关于图参数的精确可学习性：配分函数的情形
On the exact learnability of graph parameters: The case of partition functions
摘要：
我们研究了实值图参数$f$的精确可学习性，这些参数可以表示为划分函数，该划分函数计算图$H$中具有顶点权重$α$和边权重$β$的加权同态的数量。M.弗里德曼湖Lovász和A. Schrijver用$f$的$k$ -连接矩阵$C(f,k)$给出了这些图参数的一个特征.我们的可学习性模型是基于D. Angluin使用成员和等价查询的精确学习模型。给定这样的图参数$f$，学习者可以针对他们选择的图询问$f$的值，并且他们可以根据$f$的连接矩阵$C(f,k)$来制定假设。教师可以接受这个假设是正确的，或者提供一个由图表组成的反例。我们的主要结果表明，在这种情况下，一个非常大的类的配分函数，刚性配分函数，可以学习的时间多项式的大小为$H$和最大的反例的大小在Blum-Shub-Smale模型的计算在实数与单位成本。
英文摘要：
We study the exact learnability of real valued graph parameters $f$ which are known to be representable as partition functions which count the number of weighted homomorphisms into a graph $H$ with vertex weights $α$ and edge weights $β$. M. Freedman, L. Lovász and A. Schrijver have given a characterization of these graph parameters in terms of the $k$-connection matrices $C(f,k)$ of $f$. Our model of learnability is based on D. Angluin's model of exact learning using membership and equivalence queries. Given such a graph parameter $f$, the learner can ask for the values of $f$ for graphs of their choice, and they can formulate hypotheses in terms of the connection matrices $C(f,k)$ of $f$. The teacher can accept the hypothesis as correct, or provide a counterexample consisting of a graph. Our main result shows that in this scenario, a very large class of partition functions, the rigid partition functions, can be learned in time polynomial in the size of $H$ and the size of the largest counterexample in the Blum-Shub-Smale model of computation over the reals with unit cost.
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