The Probabilistic Watershed isasemi-supervised learning algorithm applied on undirected graphs.

In equilibrium statistical mechanics, phases are parameter regions in which the free energy evolves smoothly. This equilibrium perspective is well suited to to conventional solid-state experiments, in which the system of interest (e.g., the electron fluid in a metal) is well coupled to a heat bath. Present-day experiments in quantum devices necessitate a more general concept of phases: in these devices, systems are driven far from thermal equilibrium and are either isolated from the environment or coupled to engineered dissipative environments. A key step toward this general concept came from the development of quasi-adiabatic continuation [1], for pure quantum states at zero temperature. According to this definition, phases are equivalence classes of quantum states such that two states in the same phase can be prepared from one another by an efficient process--specifically, a finite-depth local unitary (FDLU) circuit. This concept of pure-state phases (called FDLU-equivalence) reduces to the conventional one for ground states of gapped local Hamiltonians, but extends to any quantum state, and connects naturally to questions in computational complexity thoery [2-5]. So far, this "preparability" perspective is only fully developed for pure quantum states and strictly unitary evolutions; thus, a natural task, which has seen intense recent activity, is its generalization to more general classes of mixed states and evolutions involving noise, measurement, and feedback [6-14]. As an important special case, a classification of mixed states from the perspective of preparability would naturally extend to general classical probability distributions, of the type that routinely arise in machine learning, and that also seem to exhibit phase transitions [15].

Technically, our online quantum algorithm "quantizes" classical algorithms based

Developing procedures for sampling higher-quality translations is therefore in high demand.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper proposes several approaches to sample from a Gibbs distribution over a discrete space by solving randomly perturbed combinatorial optimization problems (MAP inference) over the same space. The starting point is a known result [5] that allows to do sampling (in principle, using high dimensional perturbations with exponential complexity) by solving a single optimization problem. In this paper they propose to 1) use more efficient low-dimensional random perturbations to do approximate sampling (with probabilistic accuracy guarantees on tree structured models) 2) estimate (conditional) marginals using ratios of partition function estimates, and sequentially sample variables. They propose a clever rejection strategy based on self reduction that guarantees unbiasedness of the samples.