L10: k-Means Clustering
... k = O(n )) number of possible distinct cluster centers. But it could be exponential in k and d (the dimension when Euclidean distance used). • However, usually R = 10 is fine. • Smoothed analysis: if data perturbed randomly slightly, then R = O(n35 k 34 d8 ). This is “polynomial,” but still ridiculo ...
... k = O(n )) number of possible distinct cluster centers. But it could be exponential in k and d (the dimension when Euclidean distance used). • However, usually R = 10 is fine. • Smoothed analysis: if data perturbed randomly slightly, then R = O(n35 k 34 d8 ). This is “polynomial,” but still ridiculo ...
Enhanced Traveling Salesman Problem Solving by Genetic
... the principles of natural selection. The formal theory was initially developed by John Holland and his students in the 1970’s [1, 2]. The continuing improvement in the price/performance value of GA’s has made them attractive for many types of problem solving optimization methods. In particular, gene ...
... the principles of natural selection. The formal theory was initially developed by John Holland and his students in the 1970’s [1, 2]. The continuing improvement in the price/performance value of GA’s has made them attractive for many types of problem solving optimization methods. In particular, gene ...
Continuous Distributions - Department of Statistics, Yale
... giving probabilities for U taking on discrete values, we must specify probabilities for U to lie in various subintervals of its range. Indeed, if you put a equal to b you will find that P{U = b} = 0 for each b in the interval [0,1]. To distinguish more clearly between continuous distributions and th ...
... giving probabilities for U taking on discrete values, we must specify probabilities for U to lie in various subintervals of its range. Indeed, if you put a equal to b you will find that P{U = b} = 0 for each b in the interval [0,1]. To distinguish more clearly between continuous distributions and th ...
Bertrand`s Paradox
... common sense that the solution is the same no matter whether we take a smaller coin or a larger coin (the problem is scale invariant), and no matter whether the coin is placed a bit more to the left or a bit more to the right (the problem is translational invariant). With these two invariants, the r ...
... common sense that the solution is the same no matter whether we take a smaller coin or a larger coin (the problem is scale invariant), and no matter whether the coin is placed a bit more to the left or a bit more to the right (the problem is translational invariant). With these two invariants, the r ...
Pattern Recognition
... Further topics The model Let two random variables be given: • The first one is typically discrete (i.e. • The second one is often continuous ( “observation” Let the joint probability distribution As is discrete it is often specified by ...
... Further topics The model Let two random variables be given: • The first one is typically discrete (i.e. • The second one is often continuous ( “observation” Let the joint probability distribution As is discrete it is often specified by ...
10. Hidden Markov Models (HMM) for Speech Processing
... do we choose a corresponding state sequence Q = q1 q2 … qT which is optimal in some meaningful sense (i.e., best “explains” the observations)? i.e. maximizes P(Q, O|λ) The forward algorithm provides the total probability through all paths, but not the optimum path sequence Several alternative soluti ...
... do we choose a corresponding state sequence Q = q1 q2 … qT which is optimal in some meaningful sense (i.e., best “explains” the observations)? i.e. maximizes P(Q, O|λ) The forward algorithm provides the total probability through all paths, but not the optimum path sequence Several alternative soluti ...
Simulated annealing
Simulated annealing (SA) is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more efficient than exhaustive enumeration — provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution.The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. Both are attributes of the material that depend on its thermodynamic free energy. Heating and cooling the material affects both the temperature and the thermodynamic free energy. While the same amount of cooling brings the same amount of decrease in temperature it will bring a bigger or smaller decrease in the thermodynamic free energy depending on the rate that it occurs, with a slower rate producing a bigger decrease.This notion of slow cooling is implemented in the Simulated Annealing algorithm as a slow decrease in the probability of accepting worse solutions as it explores the solution space. Accepting worse solutions is a fundamental property of metaheuristics because it allows for a more extensive search for the optimal solution.The method was independently described by Scott Kirkpatrick, C. Daniel Gelatt and Mario P. Vecchi in 1983, and by Vlado Černý in 1985. The method is an adaptation of the Metropolis–Hastings algorithm, a Monte Carlo method to generate sample states of a thermodynamic system, invented by M.N. Rosenbluth and published in a paper by N. Metropolis et al. in 1953.