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Learning to Decouple Complex Systems

Zihan Zhou, Tianshu Yu

TL;DR

The paper tackles learning dynamics from cluttered, irregular sequential data by decoupling a complex system into latent sub-systems and a meta-system that captures time-evolving interactions. It introduces a decoupling-based neural system (DNS) where sub-systems evolve under Neural Controlled Dynamics and a meta-system of interactions is constrained by projected differential equations on a simplex, using neural-friendly projections based on congestion-aware Bregman divergences. The core contributions are the explicit decoupling mechanism, the ProjDE-based interpretation of dynamic interactions, and extensive experiments showing superior performance over state-of-the-art methods on synthetic and real-world tasks with irregular sampling. This approach offers robust handling of irregular data and cluttered observations, with flexible integration into diverse sequential modeling tasks and potential extensions to learn latent entities from data.

Abstract

A complex system with cluttered observations may be a coupled mixture of multiple simple sub-systems corresponding to latent entities. Such sub-systems may hold distinct dynamics in the continuous-time domain; therein, complicated interactions between sub-systems also evolve over time. This setting is fairly common in the real world but has been less considered. In this paper, we propose a sequential learning approach under this setting by decoupling a complex system for handling irregularly sampled and cluttered sequential observations. Such decoupling brings about not only subsystems describing the dynamics of each latent entity but also a meta-system capturing the interaction between entities over time. Specifically, we argue that the meta-system evolving within a simplex is governed by projected differential equations (ProjDEs). We further analyze and provide neural-friendly projection operators in the context of Bregman divergence. Experimental results on synthetic and real-world datasets show the advantages of our approach when facing complex and cluttered sequential data compared to the state-of-the-art.

Learning to Decouple Complex Systems

TL;DR

The paper tackles learning dynamics from cluttered, irregular sequential data by decoupling a complex system into latent sub-systems and a meta-system that captures time-evolving interactions. It introduces a decoupling-based neural system (DNS) where sub-systems evolve under Neural Controlled Dynamics and a meta-system of interactions is constrained by projected differential equations on a simplex, using neural-friendly projections based on congestion-aware Bregman divergences. The core contributions are the explicit decoupling mechanism, the ProjDE-based interpretation of dynamic interactions, and extensive experiments showing superior performance over state-of-the-art methods on synthetic and real-world tasks with irregular sampling. This approach offers robust handling of irregular data and cluttered observations, with flexible integration into diverse sequential modeling tasks and potential extensions to learn latent entities from data.

Abstract

A complex system with cluttered observations may be a coupled mixture of multiple simple sub-systems corresponding to latent entities. Such sub-systems may hold distinct dynamics in the continuous-time domain; therein, complicated interactions between sub-systems also evolve over time. This setting is fairly common in the real world but has been less considered. In this paper, we propose a sequential learning approach under this setting by decoupling a complex system for handling irregularly sampled and cluttered sequential observations. Such decoupling brings about not only subsystems describing the dynamics of each latent entity but also a meta-system capturing the interaction between entities over time. Specifically, we argue that the meta-system evolving within a simplex is governed by projected differential equations (ProjDEs). We further analyze and provide neural-friendly projection operators in the context of Bregman divergence. Experimental results on synthetic and real-world datasets show the advantages of our approach when facing complex and cluttered sequential data compared to the state-of-the-art.
Paper Structure (39 sections, 2 theorems, 31 equations, 6 figures, 10 tables)

This paper contains 39 sections, 2 theorems, 31 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Sequential Learning.
  4. Learning with Independence.
  5. Methodology
  6. Background
  7. Neural Controlled Dynamics.
  8. Self-attention.
  9. Projected DEs.
  10. Learning to Decouple
  11. Learning Sub-systems.
  12. Learning Interactions.
  13. Experiments
  14. Three Body
  15. Spring
  16. ...and 24 more sections

Key Result

Proposition 1

The solution of Eq. (eq: sparsemax) is of the form: where $\tau: \mathbb{R}^K \rightarrow \mathbb{R}$ is the unique function that satisfies $\sum_j [\mathbf{z}_j - \tau(\mathbf{z})]_{+} = 1$ for every $\mathbf{z}$. Furthermore, $\tau$ can be expressed as follows. Let $\mathbf{z}_{(1)} \geq \mathbf{z}_{(2)} \geq \dots \geq \mathbf{z}_{(K)}$ be the sor , where $S(\mathbf{z}) := \{ j \in [K] | \math

Figures (6)

  • Figure 1: Comparsion of softmax and L2 projection onto a simplex. We see that the softmax projection trends to project onto the "center" of the simplex while the L2 projection trends to project onto the corner.
  • Figure 2: A figure showing the corresponding three-body trajectory (on the top), as well as the evolution over time on interactions (at the bottom) between three latent sub-systems in a Three-Body environment. Timestamp from 5 to 12.
  • Figure 3: A figure showing the focus of 3 sub-systems on 9-dimensional input of Three Body. The strength of focus is reflected by the thickness of the lines.
  • Figure 4: Visualization of the evolution of the meta-systems of DNS and DNSG on Spring dataset. On each time stamp $t$, from top to bottom, we show the trajectory of the 5 balls, the meta-system state of DNS, and the meta-system state of DNSG, respectively.
  • Figure 5: A figure showing the importance of each feature vector entry for subsystems
  • ...and 1 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof