Compressed Sensing over ℓp\ell_p-balls: Minimax Mean Square Error

We consider the compressed sensing problem, where the object x_0 \in \bR^N is to be recovered from incomplete measurements $y = Ax_0 + z$; here the sensing matrix $A$ is an $n \times N$ random matrix with iid Gaussian entries and $n < N$. A popular method of sparsity-promoting reconstruction is $\ell^1$-penalized least-squares reconstruction (aka LASSO, Basis Pursuit). It is currently popular to consider the strict sparsity model, where the object $x_0$ is nonzero in only a small fraction of entries. In this paper, we instead consider the much more broadly applicable $\ell_p$-sparsity model, where $x_0$ is sparse in the sense of having $\ell_p$ norm bounded by $\xi \cdot N^{1/p}$ for some fixed $0 0$. We study an asymptotic regime in which $n$ and $N$ both tend to infinity with limiting ratio $n/N = \delta \in (0,1)$, both in the noisy ($z \neq 0$) and noiseless ($z=0$) cases. Under weak assumptions on $x_0$, we are able to precisely evaluate the worst-case asymptotic minimax mean-squared reconstruction error (AMSE) for $\ell^1$ penalized least-squares: min over penalization parameters, max over $\ell_p$-sparse objects $x_0$. We exhibit the asymptotically least-favorable object (hardest sparse signal to recover) and the maximin penalization. Our explicit formulas unexpectedly involve quantities appearing classically in statistical decision theory. Occurring in the present setting, they reflect a deeper connection between penalized $\ell^1$ minimization and scalar soft thresholding. This connection, which follows from earlier work of the authors and collaborators on the AMP iterative thresholding algorithm, is carefully explained. Our approach also gives precise results under weak-$\ell_p$ ball coefficient constraints, as we show here.Comment: 41 pages, 11 pdf figure

Message Passing Algorithms for Compressed Sensing

Compressed sensing aims to undersample certain high-dimensional signals, yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex optimization -- which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity-undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity-undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity-undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity-undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this new, apparently very different theoretical formalism.Comment: 6 pages paper + 9 pages supplementary information, 13 eps figure. Submitted to Proc. Natl. Acad. Sci. US

How do we remember the past in randomised strategies?

Graph games of infinite length are a natural model for open reactive processes: one player represents the controller, trying to ensure a given specification, and the other represents a hostile environment. The evolution of the system depends on the decisions of both players, supplemented by chance. In this work, we focus on the notion of randomised strategy. More specifically, we show that three natural definitions may lead to very different results: in the most general cases, an almost-surely winning situation may become almost-surely losing if the player is only allowed to use a weaker notion of strategy. In more reasonable settings, translations exist, but they require infinite memory, even in simple cases. Finally, some traditional problems becomes undecidable for the strongest type of strategies

Threshold values of Random K-SAT from the cavity method

Using the cavity equations of \cite{mezard:parisi:zecchina:02,mezard:zecchina:02}, we derive the various threshold values for the number of clauses per variable of the random $K$-satisfiability problem, generalizing the previous results to $K \ge 4$. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large $K$. The stability of the solution is also computed. For any $K$, the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the conjecture that this computation gives the exact satisfiability threshold.Comment: 38 pages; extended explanations and derivations; this version is going to appear in Random Structures & Algorithm

The Long-Baseline Neutrino Facility

The Deep Underground Neutrino Experiment (DUNE) collaboration is developing an international multi-kiloton Long-Baseline Neutrino experiment to be located about a mile underground at the Sanford Underground Research Facility (SURF), in Lead, SD, USA. In the current configuration four cryostats will contain a modular detector and a total of 68,400 ton of ultra pure liquid argon, with a level of impurities lower than 100 parts per trillion (ppt) of oxygen equivalent contamination. The Long-Baseline Neutrino Facility (LBNF) provides the conventional facilities and the cryogenic infrastructure (including the cryostats housing the detector) to support DUNE. This contribution presents the modes of operations, layout and main features of the LBNF cryogenic system. The system is comprised of three sub-systems: External/Infrastructure (or LN2), Proximity (or LAr) and Internal cryogenics. The External/Infrastructure provides the infrastructure and equipment to store, produce and distribute the cryogenic fluids needed for the operation of the Proximity Cryogenics, which delivers them to the Internal at the pressure, temperature, mass flow rate, quality and purity required by the detector inside the cryostat. The External/Infrastructure cryogenics includes the LN2 refrigeration system and the surface facilities, with the receiving stations, the LN2 and LAr storage tanks and the vaporizers. The Proximity Cryogenics includes the LAr and GAr purification systems, the phase separators, the condensers, and the piping connecting the various parts. The Internal Cryogenics consists of all the cryogenic equipment located inside the cryostat, namely the GAr and LAr distribution systems and the systems to cool down the cryostats and the detectors. An international engineering team will design, manufacture, commission, and qualify the LBNF cryogenic system. The expected performance, the functional requirements and the status of the design are presented in this contribution