No abstract

Submodular function minimization (SFM) is a fundamental discrete optimization problem which generalizes many well known problems, has applications in various fields, and can be solved in polynomial time. Owing to applications in computer vision and machine learning, fast SFM algorithms are highly desirable. The current fastest algorithms [36] run in O(n 2 log nM · EO + n 3 log O(1) nM ) time and O(n 3 log 2 n·EO+n 4 log O(1) n) time respectively, where M is the largest absolute value of the function (assuming the range is integers) and EO is the time taken to evaluate the function on any set. Although the best known lower bound on the query complexity is only Ω(n) [23], the current shortest non-deterministic proof [10] certifying the optimum value of a function requires Ω(n 2 ) function evaluations.The main contribution of this paper are subquadratic SFM algorithms. For integer-valued submodular functions, we give an SFM algorithm which runs in O(nM 3 log n · EO) time giving the first nearly linear time algorithm in any known regime. For real-valued submodular functions with range in [−1, 1], we give an algorithm which inÕ(n 5/3 · EO/ε 2 ) time returns an ε-additive approximate solution. At the heart of it, our algorithms are projected stochastic subgradient descent methods on the Lovasz extension of submodular functions where we crucially exploit submodularity and data structures to obtain fast, i.e. sublinear time subgradient updates. The latter is crucial for beating the n 2 bound -we show that algorithms which access only subgradients of the Lovasz extension, and these include the empirically fast Fujishige-Wolfe heuristic [48,15] and the theoretically best cutting plane methods [36] , must make Ω(n) subgradient calls (even for functions whose range is {−1, 0, 1}).

No abstract

No abstract

Generalized pustular psoriasis (GPP) is uncommon in children, and serious renal and liver complications arising from GPP are rarely reported. We describe a Chinese boy who had suffered from recurrent exacerbations of GPP from the age of 1 year. He developed IgA nephropathy at the age of 9 years. He also had recurrent episodes of oliguric renal failure, hepatomegaly and cholestasis associated with severe exacerbations of GPP. These complications progressed despite early antibiotics and supportive therapy, but responded promptly to intravenous methylprednisolone therapy. Ultimately, acitretin was given and he has successfully been in remission for a year.