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Learning the Influence Graph of a High-Dimensional Markov Process with Memory

Smita Bagewadi, Avhishek Chatterjee

TL;DR

This work extends an existing algorithm for learning i.i.d. graphical models to this Markovian setting with memory and proves that it can learn the influence graph based on the binary observations using logarithmic samples when the degree of the influence graph is bounded.

Abstract

Motivated by multiple applications in social networks, nervous systems, and financial risk analysis, we consider the problem of learning the underlying (directed) influence graph or causal graph of a high-dimensional multivariate discrete-time Markov process with memory. At any discrete time instant, each observed variable of the multivariate process is a binary string of random length, which is parameterized by an unobservable or hidden [0,1]-valued scalar. The hidden scalars corresponding to the variables evolve according to discrete-time linear stochastic dynamics dictated by the underlying influence graph whose nodes are the variables. We extend an existing algorithm for learning i.i.d. graphical models to this Markovian setting with memory and prove that it can learn the influence graph based on the binary observations using logarithmic (in number of variables or nodes) samples when the degree of the influence graph is bounded. The crucial analytical contribution of this work is the derivation of the sample complexity result by upper and lower bounding the rate of convergence of the observed Markov process with memory to its stationary distribution in terms of the parameters of the influence graph.

Learning the Influence Graph of a High-Dimensional Markov Process with Memory

TL;DR

This work extends an existing algorithm for learning i.i.d. graphical models to this Markovian setting with memory and proves that it can learn the influence graph based on the binary observations using logarithmic samples when the degree of the influence graph is bounded.

Abstract

Motivated by multiple applications in social networks, nervous systems, and financial risk analysis, we consider the problem of learning the underlying (directed) influence graph or causal graph of a high-dimensional multivariate discrete-time Markov process with memory. At any discrete time instant, each observed variable of the multivariate process is a binary string of random length, which is parameterized by an unobservable or hidden [0,1]-valued scalar. The hidden scalars corresponding to the variables evolve according to discrete-time linear stochastic dynamics dictated by the underlying influence graph whose nodes are the variables. We extend an existing algorithm for learning i.i.d. graphical models to this Markovian setting with memory and prove that it can learn the influence graph based on the binary observations using logarithmic (in number of variables or nodes) samples when the degree of the influence graph is bounded. The crucial analytical contribution of this work is the derivation of the sample complexity result by upper and lower bounding the rate of convergence of the observed Markov process with memory to its stationary distribution in terms of the parameters of the influence graph.
Paper Structure (13 sections, 11 theorems, 42 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 13 sections, 11 theorems, 42 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contribution
  3. Organization
  4. Related Literature
  5. Data Driven Applications
  6. Analytical Guarantees
  7. Model
  8. Useful Mathematical Structures
  9. Algorithm
  10. Proofs
  11. Detailed Proofs
  12. Simulations
  13. Conclusion

Key Result

Lemma 1

The process $\mathbf{Y}^{(d)}(t)$ is a Markov chain. Moreover, for any node $v$ and at any time $t$, given $\mathbf{Y}^{(d)}_{\mathcal{N}_v \cup \{v\}}(t)$, $\mathbf{Y}^{(d)}_v(t+1)$ is independent of all other past process values $\{\mathbf{Y}^{(d)}_{V\setminus \{\mathcal{N}_v \cup \{v\}\}}(\tau):

Figures (2)

  • Figure 1: (a) Probability of perfect recovery for $|V|=7$,$d=1$, $\bar{M} = 0$, (b)Probability of perfect recovery for $|V|=7,d=1$, $\bar{M}=2$ and (c) Probability of perfect recovery for $|V|=7,d=2, \bar{M}=2$
  • Figure 2: Probability of perfect recovery for $|V|=7,\bar{M} = 2, d=2$ when $2(\Bar{\mu}+L)|\rho(\Tilde{A})|^{\frac{1}{d}} > 1$

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:TVepsilonLB']}
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • Lemma 6
  • proof : Proof of Theorem \ref{['thm:2']}:
  • proof : Proof of Lemma \ref{['lem:CondEntMarkov']}
  • ...and 11 more