Persistent Homology in Sparse Regression and its Application to Brain Morphometry

Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is introduced and explored. The structures are then further exploited in speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region.Comment: submitted to IEEE Transactions on Medical Imagin

Characterization of complex networks: A survey of measurements

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the proper application and interpretation of measurements.Comment: A working manuscript with 78 pages, 32 figures. Suggestions of measurements for inclusion are welcomed by the author

Scheme for constructing graphs associated with stabilizer quantum codes

We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.Comment: 42 pages, 12 figure

Filtering graphs to check isomorphism and extracting mapping by using the Conductance Electrical Model

© 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper presents a new method of filtering graphs to check exact graph isomorphism and extracting their mapping. Each graph is modeled by a resistive electrical circuit using the Conductance Electrical Model (CEM). By using this model, a necessary condition to check the isomorphism of two graphs is that their equivalent resistances have the same values, but this is not enough, and we have to look for their mapping to find the sufficient condition. We can compute the isomorphism between two graphs in O(N-3), where N is the order of the graph, if their star resistance values are different, otherwise the computational time is exponential, but only with respect to the number of repeated star resistance values, which usually is very small. We can use this technique to filter graphs that are not isomorphic and in case that they are, we can obtain their node mapping. A distinguishing feature over other methods is that, even if there exists repeated star resistance values, we can extract a partial node mapping (of all the nodes except the repeated ones and their neighbors) in O(N-3). The paper presents the method and its application to detect isomorphic graphs in two well know graph databases, where some graphs have more than 600 nodes. (C) 2016 Elsevier Ltd. All rights reserved.Postprint (author's draft