SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Moreover, it is worth noting that the aggregate discrepancy of SF-CCA exhibits a certain degree of sensitivity beyond a threshold of six subgroups. Beyond this point, the effectiveness of SF-CCA in reducing disparity begins to diminish. Conversely, MF-CCA remains consistently unaffected by variations in the number of subgroups, consistently maintaining its fairness performance. This analysis underscores the robustness and scalability of MF-CCA in handling an increasing number of subgroups, as it consistently maintains fairness irrespective of this parameter. On the other hand, SF-CCA demonstrates effective fairness mitigation but exhibits limitations when faced with a large number of subgroups beyond a certain threshold.
B.2.2 Sensitivity of Correlation and Disparity Error to  $r$
Table 4 presents the numerical results of three measures described in Section 4.1, namely correlation  $(\rho_r)$ , maximum disparity  $(\Delta_{\mathrm{max},r})$ , and aggregate disparity  $(\Delta_{\mathrm{sum},r})$ . The results are with respect to
(a) Correlation of CCA
(b) Correlation of MF-CCA
(c) Correlation of SF-CCA
Figure 7: Visualization of the canonical correlation results on synthetic data for the total five projection dimensions  $(r)$ . All the methods are applied to both the entire dataset and individual subgroups. The closer each subgroup's curve is to the overall curve, the better.
(b) Correlation of MF-CCA on NHANES (Education)
(a) Correlation of CCA on NHANES (Education)
(d) Correlation of CCA on NHANES (Race)
Figure 8: Visualization of the canonical correlation results of NHANES (Education & Race) for the total five projection dimensions  $(r)$ . All the methods are applied to both the entire dataset and individual subgroups. The closer each subgroup's curve is to the overall curve, the better.
(c) Correlation of SF-CCA on NHANES (Education)
(e) Correlation of MF-CCA on NHANES (Race)
(f) Correlation of SF-CCA on NHANES (Race)
the first  $r$  projection dimensions. It is an extension of Table 1. Specifically, we present the results of the first seven projection dimensions for synthetic data; the first five projection dimensions for NHANES and ADNI; and the first two projection dimensions for MHAAPS, respectively. Overall, we have consistent results and conclusions in Table 1. Firstly, MF-CCA and SF-CCA show substantial improvements in fairness compared to CCA with mild compromises of correlation. Secondly, SF-CCA outperforms MF-CCA in terms of fairness improvement, although it sacrifices correlation. This highlights the effectiveness of the single-objective optimization approach in SF-CCA. F-CCA consistently performs well across these datasets, confirming its inherent scalability.
(a) Correlation of CCA on MHAAPS (Sex)
(b) Correlation of MF-CCA on MHAAPS (Sex)
(c) Correlation of SF-CCA on MHAAPS (Sex)
Figure 9: Visualization of the canonical correlation results of MHAAPS (Sex) for the total two projection dimensions  $(r)$ . All the methods are applied to both the entire dataset and individual subgroups. The closer each subgroup's curve is to the overall curve, the better.
Table 5 quantifies the improvement in fairness and the minor compromises in accuracy presented in Table 4 by depicting the percentage performance shift from baseline CCA to MF-CCA and SF-CCA. The table is a reference and an extension of Figure 4 in Section 4.1. Figure 4 only visualizes the percentage shifts of the second and fifth projection dimension for Synthetic Data, NHANES, and ADNI and the first two projection dimensions for MHAAPS regarding the results presented in Table 1 for coherence. In Table 5, the metrics are formulated as:  $P\rho_{r} = \frac{\rho_{r}\text{of F - CCA} - \rho_{r}\text{of CCA}}{\rho_{r}\text{of CCA}}\times 100$ ,  $P\Delta_{\max ,r} = -\frac{\Delta_{\max,r}\text{of F - CCA} - \Delta_{\max,r}\text{of CCA}}{\Delta_{\max,r}\text{of CCA}}\times 100$ ,  $P\Delta_{\mathrm{sum},r} = -\frac{\Delta_{\mathrm{sum},r}\text{of F - CCA} - \Delta_{\mathrm{sum},r}\text{of CCA}}{\Delta_{\mathrm{sum},r}\text{of CCA}}\times 100$ , where F-CCA is switched to MF-CCA and SF-CCA to obtain the corresponding shift results. The formulations are designed regarding the properties of the metrics where  $\rho_{r}$  is the larger the better while  $\Delta_{\mathrm{max},r}$  and  $\Delta_{\mathrm{sum},r}$  are the smaller the better. According to the table, compared to CCA, MF-CCA, and SF-CCA sacrifice slightly the performance in correlation  $(\rho_{r})$  in exchange for significant fairness improvements in terms of maximum disparity  $(\Delta_{\mathrm{max},r})$  and aggregate disparity  $(\Delta_{\mathrm{sum},r})$  across all datasets in general, which demonstrate the good performance of F-CCA.
We also provide visualizations demonstrating how the correlation changes as the dimensionality changes. Figures 7, 8, 9, and Figure 10 display these correlation curves. Our expectation is that the correlation curves of our methods will show a stronger tendency to concentrate around the overall correlation curve compared to the baseline method. The findings indicate that the utilization of both MF-CCA and SF-CCA approaches yields a convergence of subgroup performance towards the overall canonical correlation. Furthermore, the overall canonical correlations derived from these two models exhibit a comparable level of performance to that of the CCA model.
Upon examining the synthetic data in Figure 7, we observe a clear concentration tendency. However, in the case of real data, this tendency is less pronounced. This is due to the fact that the disparities between different groups in the real data are already quite small. As a result, all the dashed curves in the CCA are already close to the solid red curve.
B.2.3 Sensitivity of Correlation and Disparity of SF-CCA to  $\lambda$
The SF-CCA experiment incorporates the hyperparameter  $\lambda$  to strike a balance between correlation and fairness considerations. This hyperparameter allows us to modulate the emphasis placed on fairness within the SF-CCA model. Figure 11 provides a visual representation of the effects of different  $\lambda$  values on both correlation and fairness metrics.
The illustration clearly demonstrates that increasing the magnitude of  $\lambda$  enhances the emphasis on fairness within the SF-CCA model. Consequently, this leads to a reduction in aggregate disparity, indicating improved fairness. Conversely, lower values of  $\lambda$  prioritize correlation, emphasizing the preservation of the underlying relationship between the datasets.
This experimental finding aligns with the discussion presented in Section 3 of our work, where we discussed the trade-off between correlation and fairness. By adjusting the value of  $\lambda$ , we can effectively control the balance between these two objectives in the SF-CCA model.
(a) Correlation of CCA on ADNI AV45/AV1451 (sex)
(b) Correlation of MF-CCA on ADNI AV45/AV1451 (sex)
(d) Correlation of CCA on ADNI AV45/Cognition (Sex)
(c) Correlation of SF-CCA on ADNI AV45/AV1451 (sex)
(e) Correlation of MF-CCA on ADNI AV45/Cognition (Sex)
(f) Correlation of SF-CCA on ADNI AV45/Cognition (Sex)
Figure 10: Visualization of the canonical correlation results of ADNI for the total five projection dimensions  $(r)$ . All the methods are applied to both the entire dataset and individual subgroups. The closer each subgroup's curve is to the overall curve, the better.
B.3 Runtime Sensitivity of MF-CCA and SF-CCA
When it comes to the issue of running time, it is important to consider the trade-off between computational efficiency and hyperparameter tuning effort. While SF-CCA may require more effort in tuning its hyperparameters, it still demonstrates a notable advantage in terms of time complexity compared to MF-CCA. To conduct a sensitivity analysis concerning the sample count  $(N)$ , the feature count  $(d)$ , and the subgroup count  $(K)$ , we conduct a series of experiments on synthetic datasets, ensuring the hyperparameters remained constant. These experiments are performed in three distinct settings, each replicated ten times, employing techniques including CCA, MF-CCA, and SF-CCA. The specifics of outcomes are depicted in Figures 12, 13, and 14.
B.3.1 Runtime Sensitivity of MF-CCA and SF-CCA to  $d$
In the first experimental setup, with the sample size held constant at  $N = 2000$  and the number of groups held constant at  $K = 5$ , the dimensionality of features is varied across the set [50, 100, 150, 200, 250, 300, 350, 400]. According to Figure 12, it is discerned that as feature dimensionality increases, the runtime of MF-CCA exhibits a concomitant augmentation. Conversely, the runtime associated with SF-CCA remains comparatively stable throughout the varying feature sizes.
B.3.2 Runtime Sensitivity of MF-CCA and SF-CCA to  $N$
In the subsequent experimental configuration, the feature size is held constant at  $d = 100$ , while the sample size varies across the set [600, 800, 1000, 1200, 1400, 1600, 1800, 2000]. Notably, according to Figure 13, the increment in the sample size exerts a minimal impact on the runtime duration of SF-CCA. This observation can be rationalized by considering that an increase in the number of features would lead to a covariance matrix of greater dimensionality. Consequently, the
Table 4: Numerical results regarding three metrics including Correlation  $(\rho_r)$ , Maximum Disparity  $(\Delta_{\mathrm{max},r})$ , and Aggregate Disparity  $(\Delta_{\mathrm{sum},r})$ . The best ones in each row are bold, and the second best one is underlined. The analysis focuses on the initial five projection dimensions for NHANES and ADNI, the initial seven projection dimensions for Synthetic Data, and the initial two projection dimensions for MHAAPS. “↑” means the larger the better and “↓” means the smaller the better.
Dataset	Dim. (r)	ρr↑	Δmax,r↓	Δsum,r↓
CCA	MF-CCA	SF-CCA	CCA	MF-CCA	SF-CCA	CCA	MF-CCA	SF-CCA
Synthetic Data	1	0.8003	0.7934	0.7757	0.3013	0.2386	0.1829	2.8647	2.2836	1.7258
2	0.7533	0.7475	0.7309	0.3555	0.2866	0.2241	3.3802	2.8119	2.2722
3	0.7032	0.6980	0.6853	0.3215	0.2591	0.2033	3.0974	2.5135	1.9705
4	0.5872	0.5818	0.5677	0.3784	0.2791	0.197	3.6619	2.6939	1.8134
5	0.4717	0.4681	0.4581	0.4385	0.3313	0.2424	4.1649	3.1628	2.2304
6	0.4186	0.4161	0.4131	0.1163	0.0430	0.0000	1.161	0.4142	0.0000
7	0.4012	0.3989	0.3928	0.2156	0.1336	0.0488	2.0224	1.1771	0.4117
NHANES (Education)	1	0.7360	0.7305	0.7330	0.0149	0.0113	0.0092	0.0596	0.0450	0.0366
2	0.6392	0.6360	0.6334	0.0485	0.0359	0.0245	0.1941	0.1435	0.0980
3	0.6195	0.6163	0.6147	0.0423	0.0322	0.0187	0.1691	0.1287	0.0747
4	0.5027	0.4961	0.4990	0.0506	0.0342	0.0243	0.2022	0.1367	0.0974
5	0.4416	0.4393	0.4392	0.1001	0.0818	0.0824	0.4003	0.3272	0.3297
NHANES (Race)	1	0.7376	0.7359	0.7365	0.0482	0.0479	0.0468	0.1927	0.1916	0.1874
2	0.6400	0.6374	0.6383	0.0101	0.0097	0.0048	0.0402	0.0389	0.0191
3	0.6221	0.6216	0.6195	0.0739	0.0708	0.0608	0.2955	0.2834	0.2432
4	0.5001	0.4990	0.4980	0.0887	0.0774	0.0685	0.3549	0.3096	0.2742
5	0.4459	0.4454	0.4442	0.1561	0.1451	0.1413	0.6244	0.5805	0.5653
MHAAPS (Sex)	1	0.4464	0.4451	0.4455	0.0093	0.0076	0.0044	0.0187	0.0152	0.0088
2	0.1534	0.1529	0.1526	0.0061	0.0038	0.0019	0.0122	0.0075	0.0039
ADNI AV45 and AV1451 (Sex)	1	0.8472	0.8468	0.8450	0.0190	0.0180	0.0165	0.038	0.036	0.0329
2	0.7778	0.7776	0.7753	0.0131	0.0119	0.0064	0.0263	0.0238	0.0127
3	0.7369	0.7360	0.7332	0.0460	0.0371	0.0269	0.0919	0.0743	0.0538
4	0.7022	0.7003	0.6985	0.0083	0.0046	0.0018	0.0167	0.0092	0.0037
5	0.6810	0.6798	0.6770	0.0477	0.0399	0.0324	0.0954	0.0799	0.0648
ADNI AV45 and Cognition (Sex)	1	0.7391	0.7386	0.7373	0.0146	0.0149	0.0135	0.0293	0.0299	0.0270
2	0.5232	0.5224	0.5223	0.1212	0.1127	0.1116	0.2424	0.2254	0.2233
3	0.5189	0.5185	0.5182	0.1568	0.1508	0.1497	0.3135	0.3017	0.2994
4	0.4999	0.4979	0.4985	0.1242	0.1061	0.1084	0.2485	0.2123	0.2168
5	0.4904	0.4886	0.4896	0.0249	0.0124	0.0165	0.0498	0.0248	0.0329
corresponding numerical computations, such as the resolution of eigenvalues, become increasingly time-intensive.
B.3.3 Runtime Sensitivity of MF-CCA and SF-CCA to  $K$
Finally, we examined the sensitivity of both SF-CCA and MF-CCA to the number of subgroups. Synthetic data was generated with varying numbers of subgroups, allowing us to assess the corresponding running time. The results are presented in Figure 14. From the figure, it is evident that conventional CCA is not significantly sensitive to the number of groups. However, MF-CCA exhibits a higher level of sensitivity to the number of groups compared to SF-CCA.
C Experimental Details and Hyperparameter Selection Procedure
In this section, we delve into the particulars of our experiments, which encompass the choice of the penalty function and hyperparameters. We initiate our discussion with the penalty function  $\phi$  outlined in Equation (8). This function takes on various forms, such as absolute, square, and exponential functions. For our experimental purposes, we concentrate specifically on the absolute function.
The distinctive characteristic of the absolute function lies in its robustness against minor discrepancies, as opposed to the square function, which tends to converge rapidly towards zero. It's worth noting
Table 5: Performance change from baseline CCA to F-CCA algorithms (MF-CCA and SF-CCA) in the format of percentage regarding three metrics: Correlation  $(\rho_r)$ , Maximum Disparity  $(\Delta_{\max, r})$ , and Aggregate Disparity  $(\Delta_{\mathrm{sum}, r})$ . The changes are computed using the numerical results from Table 4:  $P\rho_r = \frac{\rho_r \text{ of F-CCA} - \rho_r \text{ of CCA}}{\rho_r \text{ of CCA}} \times 100$ ,  $P\Delta_{\max, r} = -\frac{\Delta_{\max, r} \text{ of F-CCA} - \Delta_{\max, r} \text{ of CCA}}{\Delta_{\max, r} \text{ of CCA}} \times 100$ ,  $P\Delta_{\mathrm{sum}, r} = -\frac{\Delta_{\mathrm{sum}, r} \text{ of F-CCA} - \Delta_{\mathrm{sum}, r} \text{ of CCA}}{\Delta_{\mathrm{sum}, r} \text{ of CCA}} \times 100$ . Here, F-CCA is replaced with either MF-CCA or SF-CCA to obtain the results in each column. The formulations are determined based on the properties of the metrics where  $\rho_r$  is the larger the better while  $\Delta_{\max, r}$  and  $\Delta_{\mathrm{sum}, r}$  are the smaller the better.
Dataset	Dim.	Pρr	PΔmax,r	PΔsum,r
(r)	MF-CCA	SF-CCA	MF-CCA	SF-CCA	MF-CCA	SF-CCA
Synthetic Data	1	-0.8688	-3.0817	20.8021	39.2877	20.2834	39.7572
2	-0.7726	-2.9710	19.3625	36.9495	16.8132	32.7792
3	-0.7366	-2.5461	19.3916	36.7677	18.8495	36.3813
4	-0.9243	-3.3148	26.245	47.9246	26.4333	50.4784
5	-0.7660	-2.8896	24.4426	44.7114	24.0612	46.4484
6	-0.5865	-1.3126	63.0023	99.9997	64.3202	99.9997
7	-0.5767	-2.0838	38.0292	77.3821	41.7986	79.6436
NHANES (Education)	1	-0.7511	-0.4184	24.5248	38.5621	24.5248	38.5621
2	-0.5131	-0.9070	26.0899	49.5206	26.0899	49.5206
3	-0.5243	-0.7835	23.8580	55.7921	23.8580	55.7921
4	-1.3118	-0.7309	32.4151	51.833	32.4151	51.833
5	-0.5175	-0.5422	18.2615	17.6268	18.2615	17.6268
NHANES (Race)	1	-0.2329	-0.1450	0.5481	2.7539	0.5481	2.7539
2	-0.3972	-0.2625	3.3229	52.6573	3.3229	52.6573
3	-0.0931	-0.4209	4.1097	17.7082	4.1097	17.7082
4	-0.2334	-0.4317	12.7708	22.7437	12.7708	22.7437
5	-0.1113	-0.3914	7.0315	9.4561	7.0315	9.4561
MHAAPS (Sex)	1	-0.2917	-0.2084	18.6150	52.8984	18.6150	52.8984
2	-0.2724	-0.4941	38.2692	68.1768	38.2692	68.1768
ADNI AV45 and AV1451 (Sex)	1	-0.0444	-0.2604	5.3213	13.4811	5.3213	13.4811
2	-0.0222	-0.3178	9.5217	51.4446	9.5217	51.4446
3	-0.1294	-0.4999	19.1779	41.4136	19.1779	41.4136
4	-0.2648	-0.5307	44.6624	77.8152	44.6624	77.8152
5	-0.1737	-0.5918	16.288	32.0918	16.288	32.0918
ADNI AV45 and Cognition (Sex)	1	-0.0631	-0.2416	-2.0189	7.7392	-2.0189	-7.7392
2	-0.1436	-0.1690	7.0116	7.8657	7.0116	7.8657
3	-0.0832	-0.1386	3.7799	4.5125	3.7799	4.5125
4	-0.4074	-0.2842	14.5679	12.7538	14.5679	12.7538
5	-0.3663	-0.1533	50.094	33.8674	50.094	33.8674
that due to the non-differentiability of the absolute function at the origin, its identification within the MATLAB environment requires the utilization of the sign function. When the input to the penalty function reaches zero, the sign function indicates that the discrepancy error has already reached zero, leading to the termination of the training process.
C.1 Hyperparameters Selection
When it comes to selecting hyperparameters, the key decisions revolve around providing the appropriate learning rate and the parameter  $\lambda$  within the SF-CCA framework. Our approach consists of a two-step process. Firstly, we ascertain the optimal  $\lambda$  for each dataset through an extensive sensitivity analysis, as detailed in Section B.2.3. With these derived values in hand, we then conduct a comprehensive grid search to determine the best-fitting learning rate. The unique parameter combinations utilized for each experimental set are elucidated as follows:
- Synthetic Data:  $\lambda = 10$ , learning rate of 2e-2 in SF-CCA, and 4e-1 in MF-CCA.
- NHANES (Education):  $\lambda = 5$ , learning rate of 2e-3 in SF-CCA, and 5e-2 in MF-CCA.
(e) ADNI AV45/AV1451 (sex)
(d) MHAAPS (sex)
(f) ADNI AV45/Cognition (sex))
Figure 11: Sensitivity of correlation and disparity error to  $\lambda$  in SF-CCA framework. Higher  $\lambda$  emphasizes fairness over correlation (accuracy). Moving right to left, accuracy drops as fairness improves (smaller disparity). A notable trend links higher correlation with reduced fairness.
- NHANES (Race):  $\lambda = 5$ , learning rate of 1e-3 in SF-CCA, and 2e-2 in MF-CCA.
- MHAAPS:  $\lambda = 10$ , learning rate of 2e-2 in SF-CCA, and 4e-1 in MF-CCA.
- ADNI (AV45 and AV1451):  $\lambda = 5$ , learning rate of 1e-3 in SF-CCA, and 1e-2 in MF-CCA.
- ADNI (AV45 and Cognition):  $\lambda = 5$ , learning rate of 1e-3 in SF-CCA, and 2e-2 in MF-CCA.
C.2 Fairness and Correlation Measures
The correlation and fairness between  $\mathbf{X}$  and  $\mathbf{Y}$  under projections of  $\mathbf{U}$  and  $\mathbf{V}$  within  $R$ -dimensional spaces can be quantitatively measured by
$$
\rho = \frac {\operatorname {t r a c e} \left(\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y} \mathbf {V}\right)}{\sqrt {\operatorname {t r a c e} \left(\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X} \mathbf {U}\right) \operatorname {t r a c e} \left(\mathbf {V} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y} \mathbf {V}\right)}}, \tag {21a}
$$
$$
\Delta_ {\max } = \max  _ {i, j \in [ K ]} \left| \mathcal {E} ^ {i} (\mathbf {U}, \mathbf {V}) - \mathcal {E} ^ {j} (\mathbf {U}, \mathbf {V}) \right|, \tag {21b}
$$
$$
\Delta_ {\mathrm {s u m}} = \sum_ {i, j \in [ K ]} \left| \mathcal {E} ^ {i} (\mathbf {U}, \mathbf {V}) - \mathcal {E} ^ {j} (\mathbf {U}, \mathbf {V}) \right|. \tag {21c}
$$
In Section 4.1, instead of employing matrix-based measurements (21), we adopted component-wise measurements (13) to facilitate detailed observations on each individual projection dimension  $r$ . In the following discussion, we demonstrate that the component-wise measurements, across all  $r$  dimensions, provide a more aggressive measurement approach compared to the matrix variants (21). In other words, it is evident that having large component-wise correlation values  $\rho_r$  and small
Figure 12: Computation time (mean±std) of 10 repeated experiments for the total three projection dimensions on synthetic data comprising four subgroups  $(K)$ . The number of samples is fixed at  $N = 2000$ , while the number of features varies.
Figure 13: Computation time (mean±std) of 10 repeated experiments for the total three projection dimensions on synthetic data comprising four subgroups  $(K)$ . The number of features is fixed at  $d = 100$ , and the number of groups is held constant at  $K = 5$ , while the number of samples varies.
disparity errors  $\Delta_{\mathrm{max},r}$  and  $\Delta_{\mathrm{sum},r}$  for all  $r\in [R]$  can guarantee large values of  $\rho$  and small values of  $\Delta_{\mathrm{max}}$  and  $\Delta_{\mathrm{sum}}$ .
The following lemma rigorously supports this observation.
Lemma 13. Let  $\rho_r, \Delta_{\max,r}, \Delta_{\mathrm{sum},r}$  be defined as in (13), and let  $\rho, \Delta_{\max}, \Delta_{\mathrm{sum}}$  be defined as in (21). Then,
$$
\rho = \frac {1}{R} \sum_ {r = 1} ^ {R} \rho_ {r}, \quad \Delta_ {\max } \leq \sum_ {r = 1} ^ {R} \Delta_ {\max , r}, \quad \Delta_ {\operatorname {s u m}} \leq \sum_ {r = 1} ^ {R} \Delta_ {\operatorname {s u m}, r}.
$$
Figure 14: Computation time (mean±std) of 10 repeated experiments for the total seven projection dimensions on synthetic data comprising varying numbers of subgroups  $(K)$ . The number of features is fixed at  $d = 100$ .
Proof. Since  $\mathbf{U} = [\mathbf{u}_1, \dots, \mathbf{u}_R] \in \mathbb{R}^{D_x \times R}$  and  $\mathbf{V} = [\mathbf{v}_1, \dots, \mathbf{v}_R] \in \mathbb{R}^{D_y \times R}$ , we have
$$
\mathrm {t r a c e} (\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y} \mathbf {V}) = \sum_ {r = 1} ^ {R} \mathbf {u} _ {r} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y} \mathbf {v} _ {r},
$$
$$
\operatorname {t r a c e} \left(\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X} \mathbf {U}\right) = \sum_ {r = 1} ^ {R} \mathbf {u} _ {r} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X} \mathbf {u} _ {r}, \tag {22}
$$
$$
\operatorname {t r a c e} (\mathbf {V} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y} \mathbf {V}) = \sum_ {r = 1} ^ {R} \mathbf {v} _ {r} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y} \mathbf {v} _ {r}.
$$
Note that  $\mathcal{U} = \{\mathbf{U}|\mathbf{U}^{\top}\mathbf{X}^{\top}\mathbf{X}\mathbf{U} = \mathbf{I}_{R}\}$  and  $\mathcal{V} = \{\mathbf{V}|\mathbf{V}^{\top}\mathbf{Y}^{\top}\mathbf{Y}\mathbf{V} = \mathbf{I}_{R}\}$ . Using the implementation of the retraction operation on  $\mathbf{U}$  and  $\mathbf{V}$  (see, (19) and (17)), we obtain
$$
\mathbf {u} _ {r} \mathbf {X} ^ {\top} \mathbf {X} \mathbf {u} _ {r} = \mathbf {v} _ {r} \mathbf {Y} ^ {\top} \mathbf {Y} \mathbf {v} _ {r} = 1,
$$
$$
\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X} \mathbf {U} = \mathbf {V} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y} \mathbf {V} = \mathbf {I} _ {R}, \tag {23}
$$
$$
\mathrm {t r a c e} (\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X U}) \mathrm {t r a c e} (\mathbf {V} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y V}) = \mathrm {t r a c e} (\mathbf {I} _ {R}) ^ {2} = R ^ {2}.
$$
Thus, for the component-wise measure  $\rho_r$  defined in (13a) and the matrix variant  $\rho$  defined in (21a), we have
$$
\begin{array}{l} \rho = \frac {\operatorname {t r a c e} (\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y V})}{\sqrt {\operatorname {t r a c e} (\mathbf {U} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X U}) \operatorname {t r a c e} (\mathbf {V} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y V})}} \\ = \frac {\sum_ {r = 1} ^ {R} \mathbf {u} _ {r} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y} \mathbf {v} _ {r}}{\sqrt {R ^ {2}}} \\ = \frac {1}{R} \sum_ {r = 1} ^ {R} \frac {\mathbf {u} _ {r} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y} \mathbf {v} _ {r}}{1} \tag {24} \\ = \frac {1}{R} \sum_ {r = 1} ^ {R} \frac {\mathbf {u} _ {r} ^ {\top} \mathbf {X} ^ {\top} \mathbf {Y} \mathbf {v} _ {r}}{\sqrt {\mathbf {u} _ {r} ^ {\top} \mathbf {X} ^ {\top} \mathbf {X} \mathbf {u} _ {r} \mathbf {v} _ {r} ^ {\top} \mathbf {Y} ^ {\top} \mathbf {Y} \mathbf {v} _ {r}}} \\ = \frac {1}{R} \sum_ {r = 1} ^ {R} \rho_ {r}. \\ \end{array}
$$
For component-wise measure  $\Delta_{\max,r}$  and matrix measure  $\Delta_{\max}$ , we have