Scalable Data Mixer Model

• Scalable Data Mixer is a model for high-dimensional matrix time series that decomposes autoregressive effects into low-rank and sparse components.
• The approach uses convex regularization and an efficient block coordinate minimization algorithm to ensure global convergence and tractable computation.
• Empirical studies demonstrate that this method improves forecasting accuracy and recovers interpretable network structures in complex datasets.

Upscaling ARDM (Additive Regularized Matrix Autoregressive Model) refers to algorithmic, computational, and statistical strategies for extending the ARDM framework to handle high-dimensional matrix-valued time series where the ambient dimensions $d_1$, $d_2$ and time horizon $T$ are all large, and $d_1d_2 \gg T$ is a realistic regime. The methodological innovation of ARDM is the additive decomposition of autoregressive effects, explicit convex regularization for low-rank plus sparse structure in transition matrices, and a scalable block-minimization procedure with statistical consistency guarantees, as established in (Ghosh et al., 2 Jun 2025).

1. Model Formulation and Structural Decomposition

The foundational ARDM assumes observations $\{Y_t\}_{t=1}^T$ with each $Y_t \in \mathbb{R}^{d_1 \times d_2}$ and posits an additive first-order autoregressive structure: $Y_t = A Y_{t-1} + Y_{t-1} B^\top + E_t,$ where $A \in \mathbb{R}^{d_1 \times d_1}$ models row-wise temporal dependence, $B \in \mathbb{R}^{d_2 \times d_2}$ column-wise dependence, and $E_t$ is matrix-valued white noise. To induce parsimony and interpretability in the high-dimensional regime, the transition matrices are decomposed as

$d_2$0

with $d_2$1 low-rank matrices capturing dominant global modes, and $d_2$2 sparse matrices encoding idiosyncratic, fine-scale effects.

This additive structure, as opposed to multiplicative or Tucker decompositions, offers improved interpretability and computational tractability in high dimensions, and admits direct convex formulations for parameter estimation (Ghosh et al., 2 Jun 2025).

2. Convex Regularized Estimation at Scale

Upscaling ARDM utilizes convex regularization to enforce low-rank and sparse components: $d_2$3 where $d_2$4 denotes the nuclear norm (convex surrogate for rank) and $d_2$5 induces entrywise sparsity. This convexity is crucial for tractable computation in high $d_2$6, $d_2$7 settings and guarantees convergence to a global optimum.

Complexity per iteration for each block update is $d_2$8 for SVD-based low-rank proximal steps and $d_2$9 for LASSO-type sparse updates, making the procedure scalable for $T$0 up to several hundreds per (Ghosh et al., 2 Jun 2025).

3. Identifiability and Statistical Guarantees

Unique separation of low-rank and sparse effects is ensured by (a) restricted strong convexity of the empirical risk function, (b) incoherence/spikiness conditions on low-rank components,

$T$1

to prevent confounding with sparse structure, and (c) regularization terms calibrated to dominate stochastic noise present in high-dimensions.

The regularized estimator $T$2 achieves with high probability: $T$3 where $T$4, $T$5 (Ghosh et al., 2 Jun 2025). This reflects a finite-sample error bound matching minimax rates for matrix regression models with structured parameterization.

4. Algorithmic Scaling: Alternating Block Minimization

The upscaled ARDM employs a block coordinate minimization algorithm:

• Sequentially update $T$6 holding the others fixed.
• $T$7 updates via proximal SVD (efficient for low-rank).
• $T$8 updates via convex LASSO subproblems.
• Each subproblem is solved exactly and algorithmic global convergence is guaranteed by joint convexity.

Per-iteration memory and runtime are dominated by SVDs of $T$9 matrices and LASSO on $d_1d_2 \gg T$0 variables, introducing practical upper bounds for $d_1d_2 \gg T$1 in the hundreds to low thousands per computational budget.

5. Empirical Performance and Interpretation

Simulation studies and macroeconomic panel data applications demonstrate the impact of upscaling ARDM:

• In moderate to high dimensions ($d_1d_2 \gg T$2), ARDM attains low relative error and sharp support recovery (high sensitivity/specificity for sparse patterns).
• In real data (Euro-zone macroeconomics, $d_1d_2 \gg T$3), ARDM recovers interpretable network structure with ranks $d_1d_2 \gg T$4, $d_1d_2 \gg T$5, and edge densities $d_1d_2 \gg T$6 (rows), $d_1d_2 \gg T$7 (columns).
• Forecasting (RMSE) consistently improves over bilinear MAR and vectorized sparse VAR benchmarks across multiple prediction horizons (Ghosh et al., 2 Jun 2025).

Sparsity patterns in $d_1d_2 \gg T$8, $d_1d_2 \gg T$9 can be visualized as networks for structural interpretation (e.g., crisis and boom signals in macro panels).

6. Practical Implementation Considerations

Key implementation points for scaling ARDM in practice:

• Initialization: random or identity for low-rank, zero for sparse, with convergence robust to choice.
• Regularization parameters: tuned via AIC or cross-validation, with standard scaling rules for $\{Y_t\}_{t=1}^T$0 derived from theory.
• Numerical stability: algorithm can leverage proximal/SVD acceleration, sparse storage, and parallelization for large $\{Y_t\}_{t=1}^T$1.
• Extension beyond first-order: Higher-order ARDMs require extension of the loss and block structure but maintain similar convexity and scaling principles.

A plausible implication is that ARDM can serve as a backbone for interpretable, scalable modeling of temporally-dependent relational data in finance, macroeconomics, or other matrix time series settings where both global latent structure and local perturbations are of scientific interest.

7. Comparison to Alternative High-Dimensional Matrix Models

ARDM's upscaling contrasts with alternative approaches:

• Bilinear MAR and Tucker tensor models are non-convex and computationally intensive for large $\{Y_t\}_{t=1}^T$2.
• Convex vectorized VARs lose interpretability (collapse matrix structure).
• ARDM achieves convexity, structured parameter estimates, and efficient global convergence with block minimization, offering more practical scalability and interpretability than these alternatives (Ghosh et al., 2 Jun 2025).

In summary, upscaling ARDM is feasible through a combination of explicit additive model structure, convex regularized loss, block-wise optimization, and calibrated regularization, enabling tractable statistical estimation and scientific inference in high-dimensional matrix time series contexts. The approach is empirically validated to provide finite-sample statistical guarantees and improved predictive performance over both traditional and alternative modern matrix autoregressive methods.