Traffic Speed Data Imputation Method Based on Tensor Completion

Traffic speed data plays a key role in Intelligent Transportation Systems (ITS); however, missing traffic data would affect the performance of ITS as well as Advanced Traveler Information Systems (ATIS). In this paper, we handle this issue by a novel tensor-based imputation approach. Specifically, tensor pattern is adopted for modeling traffic speed data and then High accurate Low Rank Tensor Completion (HaLRTC), an efficient tensor completion method, is employed to estimate the missing traffic speed data. This proposed method is able to recover missing entries from given entries, which may be noisy, considering severe fluctuation of traffic speed data compared with traffic volume. The proposed method is evaluated on Performance Measurement System (PeMS) database, and the experimental results show the superiority of the proposed approach over state-of-the-art baseline approaches

Low rank tensor recovery via iterative hard thresholding

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.Comment: 34 page

Tsirelson's problem and Kirchberg's conjecture

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession; we provide an example of a state together with observables such that ordinary spatial correlations are local, while the spatiotemporal correlations reveal nonlocality. Moreover, we find an extended version of Tsirelson's problem which, for each nontrivial Bell scenario, is equivalent to the QWEP conjecture. This extended version can be conveniently formulated in terms of steering the system of a third party. Finally, a comprehensive mathematical appendix offers background material on complete positivity, tensor products of C*-algebras, group C*-algebras, and some simple reformulations of the QWEP conjecture.Comment: 57 pages, to appear in Rev. Math. Phy

Diffusion Tensor Imaging for Assessment of Response to Neoadjuvant Chemotherapy in Patients With Breast Cancer.

In this study, the prognostic significance of tumor metrics derived from diffusion tensor imaging (DTI) was evaluated in patients with locally advanced breast cancer undergoing neoadjuvant therapy. DTI and contrast-enhanced magnetic resonance imaging were acquired at 1.5 T in 34 patients before treatment and after 3 cycles of taxane-based therapy (early treatment). Tumor fractional anisotropy (FA), principal eigenvalues (λ1, λ2, and λ3), and apparent diffusion coefficient (ADC) were estimated for tumor regions of interest drawn on DTI data. The association between DTI metrics and final tumor volume change was evaluated with Spearman rank correlation. DTI metrics were investigated as predictors of pathological complete response (pCR) by calculating the area under the receiver operating characteristic curve (AUC). Early changes in tumor FA and ADC significantly correlated with final tumor volume change post therapy (ρ = -0.38, P = .03 and ρ = -0.71, P < .001, respectively). Pretreatment tumor ADC was significantly lower in the pCR than in the non-pCR group (P = .04). At early treatment, patients with pCR had significantly higher percent changes of tumor λ1, λ2, λ3, and ADC than those without pCR. The AUCs for early percent changes in tumor FA and ADC were 0.60 and 0.83, respectively. The early percent changes in tumor eigenvalues and ADC were the strongest DTI-derived predictors of pCR. Although early percent change in tumor FA had a weak association with pCR, the significant correlation with final tumor volume change suggests that this metric changes with therapy and may merit further evaluation