[Paper Review] MTC: Multiresolution Tensor Completion from Partial and Coarse Observations

Chaoqi Yang, Navjot Singh+4 more|arXiv (Cornell University)|Jun 14, 2021

Tensor decomposition and applications29 references4 citations

TL;DR

This paper proposes MTC, a multiresolution tensor completion method that jointly leverages partial fine-granular observations and coarse-granular aggregated data to recover low-rank tensors. By using a coupled alternating least squares framework with multiresolution initialization, MTC achieves 65.20% and 75.79% percentage of fitness (PoF) on two COVID-19 spatio-temporal datasets with only 5% observed elements, outperforming baselines by 27.96% relative improvement.

ABSTRACT

Existing tensor completion formulation mostly relies on partial observations from a single tensor. However, tensors extracted from real-world data are often more complex due to: (i) Partial observation: Only a small subset (e.g., 5%) of tensor elements are available. (ii) Coarse observation: Some tensor modes only present coarse and aggregated patterns (e.g., monthly summary instead of daily reports). In this paper, we are given a subset of the tensor and some aggregated/coarse observations (along one or more modes) and seek to recover the original fine-granular tensor with low-rank factorization. We formulate a coupled tensor completion problem and propose an efficient Multi-resolution Tensor Completion model (MTC) to solve the problem. Our MTC model explores tensor mode properties and leverages the hierarchy of resolutions to recursively initialize an optimization setup, and optimizes on the coupled system using alternating least squares. MTC ensures low computational and space complexity. We evaluate our model on two COVID-19 related spatio-temporal tensors. The experiments show that MTC could provide 65.20% and 75.79% percentage of fitness (PoF) in tensor completion with only 5% fine granular observations, which is 27.96% relative improvement over the best baseline. To evaluate the learned low-rank factors, we also design a tensor prediction task for daily and cumulative disease case predictions, where MTC achieves 50% in PoF and 30% relative improvements over the best baseline.

Motivation & Objective

• To address tensor completion in real-world scenarios where data is both partially observed and available in coarse-grained forms.
• To fuse heterogeneous data sources—fine-granular partial observations and coarse-granular aggregated views—into a unified low-rank tensor completion framework.
• To improve computational efficiency and optimization accuracy by designing a multiresolution initialization and a compact coupled-ALS solver.
• To evaluate the model on real-world spatio-temporal health data, particularly for disease case prediction and imputation.
• To demonstrate that coarse data can significantly enhance completion performance when combined with sparse fine-grained observations.

Proposed method

• MTC formulates a coupled tensor completion problem using Frobenius norm minimization over both partial and coarse observations.
• It employs multiresolution factorization to recursively initialize low-rank components using coarse-to-fine tensor views.
• The optimization uses an alternating least squares (ALS) solver that packs multiple linear systems into joint normal equations for computational efficiency.
• The method leverages mode-specific aggregation patterns and hierarchical resolution structures to guide initialization and improve convergence.
• A two-stage optimization process is used: stage 1 performs Jacobi iterations on the coarse-grained system, and stage 2 applies Cholesky-based least squares on the fine-grained data.
• The model ensures low space and time complexity by avoiding full gradient computation and using iterative, sparse-aware solvers.

Experimental results

Research questions

• RQ1 Can coarse-granular tensor data significantly improve the accuracy of tensor completion when fine-grained data is partially observed?
• RQ2 How can multiresolution tensor factorization be used to generate effective initializations for low-rank tensor completion?
• RQ3 What is the impact of combining partial and coarse observations on completion performance compared to using only partial data?
• RQ4 How does MTC’s optimization strategy compare in convergence speed and accuracy to gradient-based baselines?
• RQ5 To what extent can the learned low-rank factors be reused for downstream prediction tasks like disease case forecasting?

Key findings

• MTC achieves 65.20% and 75.79% percentage of fitness (PoF) on two real-world COVID-19 spatio-temporal datasets with only 5% fine-grained observations, representing a 27.96% relative improvement over the best baseline.
• On the tensor prediction task for daily and cumulative disease cases, MTC achieves 50% PoF, a 30% relative improvement over the best baseline.
• The optimization of MTC runs approximately 1/8th the time of gradient-based baselines due to reduced floating-point operations and avoidance of line search.
• The multiresolution initialization enables MTC to surpass all baselines—including HOSVD and TendiB—within just 30 optimization iterations, even with only 1% observed data.
• The method shows greater advantage under higher missingness: performance gains are most pronounced when partial observations are sparse (e.g., 2.5% or 5%);
• The optimization landscape shows faster convergence and better final results for MTC compared to gradient-based methods, especially when data is scarce.