Thank you for your detailed and thorough feedback on our work. We greatly appreciate your insightful comments. Due to space constraints, we are unable to incorporate all of this content into the main text. Since URLs are not permitted (except for figures), we have instead included an updated PDF with detailed proofs in the supplementary materials, which has been uploaded to an anonymous repository. All referenced appendices are provided in full. We sincerely hope you will review these additional materials and reconsider our score.

W1

The idea of an indextree in references you given is to efficiently index spatial entities by hierarchically partitioning a region into proper granularity levels but coarser than entity level (containing multiple cities), implemented as a quadtree where child nodes separate the region of parent nodes, bring no additional information. Our work, neither motivated by nor following indextree, addresses the problem of using heterogeneous datasets to extrapolate knowledge to model unseen entities, a challenge indextree cannot solve due to its focus on summarizing known entities without extrapolation capabilities. To tackle this, we use finer granularity nodes (grids below city level), stores distinct information across parent-child levels from different granularity datasets (detailed discussion in Appendix A.9), and employs a segmenttree query method to fuse mixed-level embeddings, capturing different-level interactions. In summary, our Segment Quadtree is a new data structure designed for knowledge extrapolation, distinct from indextree’s indexing paradigm.

We listed more detailed foundational differences between indextree and Segment Quadtree are shown in Appendix A.7.

The "segment" in Segment Quadtree reflects its query method, inspired by segment trees, a data structure for efficiently computing range sums over continuous arrays. For example, in a segment tree with values (a1,a2, a3,a4), querying the sum of (a1, a2,a3) uses precomputed nodes like ([a1, a2]) and (a3), reducing computation. Similarly, in our Segment Quadtree in Figure 2(a), querying an yellow entity’s boundary fuses embeddings from mixed levels: one high-level node (Node3), one mid-level node (Node17), and two low-level nodes (Nodes69,71), rather than aggregating all lowest-level grids, as done for France in Figure 1(b). This multi-level fusion accelerates queries and leverages hierarchical information, distinguishing it from standard quadtrees like indextrees, where fusing across levels is less meaningful due to uniform node content.

W2

Due to constraints in a paper, we cannot conduct experiments across a wider range of scenarios. Therefore, we focused on two key cases to allow for a more thorough analysis. We acknowledge this limitation and will explore additional applications of our method in future work.

To address the air pollution datasets description concern, we have improved the dataset description for air pollution in Appendix A.1.

W3

We have added a comprehensive proof in Appendix A.5. Due to space constraints, we only summarize the key conclusions here. From the Rademacher Complexity Theorem, for any $\mathcal{H}$, the generalization error bound is $L(h) \leq L^S(h) + 2\hat{\mathcal{R}}_S(\mathcal{H}) + 3\sqrt{\frac{2\ln(2/\delta)}{n}}$, where $\hat{\mathcal{R}}_S(\mathcal{H})$ is the empirical Rademacher complexity. For Segment Quadtree Embedding, $\hat{\mathcal{R}}S(\mathcal{H}{\text{seg}}) \leq C_2 \cdot \frac{|W|^{L+1} \sqrt{p'} \sqrt{m d}}{\sqrt{n}}$, with $m = O(\log k)$ and $p' \propto m d$, giving a tighter bound for Segment Quadtree Embedding.

W4

Following your recommendation, we shifted from visualization-based analysis to a quantitative evaluation of embedding distances on all embeddings. Specifically, we computed distances between all pairs of embeddings and compared them with the corresponding time series distances of carbon emission amounts. For the carbon emission time series, we calculated the L2 norm between every pair of provinces. Similarly, for each model, we computed the L2 distances between all pairs of embeddings. We then assessed the correlation between these embedding distances and the real carbon emission distances using the Pearson correlation coefficient. The results demonstrate that the Pearson score for TimesNet w/Seg embeddings is significantly higher than that of MTGNN.

W5

We have written an related work section about Quadtree Applications in Spatial Data Management in Section 6.3.

W6

We have clarified the differences between the Index Tree and Segment Quadtree, added air pollution experiments, provided a detailed theoretical and complexity analysis, and quantative data scarcity in the original paper.

Thank you for your valuable feedback on our work. We greatly appreciate your insightful comments, including your detailed suggestions regarding the calculation of bounds, clarifications on data scarcity, and additional experiments. Due to space constraints, we are unable to incorporate all of this content into the main text. Since URLs are not permitted (except for figures), we have instead included an updated PDF with detailed proofs in the supplementary materials, which has been uploaded to an anonymous repository. All referenced appendices are provided in full. We sincerely hope you will review these additional materials and reconsider our score.

the calculation of bounds

We have added a comprehensive proof in Appendix A.5. Due to space constraints, we only summarize the key conclusions here. From the Rademacher Complexity Theorem, for any $\mathcal{H}$, the generalization error bound is $L(h) \leq L^S(h) + 2\hat{\mathcal{R}}_S(\mathcal{H}) + 3\sqrt{\frac{2\ln(2/\delta)}{n}}$, where $\hat{\mathcal{R}}_S(\mathcal{H})$ is the empirical Rademacher complexity. For Segment Quadtree Embedding, $\hat{\mathcal{R}}S(\mathcal{H}{\text{seg}}) \leq C_2 \cdot \frac{|W|^{L+1} \sqrt{p'} \sqrt{m d}}{\sqrt{n}}$, with $m = O(\log k)$ and $p' \propto m d$, giving a tighter bound for Segment Quadtree Embedding.

W1

Thank you for your valuable suggestion! We first clarify what data scarcity means in our work and then provide statistical evidence to support it. Due to space constraints, we have added detailed description in Appendix A.6.

In our paper, data scarcity refers to forecasting targets in the test set being unseen during pretraining, with no historical data available at that stage. This severely limits direct modeling of their relationships. To tackle this, SQGEF uses a large volume of heterogeneous datasets from related regions or entities, despite lacking test target historical records.To quantify this scarcity, we compare the data volume between the pretraining and test stages across three metrics: total data points, number of temporal points, and temporal duration.

From this table, it is evident that the test set contains significantly fewer data points than the pretraining set across all experiments. For China Province and Europe Country, the test set’s temporal duration is notably short(2 years). In the China City, test set has an extremely limited number of temporal records(37 points), indicating sparse sampling. Collectively, these statistics highlight the severe data scarcity in the test set, both in terms of quantity and coverage.

Moreover, Table 2 in our paper shows that this scarcity goes beyond data volume to informational richness. Models using only the scarce historical data of forecasting targets consistently lag behind SQGEF, which combines this limited target data with richer, more plentiful data from related grids and entities during pretraining.

W2

Thank you for your suggestion! This will help clarify the effect of different granularity datasets. We use only one grid dataset to train the model. As shown in the following table, using one single dataset significantly drops the performance, which verifies the validity of unifying various datasets.

Thank you for your valuable comments on our work. We greatly appreciate your feedback, including the detailed suggestions for analysis and clarification. However, due to space constraints, we cannot include all the content in the main text. Since URLs are not permitted except for figures, we have included the updated PDF with detailed proofs in the supplementary materials (uploaded to an anonymous repository). All referenced appendices are provided. We sincerely hope you will review these materials and reconsider our score.

Benefits of Multi-granularity

The ablation study w/o SE empirically shows that combining multi-granularity datasets is effective, as performance drops noticeably when this component is removed. To shed more light, we explain below the intuition behind why this approach boosts forecasting performance.

In SQGEF, combining multi-granularity datasets enhances forecasting by allowing each Quadtree node to capture info from its own and higher granularity levels. This hierarchical integration occurs during training, where a node’s queried embedding is computed by fusing its own embedding with those of all child nodes stored in the tree, as shown in Equation 1. When a high-level node gets supervision from a coarse-granularity dataset, its child nodes are updated simultaneously. This top-down approach ensures lower-level nodes benefit from the richer, global info in higher-level datasets.

This mechanism provides key benefits. First, it lets lower-level nodes gain global interactions beyond their limited local data. Second, it supports cross-scale training: lower-level nodes are improved using higher-level dataset supervision, allowing the model to capture nested relationships across granularities. Thus, SQGEF builds a richer representation of geographical entities, enhancing generalization to unseen entities or data-scarce regions.

W1

SQGEF introduces only constant additional computational complexity. During training, it maintains the same space and computational complexity as the naive method while capturing multi-granularity relationships for significant performance gains. During inference, SQGEF achieves lower complexity than the naive approach, ensuring superior scalability and efficiency.

Due to space constraints, the detailed analysis is available in the complexity analysis (Rebuttal, Reviewer cJrD) or Appendix A.8.

W2

We have added a comprehensive proof in Appendix A.5. Due to space constraints, we only summarize the key conclusions here. From the Rademacher Complexity Theorem, for any $\mathcal{H}$, the generalization error bound is $L(h) \leq L^S(h) + 2\hat{\mathcal{R}}_S(\mathcal{H}) + 3\sqrt{\frac{2\ln(2/\delta)}{n}}$, where $\hat{\mathcal{R}}_S(\mathcal{H})$ is the empirical Rademacher complexity. For Segment Quadtree Embedding, $\hat{\mathcal{R}}S(\mathcal{H}{\text{seg}}) \leq C_2 \cdot \frac{|W|^{L+1} \sqrt{p'} \sqrt{m d}}{\sqrt{n}}$, with $m = O(\log k)$ and $p' \propto m d$, giving a tighter bound for Segment Quadtree Embedding.

W3

We first clarify what data scarcity means in our work and then provide statistical evidence to support it. Due to space constraints, we have added detailed description in Appendix A.6.

In our paper, data scarcity refers to forecasting targets in the test set being unseen during pretraining, with no historical data available at that stage. This severely limits direct modeling of their relationships. To tackle this, SQGEF uses a large volume of heterogeneous datasets from related regions or entities, despite lacking test target historical records.To quantify this scarcity, we compare the data volume between the pretraining and test stages across three metrics: total data points, number of temporal points, and temporal duration.

From this table, it is evident that the test set contains significantly fewer data points than the pretraining set across all experiments. For China Province and Europe Country, the test set’s temporal duration is notably short(2 years). In the China City, test set has an extremely limited number of temporal records(37 points), indicating sparse sampling. Collectively, these statistics highlight the severe data scarcity in the test set, both in terms of quantity and coverage.

Moreover, Table 2 in our paper shows that this scarcity goes beyond data volume to informational richness. Models using only the scarce historical data of forecasting targets consistently lag behind SQGEF, which combines this limited target data with richer, more plentiful data from related grids and entities during pretraining.

Thank you for your thorough review! Your suggestion to include additional experiments and analysis will further strengthen our paper.

outdated baselines

Thank you for your suggestion! We have updated our experiments to include two state-of-the-art baselines: TimeXer (NeurIPS 2024) and iTransformer (ICLR 2024). The results demonstrate that our framework significantly enhances the performance of these cutting-edge baselines, underscoring its robustness and adaptability.

complexity analysis

Thank you for your insightful review! We agree that including an efficiency analysis strengthens our paper, and hightlights the effectiveness of our method that we only introduces a constant-level additional complexity to achieve significant performance increase.

Space Complexity: Both SQGEF and naive grid embedding method have a space complexity of O(N) (where N is the number of grid cells), SQGEF introduces only a constant-level overhead to store multi-level interaction information. In our Segment Quadtree, the total number of nodes is N (root level) + (1/4)N (next level) + ... + (1/4)^h N, where h = ⌈log₄N⌉ is the tree height. This geometric series sums to O((4/3)N), which remains linear in N. Thus, our approach scales efficiently even with higher-resolution grids, incurring minimal additional storage costs.

Training Stage Time Complexity: While both SQGEF and the naive grid embedding method have a computational complexity of O(N²) (where N is the number of grid cells), SQGEF adds only a constant overhead to process multi-granularity data. While SQGEF models interactions across all granularity levels, the additional computational cost is minor relative to the substantial performance gains it delivers. The complexity comprises two parts: (1) constructing a graph to capture spatial relations between grids, and (2) computing temporal relations across timestamps. For the graph, the naive method’s complexity is O(N²). In SQGEF, this becomes O(N² + (1/16)N² + ... + (1/16)^h N²) across tree levels, summing to O((16/15)N²), a slight increase over O(N²). Temporal relations are treated as a constant T, as no additional temporal computation is required beyond the input sequence. This modest overhead enables rich multi-level interactions, significantly enhancing forecasting accuracy, as evidenced by our experimental results.

Inference Stage Time Complexity: During inference, SQGEF outperforms the naive grid embedding method in efficiency. Using the segment tree query method [1]for a region [qX₁, qX₂] × [qY₁, qY₂], we recursively check overlaps with the current node’s region [X₁, X₂] × [Y₁, Y₂]. With a tree height of O(log₄N), and up to 4 nodes visited per level (if the query spans quadrants), the total nodes visited is bounded by 4·O(log₄N) = O(log N) (since log₄N = (1/2)log₂N). This contrasts with the naive grid method’s O(N) complexity, making our approach far more efficient.

Summary: During training, SQGEF maintains O(N) space complexity and O(N²) computational complexity, identical to the naive method, enabling the capture of multi-granularity relationships that yield significant performance improvements. In inference, our method achieves O(log N ) complexity, versus O(N) for the naive approach, ensuring superior scalability and efficiency. These trade-offs make SQGEF highly practical and advantageous, especially for numerous downstream spatio-temporal forecasting tasks, where its fast inference speed delivers substantial benefits across a wide range of applications.

[1] Lee, D. T., & Preparata, F. P. (1984). Computational Geometry: A Survey. IEEE Transactions on Computers.