课程名称︰ 机器学习特论 课程性质︰ 选修 课程教师︰ 林智仁 开课学院： 电资院 开课系所︰ 资工系 考试日期（年月日）︰ 2017/3/28 考试时限（分钟）： 2.5hr 试题 : Please give details of your answer. A direct answer without explanation is not counted. Your Answers must be in English. You can bring notes and text book. Other books or electronic devices are not allowed. (数学式用latex格式表示) Problem 1 (10 pts) 1. Consider the folloing function f(x) = x^3, x \in R (a) (5 pts) Is f convex? (b) (5 pts) Is f quasi-convex? You cannot answer this question by drawing figures. You must give mathmatical proofs. Problem 2 (15 pts) 2. Assume f_1,f_2 are convex functions. Consider f(x) = min{f_1(x),f_2(x)} (a) (7 pts) Is f convex? (b) (8 pts) Is f concave? Poblem 3 (15 pts) 3. In slide 2-19 we proved the existance of a separating hyperplane. In the proof, we use frac(d,dt)||d- c + t(u -d)|| < 0 to argue the existance of a small t \in (0,1) such that ||d - c + t(u-d)|| \leq ||d-c|| Instead of using (1), can you specifically derive a t^* > 0 so that ||d - c + t(u-d)|| \leq ||d-c||, \forall t \in [0.t^*] ? Problem 4 (30 pts) 4. Let g(x): R^n \rightarrow R (a) (10 pts) Assume g(x) is convex, Is f(x) = g(x)^2 convex or not ? (b) (20 pts) Assume g(x) is strictly convex and g(x) \geq 0, \forall x \in R ^n Is f(x) = g(x)^2 strictly convwx or not? Problem 5 (15 pts) 5. Consider g(x_1,x_2) = x_1 x_2, where x_1 > 0, x_2 > 0 Use the first-order condition to checke if this function is quasi-convex or not. Problem 6 (15 pts) 6. Consider f(x) = 2e^x + e^{-x} What is the conjugate function of f(x) ? --

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