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Functional Bayesian Tucker Decomposition for Continuous-indexed Tensor Data
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Proposes FunBaT, a Bayesian Tucker decomposition for continuous-indexed tensors that places state-space GP (SDE) priors on mode-wise latent functions and infers with conditional expectation propagation plus Kalman filtering/RTS smoothing.
Problem
Standard Tucker decomposition assumes discrete, finite indexes per mode, but many real tensors (e.g., latitude, longitude, time) have continuous indexes; discretization loses information and existing functional tensor methods are deterministic or limited to TT/CP.
Approach
Model each Tucker factor as a vector-valued latent function with Matérn GP priors converted to state-space (SDE) form, and infer the posterior efficiently with conditional expectation propagation (CEP) message passing combined with a Kalman filter and RTS smoother, yielding linear-cost probabilistic inference and interpolation at arbitrary indexes.
Key results
- On BeijingAir-PM2.5 with R=2 (original resolution), FunBaT reports RMSE 0.288±0.008 versus 0.942 for the best FTT baseline (FTT-cross) and about 1.0 for standard Tucker baselines.
- On US-TEMP, FunBaT-CP reaches RMSE 0.548 at R=5 versus 1.415 to 2.959 for baselines at comparable ranks, while the dense-core FunBaT is worse (1.182 at R=5), which the authors attribute to overfitting on sparse data.
- Standard discrete Tucker baselines vary strongly with discretization resolution (e.g., P-Tucker RMSE 0.805 at 50x50x150 vs 1.256 at original resolution on PM2.5 at R=2).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use continuous-index FunBaT rather than the strongest listed continuous-index baseline, FTT-cross | FunBaT on BeijingAir-PM2.5 · BeijingAir-PM2.5 | RMSE: 0.942±0.025 -> 0.288±0.008 | Helped |
| Use continuous-index FunBaT rather than the strongest listed continuous-index baseline, FTT-cross | FunBaT on BeijingAir-PM10 · BeijingAir-PM10 | RMSE: 0.933±0.012 -> 0.328±0.004 | Helped |
| Use continuous-index FunBaT rather than the strongest listed continuous-index baseline, FTT-cross | FunBaT on BeijingAir-SO2 · BeijingAir-SO2 | RMSE: 0.844±0.026 -> 0.386±0.01 | Helped |
| Use continuous-index FunBaT rather than P-Tucker | FunBaT on US-TEMP · US-TEMP (R=3) | RMSE: 1.306±0.02 -> 1.255±0.108 | Helped |
| Use a diagonal CP core instead of the dense Tucker core (ablation) | FunBaT on US-TEMP · US-TEMP (R=3) | RMSE: 1.255±0.108 -> 0.805±0.06 | Helped |
| Use a diagonal CP core instead of the dense Tucker core (ablation) | FunBaT on BeijingAir-PM2.5 · BeijingAir-PM2.5 | RMSE: 0.288±0.008 -> 0.296±0.018 | Hurt |
| Use a diagonal CP core instead of the dense Tucker core (ablation) | FunBaT on BeijingAir-PM10 · BeijingAir-PM10 | RMSE: 0.328±0.004 -> 0.343±0.028 | Hurt |
| Use a diagonal CP core instead of the dense Tucker core (ablation) | FunBaT on BeijingAir-SO2 · BeijingAir-SO2 | RMSE: 0.386±0.01 -> 0.386±0.009 | No clear effect |
Surprising findings
- The dense Tucker-core model (FunBaT) underperforms its CP variant (FunBaT-CP) on the sparse US-TEMP dataset, suggesting overfitting from the dense core.
- Standard Tucker methods on discretized tensors do not consistently improve with finer resolution, and Tucker-ALS fails (RMSE >10) on US-TEMP at higher ranks.
Limitations
- Dense Tucker core is prone to overfitting on sparse data, where the CP variant performs better.
- Inference is approximate (CEP with delta-method moment approximation) and the Matérn kernel hyperparameters are selected per dataset rather than learned fully.
Experimental setup
Datasets: BeijingAir-PM2.5 BeijingAir-PM10 BeijingAir-SO2 US-TEMP synthetic rank-1 two-mode tensor
Benchmarks: BeijingAir-PM2.5 BeijingAir-PM10 BeijingAir-SO2 US-TEMP
Metrics: RMSE MAE
Baselines: P-Tucker Tucker-ALS Tucker-SVI FTT-ALS FTT-ANOVA FTT-cross RBF-SVM BLR
Builds on: Bayesian continuous-time tensor train (BCTT) conditional expectation propagation state-space Gaussian processes Tucker decomposition
Closest papers by meaning
- Bayesian Continuous-Time Tucker Decomposition (ICML 2022 · similarity 0.91)
- DinTucker: Scaling Up Gaussian Process Models on Large Multidimensional Arrays (AAAI 2016 · similarity 0.89)
- Functional Complexity-adaptive Temporal Tensor Decomposition (NeurIPS 2025 · similarity 0.88)
- Efficient Nonparametric Tensor Decomposition for Binary and Count Data (AAAI 2024 · similarity 0.86)
- Geometry-aware training of factorized layers in tensor Tucker format (NeurIPS 2024 · similarity 0.86)
- Deep Rank-One Tensor Functional Factorization for Multi-Dimensional Data Recovery (AAAI 2025 · similarity 0.85)
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