On Gaussian Comparison Inequality and Its Application to Spectral Analysis of Large Random Matrices

Recently, Chernozhukov, Chetverikov, and Kato [Ann. Statist. 42 (2014) 1564--1597] developed a new Gaussian comparison inequality for approximating the suprema of empirical processes. This paper exploits this technique to devise sharp inference on spectra of large random matrices. In particular, we show that two long-standing problems in random matrix theory can be solved: (i) simple bootstrap inference on sample eigenvalues when true eigenvalues are tied; (ii) conducting two-sample Roy's covariance test in high dimensions. To establish the asymptotic results, a generalized $\epsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent interest.Comment: to appear in Bernoull

Stabilized Structure from Motion without Disparity Induces Disparity Adaptation

3D structures can be perceived based on the patterns of 2D motion signals [1, 2]. With orthographic projection of a 3D stimulus onto a 2D plane, the kinetic information can give a vivid impression of depth, but the depth order is intrinsically ambiguous, resulting in bistable or even multistable interpretations [3]. For example, an orthographic projection of dots on the surface of a rotating cylinder is perceived as a rotating cylinder with ambiguous direction of rotation [4]. We show that the bistable rotation can be stabilized by adding information, not to the dots themselves, but to their spatial context. More interestingly, the stabilized bistable motion can generate consistent rotation aftereffects. The rotation aftereffect can only be observed when the adapting and test stimuli are presented at the same stereo depth and the same retinal location, and it is not due to attentional tracking. The observed rotation aftereffect is likely due to direction-contingent disparity adaptation, implying that stimuli with kinetic depth may have activated neurons sensitive to different disparities, even though the stimuli have zero relative disparity. Stereo depth and kinetic depth may be supported by a common neural mechanism at an early stage in the visual system

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

We present a non-variational, kinetic energy operator approach to the solution of quantum three-body problem with Coulomb interactions, based on the utilization of symmetries intrinsic to the kinetic energy operator, i.e., the three-body Laplacian operator with the respective masses. Through a four-step reduction process, the nine dimensional problem is reduced to a one dimensional coupled system of ordinary differential equations, amenable to accurate numerical solution as an infinite-dimensional algebraic eigenvalue problem. A key observation in this reduction process is that in the functional subspace of the kinetic energy operator where all the rotational degrees of freedom have been projected out, there is an intrinsic symmetry which can be made explicit through the introduction of Jacobi-spherical coordinates. A numerical scheme is presented whereby the Coulomb matrix elements are calculated to a high degree of accuracy with minimal effort, and the truncation of the linear equations is carried out through a systematic procedureComment: 56 pages, 11 figure

Knowledge-based document retrieval with application to TEXPROS

Document retrieval in an information system is most often accomplished through keyword search. The common technique behind keyword search is indexing. The major drawback of such a search technique is its lack of effectiveness and accuracy. It is very common in a typical keyword search over the Internet to identify hundreds or even thousands of records as the potentially desired records. However, often few of them are relevant to users\u27 interests. This dissertation presents knowledge-based document retrieval architecture with application to TEXPROS. The architecture is based on a dual document model that consists of a document type hierarchy and, a folder organization. Using the knowledge collected during document filing, the search space can be narrowed down significantly. Combining the classical text-based retrieval methods with the knowledge-based retrieval can improve tremendously both search efficiency and effectiveness. With the proposed predicate-based query language, users can more precisely and accurately specify the search criteria and their knowledge about the documents to be retrieved. To assist users formulate a query, a guided search is presented as part of an intelligent user interface. Supported by an intelligent question generator, an inference engine, a question base, and a predicate-based query composer, the guided search collects the most important information known to the user to retrieve the documents that satisfy users\u27 particular interests. A knowledge-based query processing and search engine is presented as the core component in this architecture. Algorithms are developed for the search engine to effectively and efficiently retrieve the documents that match the query. Cache is introduced to speed up the process of query refinement. Theoretical proof and performance analysis are performed to prove the efficiency and effectiveness of this knowledge-based document retrieval approach