🤖 AI Summary
This work addresses the estimation of bilinear forms from noisy, partially observed tensors under a staggered sampling design. The authors propose a spectral algorithm that directly estimates the target functional without requiring full tensor completion, leveraging cross-layer information aggregation. Built upon the Tucker2 model and incorporating an anchored four-block reduction technique, the method accommodates general staggered missingness patterns and reveals a phase transition phenomenon linking the number of layers to estimation performance. Theoretical analysis establishes a non-asymptotic error bound that matches the local minimax lower bound. Experimental results demonstrate that, when the phase transition condition is satisfied, the proposed approach significantly outperforms strategies that forgo information aggregation.
📝 Abstract
We study the estimation of bilinear forms from noisy, partially observed tensor data. The signal follows a Tucker2 model, with shared unit and time factors across tensor layers and slice-specific cores. The missingness pattern is structured and motivated by staggered adoption designs, which are common in causal inference and related applications. We first analyse the four-block missingness pattern, the basic building block for general staggered adoption, and propose a spectral algorithm that pools information across layers and targets the functional directly, rather than completing the entire tensor. We prove a non-asymptotic mean squared error bound that exhibits a phase transition in the number of layers, showing when pooling improves estimation, and match it with a local minimax lower bound up to constants. We then extend the construction to general staggered adoption designs via an anchored four-block reduction, and derive analogous theoretical guarantees. Finally, we validate our theoretical findings through experiments on both simulated and real-world datasets.