Topology and Dynamics of Attractor Neural Networks: The Role of Loopiness

We derive an exact representation of the topological effect on the dynamics of sequence processing neural networks within signal-to-noise analysis. A new network structure parameter, loopiness coefficient, is introduced to quantitatively study the loop effect on network dynamics. The large loopiness coefficient means the large probability of finding loops in the networks. We develop the recursive equations for the overlap parameters of neural networks in the term of the loopiness. It was found that the large loopiness increases the correlations among the network states at different times, and eventually it reduces the performance of neural networks. The theory is applied to several network topological structures, including fully-connected, densely-connected random, densely-connected regular, and densely-connected small-world, where encouraging results are obtained.Comment: 6 pages, 4 figures, comments are favore

Segmentation of the evolving left ventricle by learning the dynamics

We propose a method for recursive segmentation of the left ventricle (LV) across a temporal sequence of magnetic resonance (MR) images. The approach involves a technique for learning the LV boundary dynamics together with a particle-based inference algorithm on a loopy graphical model capturing the temporal periodicity of the heart. The dynamic system state is a low-dimensional representation of the boundary, and boundary estimation involves incorporating curve evolution into state estimation. By formulating the problem as one of state estimation, the segmentation at each particular time is based not only on the data observed at that instant, but also on predictions based on past and future boundary estimates. We assess and demonstrate the effectiveness of the proposed framework on a large data set of breath-hold cardiac MR image sequences

Learning the dynamics and time-recursive boundary detection of deformable objects

We propose a principled framework for recursively segmenting deformable objects across a sequence of frames. We demonstrate the usefulness of this method on left ventricular segmentation across a cardiac cycle. The approach involves a technique for learning the system dynamics together with methods of particle-based smoothing as well as non-parametric belief propagation on a loopy graphical model capturing the temporal periodicity of the heart. The dynamic system state is a low-dimensional representation of the boundary, and the boundary estimation involves incorporating curve evolution into recursive state estimation. By formulating the problem as one of state estimation, the segmentation at each particular time is based not only on the data observed at that instant, but also on predictions based on past and future boundary estimates. Although the paper focuses on left ventricle segmentation, the method generalizes to temporally segmenting any deformable object

Linguistics and some aspects of its underlying dynamics

In recent years, central components of a new approach to linguistics, the Minimalist Program (MP) have come closer to physics. Features of the Minimalist Program, such as the unconstrained nature of recursive Merge, the operation of the Labeling Algorithm that only operates at the interface of Narrow Syntax with the Conceptual-Intentional and the Sensory-Motor interfaces, the difference between pronounced and un-pronounced copies of elements in a sentence and the build-up of the Fibonacci sequence in the syntactic derivation of sentence structures, are directly accessible to representation in terms of algebraic formalism. Although in our scheme linguistic structures are classical ones, we find that an interesting and productive isomorphism can be established between the MP structure, algebraic structures and many-body field theory opening new avenues of inquiry on the dynamics underlying some central aspects of linguistics.Comment: 17 page

The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles

It is shown that an exact solution of the transient dynamics of an associative memory model storing an infinite number of limit cycles with l finite steps by means of the path-integral analysis. Assuming the Maxwell construction ansatz, we have succeeded in deriving the stationary state equations of the order parameters from the macroscopic recursive equations with respect to the finite-step sequence processing model which has retarded self-interactions. We have also derived the stationary state equations by means of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must assume that crosstalk noise of an input to spins obeys a Gaussian distribution. On the other hand, the path-integral method does not require such a Gaussian approximation of crosstalk noise. We have found that both the signal-to-noise analysis and the path-integral analysis give the completely same result with respect to the stationary state in the case where the dynamics is deterministic, when we assume the Maxwell construction ansatz. We have shown the dependence of storage capacity (alpha_c) on the number of patterns per one limit cycle (l). Storage capacity monotonously increases with the number of steps, and converges to alpha_c=0.269 at l ~= 10. The original properties of the finite-step sequence processing model appear as long as the number of steps of the limit cycle has order l=O(1).Comment: 24 pages, 3 figure

Pedestrian flows in bounded domains with obstacles

In this paper we systematically apply the mathematical structures by time-evolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discrete-time Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Radon positive measures generated by a push-forward recursive relation. We assume that two fundamental aspects of pedestrian behavior rule the dynamics of the system: On the one hand, the will to reach specific targets, which determines the main direction of motion of the walkers; on the other hand, the tendency to avoid crowding, which introduces interactions among the individuals. The resulting model is able to reproduce several experimental evidences of pedestrian flows pointed out in the specialized literature, being at the same time much easier to handle, from both the analytical and the numerical point of view, than other models relying on nonlinear hyperbolic conservation laws. This makes it suitable to address two-dimensional applications of practical interest, chiefly the motion of pedestrians in complex domains scattered with obstacles.Comment: 25 pages, 9 figure

Sequential Monte Carlo with kernel embedded mappings: the mapping particle filter

In this work, a novel sequential Monte Carlo filter is introduced which aims at an efficient sampling of the state space. Particles are pushed forward from the prediction to the posterior density using a sequence of mappings that minimizes the Kullback-Leibler divergence between the posterior and the sequence of intermediate densities. The sequence of mappings represents a gradient flow based on the principles of local optimal transport. A key ingredient of the mappings is that they are embedded in a reproducing kernel Hilbert space, which allows for a practical and efficient Monte Carlo algorithm. The kernel embedding provides a direct means to calculate the gradient of the Kullback-Leibler divergence leading to quick convergence using well-known gradient-based stochastic optimization algorithms. Evaluation of the method is conducted in the chaotic Lorenz-63 system, the Lorenz-96 system, which is a coarse prototype of atmospheric dynamics, and an epidemic model that describes cholera dynamics. No resampling is required in the mapping particle filter even for long recursive sequences. The number of effective particles remains close to the total number of particles in all the sequence. Hence, the mapping particle filter does not suffer from sample impoverishment