Improving Deep Regression with Tightness

Thanks for the valuable comments, especially the design choices

Dear Reviewer CpDH, Thank you for your insightful and detailed comments. For the weakness: **W1: Formulating the global and local tightness would strengthen the theoretical analysis.** Thank you for pointing this out. We have revised the text to clarify this point. Please refer to the revised Section 4 with blue text. **W2: About the design choices of the proposed methods** - **For MT, is there a possibility that the multiple regressors will collapse into a single solution? Are the solution spaces orthogonal:** As discussed in Section 5.4 (Multiple $\theta$s) and Section 3.3, the multiple regressors do not collapse into a single one in reality, as the regressor directions remain almost identical throughout training. This stability arises because the representations are distributed around the regressor weight $\theta$, offsetting each other and thereby having a minimal impact on the direction of the regressor. The solution spaces are not orthogonal to each other, but they are orthogonal to the regressor weight $\theta$ if the regressor is linear. - **For ROT-Reg, is the self-entropic optimal transport the best choice? For example, one may simply minimize the gap of affinity matrices in Eq. (9).** The current design is effective but not necessarily the best. Simply minimizing the gap of affinity matrices (i.e., $T^z$ and $T^y$) can introduce optimization challenges, as $T^z$ and $T^y$ are obtained iteratively through the Sinkhorn algorithm rather than simply through gradient backpropagation. In contrast, ROT-Reg is updating $C^z$ through gradient backpropagation to minimize $L_{ot}$. We have revised Section 4.2 to clarify this point (see the corresponding blue text). **W3. The performance improvement by the proposed method is marginal. For instance, in Tab. 2, why did incorporating MT and ROT-Reg underperform MT?** Our method achieves higher improvement than various methods on three datasets. For instance, on AgeDB-DIR, our method outperforms the previous SOTA method, RankSim, with an improvement of +0.52 (ALL) compared to RankSim's +0.18 (ALL). The lower performance on some metrics comes from the trade-off between the head (with many samples) and tail (with few samples), which is common in imbalanced cases. Thus, although incorporating both MT and ROT-Reg achieves the best performance on RMSE ALL, it shows a lower performance on the head part (Many & Med.), reflecting the trade-off. **W4: It would be beneficial to visualize the improvement of tightness not only in toy datasets but also in real-world datasets with e.g., PCA.** The visualization of the tightness in Figure 3 is conducted on the real-world dataset NYUD2-DIR for depth estimation. We have revised our text to state this explicitly. **W5: Displaying the regression performance in Tab. 4 would be helpful to strengthen the efficacy of tightness on the performance.** Thanks for pointing this out, we have revised Table 4 to display the regression performance (i.e., RMSE ALL). **W6: There are many typos.** Thanks for pointing this out, we have revised some typos including those you mentioned. We appreciate your valuable comments, especially the design choices, which have helped us improve the quality and clarity of our work.

Thanks for the valuable suggestions

**Q1. Intuitively, predicting the same target from the same feature seems to make $\theta$s go to the same solution. Describing the reasons why the directions of $\theta$s remain almost identical during training would make the paper more convincing.** Thank you for your intuition, which aligns with our findings regarding the updating directions of $\theta$s (see discussed in Section 5.4). However, $\theta$s do not collapse into a single solution in practice. This is because **$\theta$s are randomly initialized**, and remain almost identical throughout training (See Figure 4(b) for empirical results), for **three reasons** we outline below. For reference, recall $\frac{\partial L_{reg}}{\partial \theta} = \frac{1}{b}\sum_{i=1}^b w_iz_i^T$, where $w_i = L'_{reg}(\theta^T z_i - y_i)$: - **(Update magnitude) The magnitude of $\frac{\partial L_{reg}}{\partial \theta}$ is 'scaled' by $w_i$**, since $w_i$ often follows a Gaussian distribution centered at the origin, as assumed in models like Bayesian linear regression. When $w$ and $z$ are independent or weak dependent, $\mathbb{E}[w_iz_i]$ will approach 0 and causing $\frac{\partial L_{reg}}{\partial \theta}$ be 'scaled' to 0. - **(Accumulated updates) According to the central limit theorem, the updates of $\theta$ follow a Gaussian distribution. This causes partial offsets between updates and results in a reduced accumulated effect.** In addition, we empirically observe the mean of this Gaussian distribution approaches 0 (see Figure 4(c)), indicating that $w$ and $z$ are independent or weakly dependent and the **accumulated updates approach 0 (i.e., $\theta$ remain almost identical throughout training)**. - **(Update Direction) The effect of $z_i$ on the direction of $\frac{\partial L_{reg}}{\partial \theta}$ over a batch of $b$ samples offsets each other** and resulting in the stability of the $\theta$ direction throughout training. This occurs because $z_i$ tends to be distributed around $\theta$, as discussed in the paragraph following Equation 3. As discussed above, the updates of $\theta$s are limited and thus $\theta$s remain almost identical throughout training. Thanks for this valuable suggestion. We have revised the text in Section 5.4 to clearly explain these reasons. **Q2. For ROT-Reg, a more straightforward loss (e.g., $||C^Z - C^Y ||_F$) could encourage consistency.** Directly minimizing $||C^Z - C^Y ||_F$ imposes an overly strict constraint on the feature manifold, forcing it to become identical to the target space. This is unnecessary, as the feature manifold only needs to be homeomorphic to the target space [1]. We once experimented with this approach, and it did not yield good results. A similar approach has also been implemented by RTD[2], where they minimize the gap between the two matrices on randomly selected edges: https://github.com/danchern97/RTD_AE/blob/714d1b78c6afa976229a08210e3870db4c69f794/src/top_ae.py#L214. Minimizing the gap between $C^Z$ and $C^Y$ only on paired edges (i.e., edges of the minimal spanning trees) is shown to be more effective, as demonstrated by the topological autoencoder. However, as discussed in Section 4.2, compared to the topological autoencoder, our ROG-Reg captures more local structures of targets when $\gamma>0$. Furthermore, ROG-Reg also performs better than the topological autoencoder, as shown in Table 3. Thanks for pointing this out, we have revised Section 4.2 to clarify this point. [1] S. Zhang, K. Kawaguchi, and A. Yao. Deep Regression Representation with Topology. ICML. 2024 [2] I. Trofimov, et al. Learning topology-preserving data representations. ICLR. 2023

Thanks for the careful review and constructive suggestions

Dear Reviewer QZxF, Thank you for the careful review and constructive suggestions. We are happy that the novelty and contribution of our work are recognized.

Thanks for the insightful feedback

Dear Reviewer uwbB, Thank you for your insightful and detailed comments. For the weakness: **W1: Although the authors discuss why minimizing Mean Squared Error (MSE) may fail to learn ordinal feature spaces, they do not provide empirical results or visualizations, such as t-SNE plots, to support this claim. Comparative visualizations between the proposed method and RankSim would strengthen the discussion.** Thanks for pointing this out. We do provide empirical results and visualizations to support this claim and have now revised our text to state this more explicitly. Specifically, Table 4 shows that regression does not effectively preserve ordinality, as indicated by lower Spearman’s and Kendall’s values. However, our proposed method and RankSim are both comparable in preserving ordinality. Figure 2 provides visualizations of the feature manifold on the toy dataset for coordinate prediction, showing that our method results in a more tightened and ordinal preserved feature manifold. Figure 3 shows similar results on a real-world dataset for depth estimation. **W2. Eq 3 and Eq 5 should both be from the batch level. For example, change N to b.** Thanks for highlighting this. We have revised the text accordingly. **W3. Regarding the multiple regressors:** - **The rationale for employing multiple regressors is unclear:** The use of multiple regressors is grounded in our theoretical analysis in Section 3. It is based on our observation that in classification, using multiple classifiers enhances the compression of representations in several directions. Multiple regressors serve to achieve a similar and analogous effect. - **Why is a multi-regressor approach potentially better than a single one:** Similar to the role of multiple classifiers in classification, multiple regressors help tighten the representations in multiple directions, as supported by our theoretical and empirical results. - **Is this analogous to an ensemble method:** No, it is not analogous, although the mean operation may introduce a potential ensemble effect. As discussed in Section 5.5 (mean $\hat{y}$ vs. single $\hat{y}$), a single $\hat{y}$ performs nearly the same as ensembling multiple regressors (i.e., mean $\hat{y}$). Therefore, the improvement is not due to the ensemble effect and the method is not analogous to an ensemble method. - **Will the different ranges of Y affect the selection of M:** No. The ranges of Y have a limited impact on the selection of M because the primary factor is the intrinsic dimension of the feature manifold, which determines how many additional constraints (i.e., $M-1$) are required to compress the manifold (see Section 4.1 for more details). Our empirical results also indicate this. While the ranges of Y are quite different on NYUD2-DIR (y $\in$ [0.7, 10]) and AgeDB-DIR ( y $\in$ [0, 101]), $M=8$ works well on both datasets. We have revised our text to state the above points more explicitly. Please refer to the revised Section 5.5 with blue text. **W4. In [1], the authors discuss the neural collapse phenomenon can be similarly achieved by deep networks when minimizing MSE, which seems somewhat contradictory to statements made in the current paper. Is this contrast due to the nature of the target label (discrete versus continuous)? Expanding on this point with further discussion and comparison would add clarity.** Our statements do not contradict the neural collapse phenomenon in [1]. Although the model in [1] uses MSE, it is still like a classification model with multiple classifiers/regressors, which allow representation compression in multiple directions. In cases of neural collapse of regression, the feature manifold will collapse into a single line when the target space is a line and the compression is maximized [2]. This trend can be observed in Figure 6, where the feature manifold looks like a thick line and evolves toward a thinner line over training. However, standard regression’s limited ability to tighten representations results in a slower collapse. In contrast, our proposed method and RankSim both accelerate this collapse, as shown in Figure 3. We appreciate your insightful feedback, especially the neural collapse, and have included the above discussion in our Section C.2 to improve the clarity and depth of this work. [1] X.Y. Han, et al. Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path. ICLR. 2022 [2] S. Zhang, K. Kawaguchi, and A. Yao. Deep Regression Representation with Topology. ICML. 2024