Unfortunately, commonly used one-way functions are either based on unproven conjectures or have known vulnerabilities. We show that instead of relying on number theory, the mesoscopic physics of coherent transport through a disordered medium can be used to allocate and authenticate unique identifiers by physically reducing the medium's microstructure to a fixed-length string of binary digits. These physical one-way functions are inexpensive to fabricate, prohibitively difficult to duplicate, admit no compact mathematical representation, and are intrinsically tamper-resistant. We provide an authentication protocol based on the enormous address space that is a principal characteristic of physical one-way functions.

Abstract. Recent SVD-free matrix factorization formulations have enabled rank minimization for systems with millions of rows and columns, paving the way for matrix completion in extremely large-scale applications, such as seismic data interpolation.In this paper, we consider matrix completion formulations designed to hit a target data-fitting error level provided by the user, and propose an algorithm called LR-BPDN that is able to exploit factorized formulations to solve the corresponding optimization problem. Since practitioners typically have strong prior knowledge about target error level, this innovation makes it easy to apply the algorithm in practice, leaving only the factor rank to be determined.Within the established framework, we propose two extensions that are highly relevant to solving practical challenges of data interpolation. First, we propose a weighted extension that allows known subspace information to improve the results of matrix completion formulations. We show how this weighting can be used in the context of frequency continuation, an essential aspect to seismic data interpolation. Second, we propose matrix completion formulations that are robust to large measurement errors in the available data.We illustrate the advantages of LR-BPDN on collaborative filtering problem using the MovieLens 1M, 10M, and Netflix 100M datasets. Then, we use the new method, along with its robust and subspace re-weighted extensions, to obtain high-quality reconstructions for large scale seismic interpolation problems with real data, even in the presence of data contamination.1. Introduction. Sparsity-and rank-regularization have had significant impact in many areas over the last several decades. Sparsity in certain transform domains has been exploited to solve underdetermined linear systems with applications to compressed sensing [11,9], natural image denoising/inpainting [41,28,30], and seismic image processing [20,33,18,31]. Analogously, low-rank structure has been used to efficiently solve matrix completion problems, such as the Netflix Prize problem, along with many other applications, including control, system identification, signal processing, and combinatorial optimization [12,35,7], and seismic data interpolation and denoising [34].Regularization formulations for both types of problems introduce a regularization functional of the decision variable, either by adding an explicit penalty to the data-fitting term

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We derive a cost functional for estimating the relationship between highdimensional observations and the low-dimensional process that generated them with no input-output examples. Limiting our search to invertible observation functions confers numerous benefits, including a compact representation and no suboptimal local minima. Our approximation algorithms for optimizing this cost functional are fast and give diagnostic bounds on the quality of their solution. Our method can be viewed as a manifold learning algorithm that utilizes a prior on the low-dimensional manifold coordinates. The benefits of taking advantage of such priors in manifold learning and searching for the inverse observation functions in system identification are demonstrated empirically by learning to track moving targets from raw measurements in a sensor network setting and in an RFID tracking experiment.