Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study

Abstract

Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One such challenge is the existence of "repeated eigendirections," which obscures the detection of the true dimensionality of the underlying manifold and arises when several embedding coordinates parametrize the same direction in the intrinsic geometry of the data set. We propose an algorithm, based on local linear regression, to automatically detect coordinates corresponding to repeated eigendirections. We construct a more parsimonious embedding using only the eigenvectors corresponding to unique eigendirections, and we show that this reduced diffusion maps embedding induces a metric which is equivalent to the standard diffusion distance. We first demonstrate the utility and flexibility of our approach on synthetic data sets. We then apply our algorithm to data collected from a stochastic model of cellular chemotaxis, where our approach for factoring out repeated eigendirections allows us to detect changes in dynamical behavior and the underlying intrinsic system dimensionality directly from data.

Some manifold learning considerations toward explicit model predictive control

“…DMAPS embedding of the MPC systems showing the training data using three alternative state parametrizations (a) , (b) , and (c) and the first 10 steps of the control policy ( u *) in the distance metric (see text). (d) Residuals from local linear regression for each parametrization in (a)–(c) suggesting that the first, second, and seventh eigenvectors are the best parametrization of the underlying manifold (see Equation (17) and Reference ). Arguably, only the first two eigenvectors are necessary, though we observe improved prediction accuracy when including the next nonredundant eigenvector.…”

“…Thus, the intrinsic parametrization will organize the control policy space and the state variable space using as few variables as possible—subject to the limit in Equation —and hopefully make prediction of the high‐dimensional control policies “simpler.” The leading nonredundant eigenvectors can be used to parametrize the manifold of the outputs (the control policies) and therefore provide the coordinates important for predicting them . Therefore (nonredundant) eigenvectors—sometimes after a large gap in the spectrum—correspond to coordinates in the augmented state space that are unimportant for predicting the control policy and can be eliminated to provide a reduced order approximation of the control law.…”

Automated cellular structure extraction in biological images with applications to calcium imaging data

“…Eigenvectors can be also selected based on mutual information, local linear regression, etc. [30]. Regarding the number of clusters, at present the user inputs a maximal number of Figure 13.…”

“…Yet, for data which varies in density and cluster size, typically the first k eigenvectors will not identify k clusters. Instead several eigenvectors may "repeat" on the same cluster [20,30]. An additional limitation of traditional approaches is that they assume all points belong to a cluster, whereas in calcium imaging (and other biomedical imaging applications), there are pixels belonging to the clutter (background), which are not of interest, and whose proportion out of the full image plane can vary (see Fig.…”

Exploring Single-Cell Data with Deep Multitasking Neural Networks

“…Then, these neighborhoods are either used directly for optimizing low dimensional embeddings (e.g., in TSNE [43] and LLE [44]), or they are used to infer a global data manifold by considering relations between them (e.g., using diffusion geometry [14,15,45,46]). In the latter case, the data manifold enables several applications, including dimensionality reduction [14,46], clustering [45,[47][48][49], imputation [15], and extracting latent data features [50][51][52].…”