Semi-supervised learning has been of growing interest over the past few years and many methods have been proposed. Although various algorithms are provided to implement semi-supervised learning,there are still gaps in our understanding of the dependence of generalization error on the numbers of labeled and unlabeled data. In this paper,we consider a graph-based semi-supervised classification algorithm and establish its generalization error bounds. Our results show the close relations between the generalization performance and the structural invariants of data graph.
Let C＇（α,β） be the class of functions f（z） =analytic in D ={z： ｜z｜ 〈 1}, satisfying for some convex function g（z） with g（O） = g＇（O） - 1 =- 0 and for allz in D the condition zf＇（z）-1/g（z）/zf＇（z）/g（z）＋1-2α）｜ 〈β for some α β （0 ≤ α〈1,0 〈 β 〈 1）. A sharp coefficient estimate, distortion theorems and radius of convexity are determined for the class C＇（α ,β ）. The results extend the work of C. Selvaraj.
设H是一个Hilbert空间.B(H)表示所有H到H的有界线性算子构成的Banach空间.设T={f(z):f(z)=zI-sum from n=2 to∞z^n A_n在单位圆盘|z|<1上解析,其中系数A_n是H到H的紧正Hermitian算子,I表示H上的恒等算子,sum from n=2 to∞n(A_nx,x)≤1对所有x∈H,‖x‖=1成立}.该文研究了函数族T的极值点.
The problem of reconstructing a signalφ（x） from its magnitude ｜φ（x）] isof considerable interest to engineers and physicists. This article concerns the problem of determining a time-limited signal f with period 2π when ｜f（eix）l is known for x∈[-π,π]. It is shown that the conditions ｜g（eix）｜ = ｜f（eix）｜ and ｜g（ci（x＋b）） -g（eix）｜ =f（ei（x＋b）） - f（eix）｜, b ≠ 27π, together imply that either g = wf or g = v f, where both w and v have period b. Furthermore, if b/2π is irrational then the functions w and v b is rational then w takes the form reduce to some constants c1 and c2, respectively; ifb/2π is rational then w takes the form w=elexB1（e1x）B2（elx）and v takes the form ei（x2πN/b＋a）B1（elx）B2（elx）,where B1 and B2 are Blaschke products.
