Orthogonal Trace-Sum Maximization: Applications, Local Algorithms, and Global Optimality

This paper studies a problem of maximizing the sum of traces of matrix quadratic forms on a product of Stiefel manifolds. This orthogonal trace-sum maximization (OTSM) problem generalizes many interesting problems such as generalized canonical correlation analysis (CCA), Procrustes analysis, and cryo-electron microscopy of the Nobel prize fame. For these applications finding global solutions is highly desirable but has been out of reach for a long time. For example, generalizations of CCA do not possess obvious global solutions unlike their classical counterpart to which a global solution is readily obtained through singular value decomposition; it is also not clear how to test global optimality. We provide a simple method to certify global optimality of a given local solution. This method only requires testing the sign of the smallest eigenvalue of a symmetric matrix, and does not rely on a particular algorithm as long as it converges to a stationary point. Our certificate result relies on a semidefinite programming (SDP) relaxation of OTSM, but avoids solving an SDP of lifted dimensions. Surprisingly, a popular algorithm for generalized CCA and Procrustes analysis may generate oscillating iterates. We propose a simple modification of this standard algorithm and prove that it reliably converges. Our notion of convergence is stronger than conventional objective value convergence or subsequence convergence.The convergence result utilizes the Kurdyka-Lojasiewicz property of the problem.Comment: 22 pages, 1 figur

On the Correctness of the Generalized Isotonic Recursive Partitioning Algorithm

This paper presents an in-depth analysis of the generalized isotonic recursive partitioning (GIRP) algorithm for fitting isotonic models under separable convex losses, proposed by Luss and Rosset [J. Comput. Graph. Statist., 23 (2014), pp. 192--201] for differentiable losses and extended by Painsky and Rosset [IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), pp. 308-321] for nondifferentiable losses. The GIRP algorithm poseses an attractive feature that in each step of the algorithm, the intermediate solution satisfies the isotonicity constraint. The paper begins with an example showing that the GIRP algorithm as described in the literature may fail to produce an isotonic model, suggesting that the existence and uniqueness of the solution to the isotonic regression problem must be carefully addressed. It proceeds with showing that, among possibly many solutions, there indeed exists a solution that can be found by recursive binary partitioning of the set of observed data. A small modification of the GIRP algorithm suffices to obtain a correct solution and preserve the desired property that all the intermediate solutions are isotonic. This proposed modification includes a proper choice of intermediate solutions and a simplification of the partitioning step from ternary to binary.Comment: 19 pages, 1 figur

Reformulating the Developmental State Theory to Explain Chinese Spatial Planning

Developmental state literature almost completely neglects the fact that one of the unique features of the developmental state is its capacity to reorganize its territory, and the literature on the Chinese developmental state repeats the same oversight. Against this backdrop, this study attempts to retheorize China's spatial planning from a developmental state perspective. In light of the theoretical discussion in this study, we argue that the developmentalist spatial planning has five main characteristics of the developmentalist spatial planning: 1) The state sees its territory as a means of production, not as a living environment. 2) Industrial location policies were market-conforming. 3) The spatial planning was controlled or strongly influenced by the elite economic agency that formulates industrial policies and guides the market. 4) The bureaucracy is more or less insulated from local growth coalitions. 5) Spatial planning creates rather than responds to economic changes. These five characteristics are apparent in China’s spatial planning as much as in South Korea’s

Second language English listening comprehension using different presentations of pictures and video cues

The study tested the effects of different presentations using pictures and video cues for improving listening comprehension of English news programs. Four experiments are reported, studying listening comprehension of English as a second/foreign language with 687 Korean secondary students. Comparisons on listening comprehension showed better performance with visual cues than with no visual cues. Listening comprehension with video cues was more successful than that with pictures. The advantage of the combination of verbal and visual information over the presentation of verbal information alone was in accord with dual coding theory. When contextual information presented using priming techniques was compared to using feedback and simultaneous presentations, listening comprehension was better using priming. In the comparison of feedback with simultaneous presentations, listening comprehension was improved more when pictures with headlines were presented using feedback than using simultaneous presentations. In contrast, no differences were found between feedback and simultaneous presentations when video cues with headlines were presented. Visual cues with headlines presented using priming might enable learners to activate prior knowledge or schemata to improve listening comprehension. Headlines presented at the beginning stage of listening were effective for listening comprehension. In addition, the effects of presentations were enlarged by adding headlines to visuals. Applying the priming presentation along with the enrichment of contextual cues resulted in improved listening comprehension. Less proficient students benefited relatively more from the contextual cues with headlines and pictorial cues for comprehending the news than more proficient students. In particular, for less proficient students, video cues with headlines were more helpful in listening comprehension than pictures with headlines. This was because more abundant visual cues such as paralinguistic cues were more likely to be provided in video than in picture formats. The best listening comprehension occurred when presenting pictorial cues with headlines using priming presentation. The present study concluded that more abundant pictorial cues were useful for improving listening comprehension. Headlines added to the pictorial cues improved performance, especially for less proficient students, who benefited relatively more. The pictorial cues with headlines presented using a 'priming' technique were most effective in improving listening comprehension, probably because they activated prior knowledge or schemata