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Time Series Generative Learning with Application to Brain Imaging Analysis

Zhenghao Li, Sanyou Wu, Long Feng

TL;DR

Time Series Generative Learning with Application to Brain Imaging Analysis develops a nonparametric framework to generate future brain image sequences by learning a generator under a Markov and conditional invariance assumption. It formulates iterative generation $\widehat{X}_{T+s} = g(\eta_{T+s-1},\widehat{X}_{T+s-1})$ (with $\widehat{X}_T = X_T$) and a parallel $s$-step form $\widetilde{X}_{T+s} = G(\eta_T,X_T,s)$, trained via an $f$-divergence–based min–max objective implemented with deep networks. The authors prove convergence of both joint and pairwise distributions of generated sequences to the true distributions under mild regularity, extend the approach to lag-$k$ and panel data settings, and validate the method on ADNI MRI data, demonstrating its potential for data augmentation in Alzheimer’s disease analysis. Across simulations and real data, iterative and $s$-step generations show strong performance in nonlinear settings, and the ADNI study reveals brain-aging patterns that the generated sequences can help augment downstream diagnostic tasks.

Abstract

This paper focuses on the analysis of sequential image data, particularly brain imaging data such as MRI, fMRI, CT, with the motivation of understanding the brain aging process and neurodegenerative diseases. To achieve this goal, we investigate image generation in a time series context. Specifically, we formulate a min-max problem derived from the $f$-divergence between neighboring pairs to learn a time series generator in a nonparametric manner. The generator enables us to generate future images by transforming prior lag-k observations and a random vector from a reference distribution. With a deep neural network learned generator, we prove that the joint distribution of the generated sequence converges to the latent truth under a Markov and a conditional invariance condition. Furthermore, we extend our generation mechanism to a panel data scenario to accommodate multiple samples. The effectiveness of our mechanism is evaluated by generating real brain MRI sequences from the Alzheimer's Disease Neuroimaging Initiative. These generated image sequences can be used as data augmentation to enhance the performance of further downstream tasks, such as Alzheimer's disease detection.

Time Series Generative Learning with Application to Brain Imaging Analysis

TL;DR

Time Series Generative Learning with Application to Brain Imaging Analysis develops a nonparametric framework to generate future brain image sequences by learning a generator under a Markov and conditional invariance assumption. It formulates iterative generation (with ) and a parallel -step form , trained via an -divergence–based min–max objective implemented with deep networks. The authors prove convergence of both joint and pairwise distributions of generated sequences to the true distributions under mild regularity, extend the approach to lag- and panel data settings, and validate the method on ADNI MRI data, demonstrating its potential for data augmentation in Alzheimer’s disease analysis. Across simulations and real data, iterative and -step generations show strong performance in nonlinear settings, and the ADNI study reveals brain-aging patterns that the generated sequences can help augment downstream diagnostic tasks.

Abstract

This paper focuses on the analysis of sequential image data, particularly brain imaging data such as MRI, fMRI, CT, with the motivation of understanding the brain aging process and neurodegenerative diseases. To achieve this goal, we investigate image generation in a time series context. Specifically, we formulate a min-max problem derived from the -divergence between neighboring pairs to learn a time series generator in a nonparametric manner. The generator enables us to generate future images by transforming prior lag-k observations and a random vector from a reference distribution. With a deep neural network learned generator, we prove that the joint distribution of the generated sequence converges to the latent truth under a Markov and a conditional invariance condition. Furthermore, we extend our generation mechanism to a panel data scenario to accommodate multiple samples. The effectiveness of our mechanism is evaluated by generating real brain MRI sequences from the Alzheimer's Disease Neuroimaging Initiative. These generated image sequences can be used as data augmentation to enhance the performance of further downstream tasks, such as Alzheimer's disease detection.
Paper Structure (30 sections, 19 theorems, 178 equations, 4 figures, 3 tables)

This paper contains 30 sections, 19 theorems, 178 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Existence and estimation
  3. Convergence analysis for Lag-1 time series
  4. General bounds for iterative generation
  5. General bounds for $s$-step generation
  6. Analysis of deep neural network spaces
  7. Generalizations to lag-k time series
  8. Further generalizations to panel data
  9. Convergence analysis for $T_i\to \infty$
  10. Convergence analysis for $n\to\infty$ and $T$ is finite
  11. Simulation studies
  12. Study I: Single Time Series
  13. Study II: Multiple Time Series
  14. The ADNI study
  15. Proofs
  16. ...and 15 more sections

Key Result

Theorem 1

Let $X_0,X_1,\cdots,$ be a sequence of random variables which satisfy: for any $t \ge 1$. Suppose $\widehat{X}_0^0 \overset{\text{d}}{=} X_0$ and $\eta_0,\eta_1,\cdots$ be a sequence of independent $m$-dimensional Gaussian vector. Then there exist a a measurable function $g$ such that the sequence satisfies that for any $s\ge 1$,

Figures (4)

  • Figure 1: The original, iterative and s-step generated MRI sequence $s=0,1,\ldots, 7$ along with their difference to $X_0$ for one subject.
  • Figure 2: An illustration of the generated samples for three subjects in the testing group. For each subject, we present from left to right: starting image $X_{0}$, target $X_{s}$, and two generated samples $\widehat{X}_{s}$ and ${\widetilde{X}}_{s}$ by iterative and s-step generation respectively. Moreover, we highlight (with different colors) four different regions that $X_{s}$ (along with $\widehat{X}_{s}$ and ${\widetilde{X}}_{s}$) are most different from $X_0$, including: a) cortical sulci, b) ventricles, c) edge of ventricles, d) anterior interhemispheric fissure.
  • Figure 3: The mean SSIM, PSNR, and NRMSE over all testing subjects in the ADNI data analysis for $s=1,\ldots, 9$.
  • Figure 4: The visualization of $\phi_1$, $\phi_e$, and $\Sigma_{\infty}$ of Case 1 in the simulation study.

Theorems & Definitions (41)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Remark
  • Remark
  • Proposition 1
  • Example 1
  • Theorem 3
  • Corollary 1
  • Remark
  • ...and 31 more