Improving compressed sensing with the diamond norm

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio

Design thinking support: information systems versus reasoning

Numerous attempts have been made to conceive and implement appropriate information systems to support architectural designers in their creative design thinking processes. These information systems aim at providing support in very diverse ways: enabling designers to make diverse kinds of visual representations of a design, enabling them to make complex calculations and simulations which take into account numerous relevant parameters in the design context, providing them with loads of information and knowledge from all over the world, and so forth. Notwithstanding the continued efforts to develop these information systems, they still fail to provide essential support in the core creative activities of architectural designers. In order to understand why an appropriately effective support from information systems is so hard to realize, we started to look into the nature of design thinking and on how reasoning processes are at play in this design thinking. This investigation suggests that creative designing rests on a cyclic combination of abductive, deductive and inductive reasoning processes. Because traditional information systems typically target only one of these reasoning processes at a time, this could explain the limited applicability and usefulness of these systems. As research in information technology is increasingly targeting the combination of these reasoning modes, improvements may be within reach for design thinking support by information systems

XML content warehousing: Improving sociological studies of mailing lists and web data

In this paper, we present the guidelines for an XML-based approach for the sociological study of Web data such as the analysis of mailing lists or databases available online. The use of an XML warehouse is a flexible solution for storing and processing this kind of data. We propose an implemented solution and show possible applications with our case study of profiles of experts involved in W3C standard-setting activity. We illustrate the sociological use of semi-structured databases by presenting our XML Schema for mailing-list warehousing. An XML Schema allows many adjunctions or crossings of data sources, without modifying existing data sets, while allowing possible structural evolution. We also show that the existence of hidden data implies increased complexity for traditional SQL users. XML content warehousing allows altogether exhaustive warehousing and recursive queries through contents, with far less dependence on the initial storage. We finally present the possibility of exporting the data stored in the warehouse to commonly-used advanced software devoted to sociological analysis