We study the problem of generating synthetic data under differential privacy. We establish fixed-parameter tractability (FPT) for this problem where the parameter is the treewidth of the query family's incidence graph. Our algorithms attain optimal error rates across all regimes and are realized by two different approaches: the first is based on linear programming (LP) and the FPT of the separation problem for the LP dual; the second is based on a subsampled private multiplicative weights method, where we obtain FPT for sampling from Gibbs distributions. Both approaches are unified by a dynamic programming framework over a tree decomposition.

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.