Grad Student Seminar: Yifan Cui and Melody Zhu
Some asymptotic results of survival tree and forest models
We develop atheoretical framework and asymptotic results for survival tree and forest models under right censoring. We first investigate the method from the aspect of splitting rules, where the survival curves of the two potential child nodes are calculated and compared. We show that existing approaches lead to a potentially biased estimation of the within-node survival and cause non-optimal selection of the splitting rules. This bias is due to the censoring distribution and the non i.i.d. sample structure within each node. Based on this observation, we develop an adaptive concentration bound result for both tree and forest versions of the survival tree models. The result quantifies the variance component for survival forest models. Furthermore, we show with three specific examples how these concentration bounds, combined with properly designed splitting rules, yield consistency results. The three examples are: 1) a finite dimensional setting with random splitting rules; 2) an infinite dimensional case with marginal signal checking; and 3) an infinite dimensional setting with principled Cox screening splitting rule. The development of these results serves as a general framework for showing the consistency of tree- and forest-based survival models.
This talk is based on a joint work with Dr. Ruoqing Zhu, Dr. Mai Zhou, Dr. Michael Kosorok, and Dr. Jan Hannig.
Yuzixuan (Melody) Zhu
Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs.
We introduce Sieve-SDP, a simple algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP belongs to the class of facial reduction algorithms. It inspects the constraints of the problem, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on several problem collections from the literature.
We also highlight an issue arising in SDPs with positive duality gap: on such problems SDP solvers may compute a “fake” solution with an arbitrarily small constraint violation, and arbitrarily small duality gap.