Sparse Power Factorization: Balancing peakiness and sample complexity

In many applications, one is faced with an inverse problem, where the known signal depends in a bilinear way on two unknown input vectors. Often at least one of the input vectors is assumed to be sparse, i.e., to have only few non-zero entries. Sparse Power Factorization (SPF), proposed by Lee, Wu, and Bresler, aims to tackle this problem. They have established recovery guarantees for a somewhat restrictive class of signals under the assumption that the measurements are random. We generalize these recovery guarantees to a significantly enlarged and more realistic signal class at the expense of a moderately increased number of measurements.Comment: 18 page

Phase Retrieval of Quaternion Signal via Wirtinger Flow

The main aim of this paper is to study quaternion phase retrieval (QPR), i.e., the recovery of quaternion signal from the magnitude of quaternion linear measurements. We show that all $d$-dimensional quaternion signals can be reconstructed up to a global right quaternion phase factor from $O(d)$ phaseless measurements. We also develop the scalable algorithm quaternion Wirtinger flow (QWF) for solving QPR, and establish its linear convergence guarantee. Compared with the analysis of complex Wirtinger flow, a series of different treatments are employed to overcome the difficulties of the non-commutativity of quaternion multiplication. Moreover, we develop a variant of QWF that can effectively utilize a pure quaternion priori (e.g., for color images) by incorporating a quaternion phase factor estimate into QWF iterations. The estimate can be computed efficiently as it amounts to finding a singular vector of a $4\times 4$ real matrix. Motivated by the variants of Wirtinger flow in prior work, we further propose quaternion truncated Wirtinger flow (QTWF), quaternion truncated amplitude flow (QTAF) and their pure quaternion versions. Experimental results on synthetic data and color images are presented to validate our theoretical results. In particular, for pure quaternion signal recovery, our quaternion method often succeeds with measurements notably fewer than real methods based on monochromatic model or concatenation model.Comment: 21 pages (paper+supplemental), 6 figure

The Aha! Experience of Spatial Reorientation

The experience of spatial re-orientation is investigated as an instance of the wellknown phenomenon of the Aha! moment. The research question is: What are the visuospatial conditions that are most likely to trigger the spatial Aha! experience? The literature suggests that spatial re-orientation relies mainly on the geometry of the environment and a visibility graph analysis is used to quantify the visuospatial information. Theories from environmental psychology point towards two hypotheses. The Aha! experience may be triggered by a change in the amount of visual information, described by the isovist properties of area and revelation, or by a change in the complexity of the visual information associated with the isovist properties of clustering coefficient and visual control. Data from participants’ exploratory behaviour and EEG recordings are collected during wayfinding in virtual reality urban environments. Two types of events are of interest here: (a) sudden changes of the visuospatial information preceding subjects' response to investigate changes in EEG power; and (b) participants brain dynamics (Aha! effect) just before the response to examine differences in isovist values at this location. Research on insights, time-frequency analysis of the P3 component and findings from navigation and orientation studies suggest that the spatial Aha! experience may be reflected by: a parietal alpha power decrease associated with the switch of the representation and a frontocentral theta increase indexing spatial processing during decision-making. Single-trial time-frequency analysis is used to classify trials into two conditions based on the alpha/theta power differences between a 3s time-period before participants’ response and a time-period of equal duration before that. Behavioural results show that participants are more likely to respond at locations with low values of clustering coefficient and high values of visual control. The EEG analysis suggests that the alpha decrease/theta increase condition occurs at locations with significantly lower values of clustering coefficient and higher values of visual control. Small and large decreases in clustering coefficient, just before the response, are associated with significant differences in delta/theta power. The values of area and revelation do not show significant differences. Both behavioural and EEG results suggest that the Aha! experience of re-orientation is more likely to be triggered by a change in the complexity of the visual-spatial environment rather than a change in the amount, as measured by the relevant isovist properties