Detecting phase transitions in collective behavior using manifold's curvature

Gajamannage, Kelum; Bollt, Erik M.

doi:10.3934/mbe.2017027

Cited by 8 publications

(11 citation statements)

References 38 publications

(38 reference statements)

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“…In addition, we are interested in DR of collective motion such as schools of fish and flocks of birds. Our recent work, [29], [30], concludes that the transitions in collective motion can be represented as switchings of the underlying manifolds which we will study further in an efficient manner by utilizing BMC technique.…”

Section: Discussionmentioning

confidence: 99%

Bounded Manifold Completion

Gajamannage¹,

Paffenroth²

2019

Preprint

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Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to detect the existence of a low-dimensional manifold of a given dimension that lies within a set of bounds derived from a given point cloud. A matrix representing the appropriately defined distances on a low-dimensional manifold is low-rank, and our method is based on current techniques for recovering a partially observed matrix from a small set of fully observed entries that can be implemented as a low-rank Matrix Completion (MC) problem. MC methods are currently used to solve challenging real-world problems, such as image inpainting and recommender systems, and we leverage extent efficient optimization techniques that use a nuclear norm convex relaxation as a surrogate for non-convex and discontinuous rank minimization. Our proposed method provides several advantages over current nonlinear dimensionality reduction techniques, with the two most important being theoretical guarantees on the detection of low-dimensional embeddings and robustness to non-uniformity in the sampling of the manifold. We validate the performance of this approach using both a theoretical analysis as well as synthetic and real-world benchmark datasets.

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Section: Discussionmentioning

confidence: 99%

Bounded Manifold Completion

Gajamannage¹,

Paffenroth²

2019

Preprint

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“…for some B (t) ∈ M, [12,5]. This linear subspace that describes the underlying structure of the data satisfies all the properties of being a linear manifold, but we use the term linear subspace to make it more general.…”

Section: W Kmentioning

confidence: 99%

“…The loss function, defined in Eqn. (12), is minimized using backpropagation [25] to determine optimum weights W 1 , . .…”

Section: Hadamard Deep Autoencodermentioning

confidence: 99%

“…This linear subspace that describes the underlying structure of the data satisfies all the properties of being a linear manifold, but we use the term linear subspace to make it more general. Moreover, these configurations of coordinated agents evolve in time according to the mapping Φ : [12,5]. Two maps in Eqns.…”

Section: W Kmentioning

confidence: 99%

“…The key motivation of employing DA-based reconstruction method is that the DAs generate a lowdimensional nonlinear representation of the low-rank collective motion data after filtering out coarse features including noise. The interactions between the agents of a system can be simple, complex, or somewhere in between, and can exist among neighbors in space or in time [12]. Trajectories of the agents in collective motion can be represented as a high-dimensional data-cloud in Euclidean space where each point in this data-cloud is called a configuration vector that is defined as the vector representing the positions of all the agents of the group at a given time-step [13].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reconstruction of Agents’ Corrupted Trajectories of Collective Motion Using Low-rank Matrix Completion

Gajamannage

Paffenroth

2019

2019 IEEE International Conference on Big Data (Big Data)

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Learning dynamics of collectively moving agents such as fish or humans is an active field in research. Due to natural phenomena such as occlusion and change of illumination, the multi-object methods tracking such dynamics might lose track of the agents where that might result fragmentation in the constructed trajectories. Here, we present an extended deep autoencoder (DA) that we train only on fully observed segments of the trajectories by defining its loss function as the Hadamard product of a binary indicator matrix with the absolute difference between the outputs and the labels. The trajectories of the agents practicing collective motion is low-rank due to mutual interactions and dependencies between the agents that we utilize as the underlying pattern that our Hadamard deep autoencoder (HDA) codes during its training. The performance of our HDA is compared with that of a low-rank matrix completion scheme in the context of fragmented trajectory reconstruction.

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Recurrent neural networks for dynamical systems: Applications to ordinary differential equations, collective motion, and hydrological modeling

Gajamannage

Jayathilake

Park

et al. 2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Classical methods of solving spatiotemporal dynamical systems include statistical approaches such as autoregressive integrated moving average, which assume linear and stationary relationships between systems’ previous outputs. Development and implementation of linear methods are relatively simple, but they often do not capture non-linear relationships in the data. Thus, artificial neural networks (ANNs) are receiving attention from researchers in analyzing and forecasting dynamical systems. Recurrent neural networks (RNNs), derived from feed-forward ANNs, use internal memory to process variable-length sequences of inputs. This allows RNNs to be applicable for finding solutions for a vast variety of problems in spatiotemporal dynamical systems. Thus, in this paper, we utilize RNNs to treat some specific issues associated with dynamical systems. Specifically, we analyze the performance of RNNs applied to three tasks: reconstruction of correct Lorenz solutions for a system with a formulation error, reconstruction of corrupted collective motion trajectories, and forecasting of streamflow time series possessing spikes, representing three fields, namely, ordinary differential equations, collective motion, and hydrological modeling, respectively. We train and test RNNs uniquely in each task to demonstrate the broad applicability of RNNs in the reconstruction and forecasting the dynamics of dynamical systems.

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Detecting phase transitions in collective behavior using manifold's curvature

Cited by 8 publications

References 38 publications

Bounded Manifold Completion

Bounded Manifold Completion

Reconstruction of Agents’ Corrupted Trajectories of Collective Motion Using Low-rank Matrix Completion

Recurrent neural networks for dynamical systems: Applications to ordinary differential equations, collective motion, and hydrological modeling

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