Most people believe that renaming is easy: simply choose a name at random; if more than one process selects the same name, then try again. We highlight the issues that occur when trying to implement such a scheme and shed new light on the read-write complexity of randomized renaming in an asynchronous environment. At the heart of our new perspective stands an adaptive implementation of a randomized test-and-set object, that has poly-logarithmic step complexity per operation, with high probability. Interestingly, our implementation is anonymous, as it does not require process identifiers. Based on this implementation, we present two new randomized renaming algorithms. The first ensures a tight namespace of n names using O(n log 4 n) total steps, with high probability. This improves on the best previously known algorithm by almost a quadratic factor. The second algorithm achieves a namespace of size k(1 +) using O(k log 4 k/ log 2 (1 +)) total steps, both with high probability, where k is the total contention in the execution. It is the first adaptive randomized renaming algorithm, and it improves on existing deterministic solutions by providing a smaller namespace, and by significantly lowering complexity.

The aim of this paper is to show that spatial coupling can be viewed not only as a means to build better graphical models, but also as a tool to better understand uncoupled models. The starting point is the observation that some asymptotic properties of graphical models are easier to prove in the case of spatial coupling. In such cases, one can then use the so-called interpolation method to transfer known results for the spatially coupled case to the uncoupled one.Our main use of this framework is for LDPC codes, where we use interpolation to show that the average entropy of the codeword conditioned on the observation is asymptotically the same for spatially coupled as for uncoupled ensembles.We give three applications of this result for a large class of LDPC ensembles. The first one is a proof of the so-called Maxwell construction stating that the MAP threshold is equal to the Area threshold of the BP GEXIT curve. The second is a proof of the equality between the BP and MAP GEXIT curves above the MAP threshold. The third application is the intimately related fact that the replica symmetric formula for the conditional entropy in the infinite block length limit is exact.

Investigations on spatially coupled codes have lead to the conjecture that, in the infinite size limit, the average input-output conditional entropy for spatially coupled low-density parity-check ensembles, over binary memoryless symmetric channels, equals the entropy of the underlying individual ensemble. We give a self-contained proof of this conjecture for the case when the variable degrees have a Poisson distribution and all check degrees are even. The ingredients of the proof are the interpolation method and the Nishimori identities. We explain why this result is an important step towards proving the Maxwell conjecture in the theory of low-density parity-check codes. I. INTRODUCTIONSpatial coupling of graphical models has emerged as a useful design principle in order to construct graphical models that show good performance under message-passing algorithms. Two areas that have experienced significant progress by using spatial coupling are coding, see [1]-[3] and references therein, and compressive sensing [4]- [7]. But there is also a large literature on a variety of other graphical models and for each of these models the same significant improvement in performance has been observed when going from the uncoupled to the coupled model. For a more thorough review, we refer the reader to [3].The above examples all take advantage of the algorithmic opportunities that spatial coupling provides. But there is a second way in which spatial coupling of graphical models can be useful, namely as a proof technique. The idea is to prove a property of the underlying graph in the following way: (i) prove that this property holds for the spatially coupled graph, and (ii), show that the coupled and the uncoupled systems behave in the same way (with respect to this property). Due to the special nature of spatially coupled graphical models, accomplishing step (i), i.e., proving the desired property for the coupled system, is sometimes easier than for the uncoupled one. Before we outline in more detail how we use spatial coupling as a proof technique, let us introduce the underlying problem.We consider the so-called Maxwell conjecture. This conjecture states that the MAP threshold of an LDPC ensemble is equal to its area threshold. The area threshold is, in turn, determined via the so-called BP-GEXIT curve. To keep things simple, consider (l, r)-regular LDPC ensembles. Figure 1 shows the case of the (3, 6)-regular LDPC ensemble when transmission takes place over the BAWGN channel. It is known, see [8, Theorem 3.120], that for regular degree

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Abstract. These notes review six lectures given by Prof. Andrea Montanari on the topic of statistical estimation for linear models. The first two lectures cover the principles of signal recovery from linear measurements in terms of minimax risk. Subsequent lectures demonstrate the application of these principles to several practical problems in science and engineering. Specifically, these topics include denoising of error-laden signals, recovery of compressively sensed signals, reconstruction of low-rank matrices, and also the discovery of hidden cliques within large networks.These are notes from the lecture of Andrea Montanari given at the autumn school "Statistical Physics, Optimization, Inference, and Message-Passing Algorithms", that took place in Les Houches,