Decoding the Encoding of Functional Brain Networks: an fMRI Classification Comparison of Non-negative Matrix Factorization (NMF), Independent Component Analysis (ICA), and Sparse Coding Algorithms

Brain networks in fMRI are typically identified using spatial independent component analysis (ICA), yet mathematical constraints such as sparse coding and positivity both provide alternate biologically-plausible frameworks for generating brain networks. Non-negative Matrix Factorization (NMF) would suppress negative BOLD signal by enforcing positivity. Spatial sparse coding algorithms ($L1$ Regularized Learning and K-SVD) would impose local specialization and a discouragement of multitasking, where the total observed activity in a single voxel originates from a restricted number of possible brain networks. The assumptions of independence, positivity, and sparsity to encode task-related brain networks are compared; the resulting brain networks for different constraints are used as basis functions to encode the observed functional activity at a given time point. These encodings are decoded using machine learning to compare both the algorithms and their assumptions, using the time series weights to predict whether a subject is viewing a video, listening to an audio cue, or at rest, in 304 fMRI scans from 51 subjects. For classifying cognitive activity, the sparse coding algorithm of $L1$ Regularized Learning consistently outperformed 4 variations of ICA across different numbers of networks and noise levels (p$<$0.001). The NMF algorithms, which suppressed negative BOLD signal, had the poorest accuracy. Within each algorithm, encodings using sparser spatial networks (containing more zero-valued voxels) had higher classification accuracy (p$<$0.001). The success of sparse coding algorithms may suggest that algorithms which enforce sparse coding, discourage multitasking, and promote local specialization may capture better the underlying source processes than those which allow inexhaustible local processes such as ICA

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

We generalize the idea of Vassiliev invariants to the spin network context, with the aim of using these invariants as a kinematical arena for a canonical quantization of gravity. This paper presents a detailed construction of these invariants (both ambient and regular isotopic) requiring a significant elaboration based on the use of Chern-Simons perturbation theory which extends the work of Kauffman, Martin and Witten to four-valent networks. We show that this space of knot invariants has the crucial property -from the point of view of the quantization of gravity- of being loop differentiable in the sense of distributions. This allows the definition of diffeomorphism and Hamiltonian constraints. We show that the invariants are annihilated by the diffeomorphism constraint. In a companion paper we elaborate on the definition of a Hamiltonian constraint, discuss the constraint algebra, and show that the construction leads to a consistent theory of canonical quantum gravity.Comment: 21 Pages, RevTex, many figures included with psfi

Bridging topological and functional information in protein interaction networks by short loops profiling

Protein-protein interaction networks (PPINs) have been employed to identify potential novel interconnections between proteins as well as crucial cellular functions. In this study we identify fundamental principles of PPIN topologies by analysing network motifs of short loops, which are small cyclic interactions of between 3 and 6 proteins. We compared 30 PPINs with corresponding randomised null models and examined the occurrence of common biological functions in loops extracted from a cross-validated high-confidence dataset of 622 human protein complexes. We demonstrate that loops are an intrinsic feature of PPINs and that specific cell functions are predominantly performed by loops of different lengths. Topologically, we find that loops are strongly related to the accuracy of PPINs and define a core of interactions with high resilience. The identification of this core and the analysis of loop composition are promising tools to assess PPIN quality and to uncover possible biases from experimental detection methods. More than 96% of loops share at least one biological function, with enrichment of cellular functions related to mRNA metabolic processing and the cell cycle. Our analyses suggest that these motifs can be used in the design of targeted experiments for functional phenotype detection.This research was supported by the Biotechnology and Biological Sciences Research Council (BB/H018409/1 to AP, ACCC and FF, and BB/J016284/1 to NSBT) and by the Leukaemia & Lymphoma Research (to NSBT and FF). SSC is funded by a Leukaemia & Lymphoma Research Gordon Piller PhD Studentship

Learning Dynamic Boltzmann Distributions as Reduced Models of Spatial Chemical Kinetics

Finding reduced models of spatially-distributed chemical reaction networks requires an estimation of which effective dynamics are relevant. We propose a machine learning approach to this coarse graining problem, where a maximum entropy approximation is constructed that evolves slowly in time. The dynamical model governing the approximation is expressed as a functional, allowing a general treatment of spatial interactions. In contrast to typical machine learning approaches which estimate the interaction parameters of a graphical model, we derive Boltzmann-machine like learning algorithms to estimate directly the functionals dictating the time evolution of these parameters. By incorporating analytic solutions from simple reaction motifs, an efficient simulation method is demonstrated for systems ranging from toy problems to basic biologically relevant networks. The broadly applicable nature of our approach to learning spatial dynamics suggests promising applications to multiscale methods for spatial networks, as well as to further problems in machine learning

Computation of Smooth Optical Flow in a Feedback Connected Analog Network

In 1986, Tanner and Mead \cite{Tanner_Mead86} implemented an interesting constraint satisfaction circuit for global motion sensing in aVLSI. We report here a new and improved aVLSI implementation that provides smooth optical flow as well as global motion in a two dimensional visual field. The computation of optical flow is an ill-posed problem, which expresses itself as the aperture problem. However, the optical flow can be estimated by the use of regularization methods, in which additional constraints are introduced in terms of a global energy functional that must be minimized. We show how the algorithmic constraints of Horn and Schunck \cite{Horn_Schunck81} on computing smooth optical flow can be mapped onto the physical constraints of an equivalent electronic network

Vassiliev invariants: a new framework for quantum gravity

We show that Vassiliev invariants of knots, appropriately generalized to the spin network context, are loop differentiable in spite of being diffeomorphism invariant. This opens the possibility of defining rigorously the constraints of quantum gravity as geometrical operators acting on the space of Vassiliev invariants of spin nets. We show how to explicitly realize the diffeomorphism constraint on this space and present proposals for the construction of Hamiltonian constraints.Comment: 15 pages, several figures included with psfi

Moment-Based Relaxation of the Optimal Power Flow Problem

The optimal power flow (OPF) problem minimizes power system operating cost subject to both engineering and network constraints. With the potential to find global solutions, significant research interest has focused on convex relaxations of the non-convex AC OPF problem. This paper investigates ``moment-based'' relaxations of the OPF problem developed from the theory of polynomial optimization problems. At the cost of increased computational requirements, moment-based relaxations are generally tighter than the semidefinite relaxation employed in previous research, thus resulting in global solutions for a broader class of OPF problems. Exploration of the feasible space for test systems illustrates the effectiveness of the moment-based relaxation.Comment: 7 pages, 4 figures. Abstract accepted, full paper in revie