Technical Summary: Direct and Efficient Estimation of Bilinear Forms in Staggered Tensor Panels

Problem Formulation

This paper addresses the estimation of bilinear forms from noisy, partially observed tensor data within the context of causal inference. The setting involves a signal tensor $M \in \mathbb{R}^{N \times T \times K}$ following a Tucker2 model with rank $(r, r, K)$. The signal is decomposed as $M = C \times_1 U \times_2 V \times_3 I_K$, where $U \in \mathbb{R}^{N \times r}$ and $V \in \mathbb{R}^{T \times r}$ are shared factor matrices across all $K$ layers (slices), and $C \in \mathbb{R}^{r \times r \times K}$ contains slice-specific core tensors.

The data is observed under a staggered adoption missingness pattern, common in causal panel data where units adopt a treatment at different times. Once treated, a unit remains treated in all subsequent periods, creating a "staircase" pattern of missing untreated potential outcomes. The authors focus on estimating general bilinear forms of the unobserved (counterfactual) blocks:
$$\mu_{xy}^{(k)} = x^\top M_{\cdot,\cdot,k}^{(d)} y$$
where $x$ and $y$ are unit vectors, and $M_{\cdot,\cdot,k}^{(d)}$ represents the unobserved bottom-right block in the $k$-th layer. This target includes causal estimands such as average treatment effects (ATE), individual counterfactuals, and policy-weighted aggregates.

The primary challenge is that existing literature largely focuses on full matrix/tensor recovery (minimizing Frobenius or entrywise error) rather than the direct estimation of specific functionals. Furthermore, while tensor methods exist for causal inference, they often lack theoretical guarantees for direct functional estimation under structured Missing Not At Random (MNAR) patterns.

Methodology

1. Four-Block Reduction and Spectral Algorithm

The authors first analyze the four-block missingness pattern, the fundamental building block of staggered designs. In this setting, each layer $j$ has observed blocks $a, b, c$ and a missing block $d$.

They propose Algorithm 1 (BilinearTensor4Block), a spectral estimator that targets the functional directly without reconstructing the full missing tensor. The algorithm leverages the shared latent subspaces ($U$ and $V$) across layers:

1. Pooled Matrix Construction: It constructs two pooled matrices by stacking observed blocks across all $K$ layers:
   • $Y^p_{\text{left}}$: Stacks the "left" observed blocks (combining blocks $a$ and $c$) to estimate the shared row subspace $U$.
   • $Y^p_{\text{up}}$: Stacks the "upper" observed blocks (combining blocks $a$ and $b$) to estimate the shared column subspace $V$.
2. SVD and Subspace Estimation: It computes rank-$r$ truncated Singular Value Decompositions (SVDs) of these pooled matrices to obtain estimators $\hat{U}_{\text{left}}$ and $\hat{V}_{\text{up}}$.
3. Direct Functional Estimation: Instead of imputing the missing block, the algorithm performs a "regress-then-denoise" procedure. It uses the estimated subspaces to map the observed block $b$ to the missing block $d$ via a spectral regression operator, specifically utilizing a clipped spectral inverse to stabilize the inversion of the Gram matrix $\hat{U}_{1k}^\top \hat{U}_{1k}$.

2. Extension to General Staggered Designs

For general staggered adoption designs (Section 5), the authors propose Algorithm 2 (BilinearTensorStaggered). This method reduces the complex staircase missingness to an auxiliary four-block problem by:

• Permuting rows to align with the target layer's adoption order.
• Identifying anchor sets (fully observed rows and columns) across all layers that correspond to the target missing block.
• Constructing auxiliary pooled matrices using only these anchor blocks, effectively isolating a four-block structure within the larger tensor.

Key Theoretical Results

1. Non-Asymptotic Error Bounds

The paper establishes a non-asymptotic mean squared error (MSE) bound for the proposed estimator (Theorem 1). The bound reveals a phase transition dependent on the number of layers $K$:

• Small $K$ Regime: The error is dominated by terms related to estimating the shared factors $U$ and $V$, which scale as $O(\sigma^2 \rho^{-1})$.
• Large $K$ Regime: As $K$ increases, the error terms related to $U$ and $V$ estimation decay (improving with pooling), but a layer-specific irreducible error term remains. This term, scaling as $O(\sigma^2 \min(N/N_{1k}, T/T_{1k}))$, reflects the difficulty of estimating the slice-specific core $C_{\cdot,\cdot,k}$, which cannot be improved by pooling across layers.

2. Local Minimax Lower Bounds

The authors derive local minimax lower bounds (Theorems 2 and 3) that match the upper bounds up to constants. These bounds confirm that the phase transition is inherent to the problem structure:

• Pooling information across layers is necessary to estimate the shared subspaces efficiently.
• However, once the shared subspaces are well-estimated, the precision of the functional estimate is limited by the signal-to-noise ratio within the specific target layer's missing block.

3. Computational Efficiency

The proposed method is computationally efficient. The dominant cost is the rank-$r$ SVD of the pooled matrices, scaling as $O(NT_{1,p}r + N_{1,p}Tr)$. Crucially, the algorithm computes the bilinear form directly, avoiding the $O(N_{1k}T_{2k}r)$ cost of constructing the full missing block. For multiple queries, the cost is further reduced by caching the pooled SVDs.

Significance and Contributions

The paper claims the following contributions:

1. Direct Functional Estimation: It shifts the focus from full tensor recovery to the direct estimation of causal functionals, which is often the actual goal in applications. This approach is shown to be statistically and computationally more efficient for these specific targets.
2. Tensor Generalization: It extends spectral methods for MNAR matrix completion (previously limited to $K=1$ or specific entrywise inference) to the tensor setting with multiple layers ($K \ge 1$), accommodating multiple treatments or outcomes.
3. Phase Transition Characterization: It provides a rigorous theoretical characterization of how pooling across layers improves estimation, identifying the precise point where the benefit of pooling saturates due to slice-specific heterogeneity.
4. Robustness to MNAR: The methodology is designed specifically for structured MNAR patterns (staggered adoption), a setting where standard matrix completion methods often fail or require stronger assumptions.

Empirical Validation

The authors validate their theory through:

• Simulations: Demonstrating the phase transition in error rates as $K$ increases and showing that the pooled estimator outperforms single-layer (no-pool) estimators, particularly in low-SNR regimes.
• Real-World Applications:
  • Castle Doctrine Laws: Estimating the effect of self-defense laws on crime rates across different outcomes (robbery, assault, etc.), showing that the tensor method reduces uncertainty compared to matrix-only methods.
  • COVID-19 Policies: Analyzing the impact of stay-at-home orders and contact tracing on deaths and cases, highlighting the method's ability to leverage information from related policy outcomes (different slices) to improve estimation in data-sparse target layers.

The paper concludes that while full tensor recovery is valuable, direct estimation of bilinear forms offers a superior trade-off between statistical efficiency and computational cost for causal inference tasks involving structured missingness.