Manifold Learning With Contracting Observers for Data-Driven Time-Series Analysis

Abstract

Analyzing signals arising from dynamical systems typically requires many modeling assumptions and parameter estimation. In high dimensions, this modeling is particularly difficult due to the "curse of dimensionality". In this paper, we propose a method for building an intrinsic representation of such signals in a purely data-driven manner. First, we apply a manifold learning technique, diffusion maps, to learn the intrinsic model of the latent variables of the dynamical system, solely from the measurements. Second, we use concepts and tools from control theory and build a linear contracting observer to estimate the latent variables in a sequential manner from new incoming measurements. The effectiveness of the presented framework is demonstrated by applying it to a toy problem and to a music analysis application. In these examples we show that our method reveals the intrinsic variables of the analyzed dynamical systems.

Diffusion Maps Kalman Filter for a Class of Systems With Gradient Flows

“…This property facilitates the use of our devised Kalman model in many applications, as demonstrated in the experimental study in Section IV, even though the model is inaccurate due to the measurement noise. We further note that the noise term in (22) can be used to represent deviations from the true model as described in [24]. However, significant measurement noise may still lead to some deterioration in the performance of the proposed method, as demonstrated in Subsection IV-A.…”

“…The remainder of this section is described as follows. In Subsection III-B and Subsection III-C, we reiterate the derivations presented in [22] for state and model recovery using diffusion maps [19]. In Subsection III-D, we present our proposed Kalman filter framework.…”

“…The framework in this paper is tightly related to [22], in which a linear observer was constructed by exploiting similar properties of the diffusion maps coordinates and their dynamics. We show that the Kalman filter framework introduced in the present work outperforms the observer framework both by reducing the amount of hyper-parameters and by significantly improving the results in certain scenarios.…”

Cell-type specific outcome representation in primary motor cortex

“…Mice were habituated to head fixation in a custom-built apparatus in dark and quiet conditions, monitored by a webcam. Mice were initially trained to retrieve food pellets (14)(15)(16)(17)(18)(19)(20)Test Diet;St Louis,MO) For tongue reach experiments, expert hand reach task mice were retrained to access the food pellet with their tongue.…”

“…In recent years, manifold learning has become a leading approach for detecting underlying structures and hidden parameters in data. Methods such as Laplacian eigenmaps (Belkin and Niyogi 2003), Hessian eigenmaps (Donoho and Grimes 2003) which is the main reason why diffusion maps is broadly and successfully applied (Mishne et al 2017, Shemesh et al 2017, Yair and Talmon 2017, Shnitzer et al 2016, Sulam et al 2017.…”