# Learning Conditional Independence Differential Graphs From Time-Dependent Data

arXiv: https://arxiv.org/abs/2512.06960 · HTML: https://mlsanity.com/paper/2512.06960/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 7 Dec 2025.

## Builds on (most cited first, 19 of 19)

- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [Large Scale Graph Learning from Smooth Signals](https://mlsanity.com/paper/1710.05654/) (2017)
- [Support recovery without incoherence: A case for nonconvex regularization](https://mlsanity.com/paper/1412.5632/) (2014)
- [Autoregressive Models for Matrix-Valued Time Series](https://mlsanity.com/paper/1812.08916/) (2018)
- [Structured Graph Learning for Clustering and Semi-supervised Classification](https://mlsanity.com/paper/2008.13429/) (2020)
- [Graphical LASSO Based Model Selection for Time Series](https://mlsanity.com/paper/1410.1184/) (2014)
- [Differential Network Analysis via the Lasso Penalized D-Trace Loss](https://mlsanity.com/paper/1511.09188/) (2015)
- [A Direct Approach for Sparse Quadratic Discriminant Analysis](https://mlsanity.com/paper/1510.00084/) (2015)
- [Estimating Differential Latent Variable Graphical Models with Applications to Brain Connectivity](https://mlsanity.com/paper/1909.05892/) (2019)
- [FuDGE: A Method to Estimate a Functional Differential Graph in a High-Dimensional Setting](https://mlsanity.com/paper/2003.05402/) (2020)
- [Testing for Differences in Gaussian Graphical Models: Applications to Brain Connectivity](https://mlsanity.com/paper/1512.08643/) (2015)
- [On Sparse High-Dimensional Graphical Model Learning For Dependent Time Series](https://mlsanity.com/paper/2111.07897/) (2021)
- [Frequency Domain Statistical Inference for High-Dimensional Time Series](https://mlsanity.com/paper/2206.02250/) (2022)
- [Sparse Graph Learning Under Laplacian-Related Constraints](https://mlsanity.com/paper/2111.08161/) (2021)
- [Fairness-Aware Estimation of Graphical Models](https://mlsanity.com/paper/2408.17396/) (2024)
- [Estimation of Large Covariance and Precision Matrices from Temporally Dependent Observations](https://mlsanity.com/paper/1412.5059/) (2014)
- [Statistical inference for high-dimensional spectral density matrix](https://mlsanity.com/paper/2212.13686/) (2022)
- [Learning High-Dimensional Differential Graphs From Multi-Attribute Data](https://mlsanity.com/paper/2312.03761/) (2023)
- [Learning Multi-Attribute Differential Graphs with Non-Convex Penalties](https://mlsanity.com/paper/2505.09748/) (2025)

## Abstract (arXiv)

Estimation of differences in conditional independence graphs (CIGs) of two time series Gaussian graphical models (TSGGMs) is investigated where the two TSGGMs are known to have similar structure. The TSGGM structure is encoded in the inverse power spectral density (IPSD) of the time series. In several existing works, one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data consisting of independent and identically distributed (i.i.d.) observations. In this paper we consider estimation of the difference in two IPSDs to characterize the underlying changes in conditional dependencies of two sets of time-dependent data. Our approach accounts for data time dependencies unlike past work. We analyze a penalized D-trace loss function approach in the frequency domain for differential graph learning, using Wirtinger calculus. We consider both convex (group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm) and graph recovery. Both synthetic and real data examples are presented in support of the proposed approaches. In synthetic data examples, our log-sum-penalized differential time-series graph estimator significantly outperformed our lasso based differential time-series graph estimator which, in turn, significantly outperformed an existing lasso-penalized i.i.d. modeling approach, with F1 score as the performance metric.