1,880 research outputs found

    Smoothed Complexity Theory

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    Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

    Cell assembly dynamics of sparsely-connected inhibitory networks: a simple model for the collective activity of striatal projection neurons

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    Striatal projection neurons form a sparsely-connected inhibitory network, and this arrangement may be essential for the appropriate temporal organization of behavior. Here we show that a simplified, sparse inhibitory network of Leaky-Integrate-and-Fire neurons can reproduce some key features of striatal population activity, as observed in brain slices [Carrillo-Reid et al., J. Neurophysiology 99 (2008) 1435{1450]. In particular we develop a new metric to determine the conditions under which sparse inhibitory networks form anti-correlated cell assemblies with time-varying activity of individual cells. We found that under these conditions the network displays an input-specific sequence of cell assembly switching, that effectively discriminates similar inputs. Our results support the proposal [Ponzi and Wickens, PLoS Comp Biol 9 (2013) e1002954] that GABAergic connections between striatal projection neurons allow stimulus-selective, temporally-extended sequential activation of cell assemblies. Furthermore, we help to show how altered intrastriatal GABAergic signaling may produce aberrant network-level information processing in disorders such as Parkinson's and Huntington's diseases.Comment: 22 pages, 9 figure

    Towards explaining the speed of kk-means

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    The kk-means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the kk-means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences

    Order and disorder in the Local Evolutionary Minority Game

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    We study a modification of the Evolutionary Minority Game (EMG) in which agents are placed in the nodes of a regular or a random graph. A neighborhood for each agent can thus be defined and a modification of the usual relaxation dynamics can be made in which each agent updates her decision scheme depending upon the options made in her immediate neighborhood. We name this model the Local Evolutionary Minority Game (LEMG). We report numerical results for the topologies of a ring, a torus and a random graph changing the size of the neighborhood. We focus our discussion in a one dimensional system and perform a detailed comparison of the results obtained from the random relaxation dynamics of the LEMG and from a linear chain of interacting spin-like variables at a finite temperature. We provide a physical interpretation of the surprising result that in the LEMG a better coordination (a lower frustration) is achieved if agents base their actions on local information. We show how the LEMG can be regarded as a model that gradually interpolates between a fully ordered, antiferromagnetic system and a fully disordered system that can be assimilated to a spin glass.Comment: 12 pages, 8 figures, RevTex; omission of a relevant reference correcte

    How to allocate Research (and other) Subsidies

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    A budget-constrained buyer wants to purchase items from a short-listed set. Items are differentiated by observable quality and sellers have private reserve prices for their items. The buyer’s problem is to select a subset of maximal quality. Money does not enter the buyer’s objective function, but only his constraints. Sellers quote prices strategically, inducing a knapsack game. We derive the Bayesian optimal mechanism for the buyer’s problem. We ?nd that simultaneous take-it-or-leave-it offers are optimal. Hence, somewhat surprisingly, ex-postcompetition is not required to implement optimality. Finally, we discuss the problem in a detail free setting

    Lower Bounds for the Average and Smoothed Number of Pareto Optima

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    Smoothed analysis of multiobjective 0-1 linear optimization has drawn considerable attention recently. The number of Pareto-optimal solutions (i.e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems is the central object of study. In this paper, we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of \Omega_d(n^(d-1)) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of lower bounding the number of Pareto optima to results in geometry connected to arrangements of hyperplanes. We use our basic result to derive (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this for the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds for other standard objective functions studied in this setting (such as, multiobjective shortest path, TSP tour, matching). (2) Smoothed lower bound of min {\Omega_d(n^(d-1.5) \phi^{(d-log d) (1-\Theta(1/\phi))}), 2^{\Theta(n)}}$ for the 0-1 knapsack problem with d profits for phi-semirandom distributions for a version of the knapsack problem. This improves the recent lower bound of Brunsch and Roeglin

    WARNING: Physics Envy May Be Hazardous To Your Wealth!

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    The quantitative aspirations of economists and financial analysts have for many years been based on the belief that it should be possible to build models of economic systems - and financial markets in particular - that are as predictive as those in physics. While this perspective has led to a number of important breakthroughs in economics, "physics envy" has also created a false sense of mathematical precision in some cases. We speculate on the origins of physics envy, and then describe an alternate perspective of economic behavior based on a new taxonomy of uncertainty. We illustrate the relevance of this taxonomy with two concrete examples: the classical harmonic oscillator with some new twists that make physics look more like economics, and a quantitative equity market-neutral strategy. We conclude by offering a new interpretation of tail events, proposing an "uncertainty checklist" with which our taxonomy can be implemented, and considering the role that quants played in the current financial crisis.Comment: v3 adds 2 reference
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