SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\operatorname {E x p l C o v} \left(L, \tau_ {1}, \tau_ {2}, \mathfrak {D}\right) = \frac {\left| \left\{x \in \mathfrak {D}: \left| M _ {\left[ \tau_ {1} , \tau_ {2} \right]} (x) \cap S (x , L) \right| > 0 \right\} \right|}{\left| \left\{x \in \mathfrak {D}: \left| M _ {\left[ \tau_ {1} , \tau_ {2} \right]} (x) \right| > 0 \right\} \right|} \tag {16}
$$
Label Masking The cosine similarity computed by comparing the neuron's activations when the model is fed by using the full input  $x$  and the masked input  $\theta(x, L)$ . A high score indicates a strong connection between the label and the activation range. Note that we only keep track of the changes in the regions identified by the activation range  $\tau_1, \tau_2$ .
$$
\operatorname {L a b M a s k} \left(L, \tau_ {1}, \tau_ {2}, \mathfrak {D}\right) = \frac {\sum_ {x \in \mathfrak {D}} \operatorname {C o s i n e S i m} \left(M _ {\left[ \tau_ {1} , \tau_ {2} \right]} ^ {k} (x) A ^ {k} (\theta (x , L)) , M _ {\left[ \tau_ {1} , \tau_ {2} \right]} ^ {k} (x) A ^ {k} (x)\right)}{| \{x \in \mathfrak {D}: | M _ {\left[ \tau_ {1} , \tau_ {2} \right]} (x) | > 0 \} |} \tag {17}
$$
For the experiments in this section, we follow the same setup of Mu and Andreas [30] with the addition of the set of thresholds used to compute the activation ranges. We fix the length of the explanation to 3, as commonly done in the current literature [28, 27]. For space reasons, in almost all the experiments in this section, we report the results using the last layer of ResNet18 [17] as a base model and Ade20k [44, 46] as a concept dataset. However, the claims hold across different architectures and concept datasets, as reported in Appendix B. We use K-Means as a clustering algorithm and fix the number of clusters to five. The choice of K-Means is motivated by the need for a fast clustering algorithm that aggregates activations associated with a shared semantic and that can be applied to a large quantity of data (see Appendix G for further details about the choice of the clustering algorithm). The number has been set to five because a higher number of clusters returns almost the same labels but repeated over more clusters, and, on average, the scores are lower (Appendix C).
4.2 Heuristic Effectiveness
This section compares the number of labels for which the algorithm computes the true IoU (visited states) needed by the vanilla CoEx algorithm, our heuristic, and alternative heuristics.
We select and test three heuristics using different amounts of the needed information to estimate the IoU score: the vanilla CoEx algorithm where no heuristics are used (CoEx); a heuristic that uses only the size of the label masks per sample (Areas); the Coordinates-Free Heuristic (CFH), an ablated version of our proposed heuristic that does not estimate the size of the label's mask; and our proposed heuristic (MMESH). Refer to Appendix A.2 for further details about the baselines.
Table 1 compares the average number of states visited during the computation of the baselines and our MMESH. The results are computed over 100 randomly selected units fixing  $T = \left\lbrack  {{\tau }^{\text{top }},\infty }\right\rbrack$  as
Table 1: Avg. and standard deviation of visited states per unit. Results are computed for 100 randomly extracted units.
Heuristic	Info from	Visited States
NetDissect	Dataset
CoEx	-	-	39656 ± 12659
+ Areas	-	✓	23602 ± 3420
+ CFH	✓	✓	5990 ± 3066
+ MMESH	✓	✓	129 ± 712
in the CoEx algorithm. We can observe that it is possible to lower the number of visited states by increasing the amount of information. The areas heuristic lowers the number of operations by a third and, since the heuristic uses only information from the dataset, it represents a valid option for making NetDissect faster. By adding the estimation of the intersection, we further reduce the number of visited states by one order of magnitude. Finally, MMESH, which adds the estimation of the label's mask, reaches the best result, reducing the number of visited states by two orders of magnitude.
Practically, this result means that, for each unit, we can generate explanations for clusters in the same amount of time (or less) as the vanilla compositional algorithm as long as the number of clusters is reasonably low. Indeed, while the CoEx algorithm takes, on average, more than  $\sim 60$  minutes per unit, our proposed MMESH takes less than 2 minutes.
4.3 Explanations Analysis
In this section, we analyze the explanations produced by NetDissect, CoEx, and our algorithm in terms of the desiderata qualities introduced in Section 3.2.
Table 2 shows that NetDissect and CoEx reach similar values in most considered scores, and the labels returned by CoEx are only slightly better than the NetDissect ones. The most significant margin can be observed in Sample Coverage, Detection Accuracy, and IoU. The higher IoU can be explained by the degenerate effect of increasing the formula's length [30]. The margin in Sample Coverage and Detection Accuracy means that the neuron fires in most of the samples annotated with labels returned by NetDissect. However, since the Detection Accuracy is lower, the overlap between activations and annotation is less consistent and more sparse. This is verified by the observation that most of the CoEx labels are connected by OR operators, thus increasing the number of candidate samples, and that most of the NetDissect labels are scene concepts (i.e., concepts that cover the full input), which are associated with a sparse overlapping due to the size of activation ranges. Looking at the average over the clusters reached by Clustered Compositional Explanations, the algorithm seems to return better labels with respect to almost all the desiderata qualities by a large margin. However, if we look at the average scores per cluster, we can note that the average is favored by Cluster 1 and Cluster 2. In Table 2, clusters are named progressively according to the activation ranges. Thus, Cluster 1 corresponds to the lowest activations and Cluster 5 to the highest activations.
First, we can note that Cluster 1 and 2 have an almost perfect Dataset and Sample Coverage. By the definition of the scores, this means that their labels cover the full dataset and that there is a strong connection between the activation range and the label. These extreme values motivated us to further investigate these clusters, which are discussed in the next section. Now, we can analyze Cluster 5, which includes the highest activations and, thus, also the range used by NetDissect and CoEx. We observe that with respect to CoEx, enlarging the range of activation has a marginal impact on Detection Accuracy and Label Masking, no effect on IoU and Sample Coverage, and a positive effect on Explanation Coverage and Activation Coverage. Combining the scores, we can infer that the larger activations (higher Activation Coverage) allow the algorithm to detect labels associated with slightly bigger concepts and better aligned to the neuron behavior (the same IoU and a higher Explanation Coverage). Finally, Cluster 3 and Cluster 4, which include intermediate activations,
Table 2: Avg. and Std Dev. of the desiderata qualities over the labels returned by NetDissect, CoEx, and Clustered Compositional Explanations (our).
IoU	ExplCov	SampleCov	ActCov	DetAcc	LabMask
NetDissect	0.05 ± 0.03	0.14 ± 0.20	0.58 ± 0.27	0.14 ± 0.17	0.15 ± 0.15	0.62± 0.26
CoEx	0.08 ± 0.03	0.14 ± 0.15	0.53 ± 0.25	0.15 ± 0.10	0.18 ± 0.10	0.57± 0.22
Our	0.15 ± 0.07	0.60 ± 0.36	0.69 ± 0.24	0.37 ± 0.17	0.20 ± 0.09	0.61± 0.16
Cluster 1	0.27 ± 0.02	0.99 ± 0.00	0.96 ± 0.02	0.54 ± 0.02	0.34 ± 0.03	0.64± 0.06
Cluster 2	0.15 ± 0.02	0.91 ± 0.16	0.80 ± 0.06	0.49 ± 0.09	0.18 ± 0.03	0.68± 0.07
Cluster 3	0.12 ± 0.03	0.56 ± 0.27	0.64 ± 0.15	0.33 ± 0.10	0.16 ± 0.05	0.62± 0.14
Cluster 4	0.10 ± 0.04	0.35 ± 0.19	0.52 ± 0.22	0.25 ± 0.12	0.15 ± 0.07	0.53± 0.19
Cluster 5	0.09 ± 0.05	0.21 ± 0.18	0.53 ± 0.26	0.22 ± 0.15	0.18 ± 0.09	0.58± 0.23
3Timing collected using a workstation powered by an NVIDIA GeForce RTX-3090 graphic card.
Figure 5: Examples of specialization (left) and polysemy (right). Neuron #455 (left) recognizes streets in more and more specific contexts through the clusters. Neuron #368 (right) recognizes unrelated concepts among different activation ranges (windmill, closet, amusement park).
behave similarly to Cluster 5, but progressively improve the Sample Coverage, IoU, and Explanation Coverage. Therefore, the insight extracted for Cluster 5 holds also for them and, as we discuss in the next section, it is connected to the property of specialization.
In summary, extracting compositional explanations at different and wider activation ranges maintains or improves the same qualities of the returned explanations. Additionally, the combination and analysis of different qualities simultaneously allow us to extract a bigger view of the compositionality of neuron activations, providing hints ready to be exploited by further analyses.
4.4 Neurons Compositionality
In this section, starting from the results reported in the previous section, we analyze the compositionality of neurons at a wider range of activation.
Unspecialized Activations. As previously discussed, Cluster 1 and Cluster 2 are associated with a high Sample Coverage, meaning that they almost cover the full dataset. Therefore, the labels should be present in almost all the samples. By inspecting the labels, we observe that they are often a union of colors (i.e., Black OR Blue OR Grey) or a mix of colors and general concepts (i.e., Black OR Blue OR Sky), and few labels are repeated over the whole range of neurons. While the first observation can suggest that they recognize general concepts or colors, the second one goes in the opposite direction. To investigate the phenomenon, we applied our algorithm on untrained networks, finding that all the clusters are associated with these kinds of labels in this case. Thus, these labels represent the default labels to which the algorithm converges when the activations are random. We call the activations ranges associated with such labels unspecialized, meaning that neurons do not recognize specific concepts or purposely ignore the concepts covered by the activation. By analyzing the clusters from one to five, we found (Table 3) that Cluster 1 is almost always associated with unspecialized
Table 3: Percentage of unspecialized and weakly specialized activation ranges in ResNet18 (ReLU Yes) and DenseNet161(ReLU No) over 128 randomly extracted units.
ReLU	Cluster 1	Cluster 2	Cluster 3	Cluster 4	Cluster 5
Unspecialized	Yes	0.93	0.37	0.01	0.00	0.00
No	0.05	0.60	0.95	0.30	0.05
Weakly Specialized	Yes	0.00	0.03	0.01	0.01	0.00
No	0.05	0.12	0.02	0.17	0.02
activations and Cluster 2 half of the time. This phenomenon can also be partial, meaning that the first part of the label is assigned to a default label, but the last part converges on a different concept. In this case, we call these activation ranges weakly specialized. They are rare, especially in ReLU networks, and usually appear only in the clusters near the unspecialized ones. We hypothesize that they are a side effect of the clustering algorithm, and a future clustering algorithm tailored to extracting the best activation ranges could reduce their number. Table 3 also shows a similar behavior of ReLU and non-ReLU networks when the activations are close to 0. In ReLU layers, activations are stored in Cluster 1, and they are unspecialized  $93\%$  of the time. This percentage becomes smaller when we approach the higher clusters. We can observe a similar behavior in the case of the layer without ReLU. In this case, since the activations can assume negative values, the activations close to zero are stored in the middle clusters, and thus, Cluster 3 includes unspecialized activations  $95\%$  of the time. And again, when we move far away from zero, the percentage starts to decrease, as in the ReLU layers.
Progressive Specialization. Progressive specialization is a phenomenon we observe in association with ReLU layers, where lower activations recognize more general concepts (e.g., building, sky, etc.), and higher ones progressively detect more complex objects. The phenomenon is similar to the one observed by Zhou et al. [43], in which the lower layers of a neural network detect more abstract concepts while the latest detect the most complex and specific ones. In the case of image data, this phenomenon seems to be also spatially aligned, meaning that lower activations capture external elements surrounding the objects detected by higher activations (Figure 4). The specialization property highlighted by Mu and Andreas [30] is an example of specialization (Figure 5).
Activation Polysemey Following Mu and Andreas [30], we manually inspected 128 randomly extracted units to analyze the relations among concepts inside the returned labels and among activation ranges. A polysemic neuron is the one that fires for unrelated concepts [30]. Mu and Andreas [30] found that  $31\%$  of neurons are polysemic in the highest activation ranges. We explore the polysemy considering the full range of activations, meaning that a neuron is considered non-polysemic only if all the labels associated with the clusters are related. While, as expected, the number of polysemic neurons is much larger  $(\sim 85\%)$ , it is interesting to note that  $\sim 15\%$  of neurons fire for related concepts at any activation range, meaning that they are highly specialized. $^4$
This paper presented a first step towards a fuller understanding of neurons compositionality. We introduced Clustered Compositional Explanation, an algorithm that clusters the activations and then applies the CoEx algorithm guided by heuristics to them. Using our algorithm, we found and analyzed novel phenomena connected to the neuron's activations, such as the unspecialization of the lowest activations in ReLU networks and the progressive specialization. Despite the progress, there are some limitations of the current design. First, the labels returned by our algorithm are deeply connected to the activation ranges identified by the clustering algorithm. Therefore, future work could analyze the differences among different clustering algorithms or develop a novel one tailored to the given task. The extracted insights refer to the image data case. While the heuristic and the approach are domain agnostic, the application on a different domain could extract different kinds of insights, thus limiting or confirming the generality of the findings of Section 4.4. We hypothesized that looking at the scores of each cluster can uncover a deeper understanding of the behavior of different activation ranges. However, an interesting direction could be the development of weighting mechanisms to weight the contribution of each cluster to the final averaged scores, which is desirable when the number of clusters is high and looking and comparing individual cluster can become problematic. Finally, the specific labels returned by these algorithms are linked to the concept dataset used to probe the neurons, as observed by Ramaswamy et al. [38]. While the general insights hold even when changing the dataset (Appendix B.2), we do not address the problem of returning the same specific labels across different datasets. Other than mitigating the above-mentioned limitations, future work could also explore the application of the heuristic to study the optimality of the returned explanations, or the application of clusters on recent methods for reducing the dependency on the concepts datasets [36, 19].
Note that the evaluation is subjective. Therefore, the reported numbers must be considered as an indication.
A Heuristics
A.1 Proof of Admissibility of MMESH
Since the heuristic aims at approximating the IoU score, we can start by expanding the denominator of the IoU formula:
$$
\begin{array}{l} I o U (L, \tau_ {1}, \tau_ {2}, \mathfrak {D}) = \frac {I}{U} \\ = \frac {\sum_ {x \in \mathfrak {D}} | M _ {[ \tau_ {1} , \tau_ {2} ]} (x) \cap S (x , L) |}{\sum_ {x \in \mathfrak {D}} | M _ {[ \tau_ {1} , \tau_ {2} ]} (x) \cup S (x , L) |} \tag {18} \\ = \frac {I _ {x}}{\sum_ {x \in \mathfrak {D}} | M _ {[ \tau_ {1} , \tau_ {2} ]} (x) | + \sum_ {x \in \mathfrak {D}} | S (x , L) | - I} \\ \end{array}
$$
Specifically, the heuristic should avoid the direct computation of  $I$  and  $\sum_{x \in \mathfrak{D}} |S(x, L)|$ , providing estimations of them. To be admissible, the heuristic cannot underestimate the intersection or overestimate the union. Thus, it must satisfy the following constraints:
$$
\left\{ \begin{array}{l} \left| \widehat {I _ {x}} \right| \geq \left| I _ {x} \right| \\ 0 \leq \left| \widehat {S (x , L)} \right| - \left| \widehat {I _ {x}} \right| \leq | S (x, L) | - I \end{array} \right. \tag {19}
$$
$\forall x \in \mathfrak{D}$ . Eq. (19) ensures that the heuristic returns an optimistic estimation for the intersection at the numerator. Eq. (20) ensures that the heuristic returns a pessimistic estimation of the denominator of eq. (18).
Equation (19) We begin by proving the first equation. Recall that MMESH estimates  $\left|\widehat{I_x}\right|$  as:
$$
\widehat {I _ {x}} = \left\{\begin{array}{l l}\min  \left(\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right| + \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|, \mid M (x) |\right)&o p = O R\\\min  \left(\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right|, \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|\right)&o p = A N D\\\min  \left(\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right|, \left| M _ {\left[ \tau_ {1}, \tau_ {2} \right]} (x) \right| - \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|\right)&o p = A N D N O T\end{array}\right. \tag {23}
$$
Given the masks  $L_{\leftarrow}$  and  $L_{\rightarrow}$ , we can distinguish between two cases: overlapping and non-overlapping masks. If the masks do not overlap, then the real intersection is given by:
$$
I _ {x} = \left\{\begin{array}{l l}\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right| + \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|&o p = O R\\0&o p = A N D\\\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right) \right|&o p = A N D N O T\end{array}\right. \tag {26}
$$
Comparing eq.(22-24) and eq. (25-27) we can verify that eq. (19) holds. Indeed, eq. (25)  $\leq$  eq. (22) due to the non-negative property of the cardinality and  $IMS_{[\tau_1,\tau_2]}(x,L\rightarrow)\subseteq M_{[\tau_1,\tau_2]}(x)$ . eq. (26)  $\leq$  eq. (23) can be proved by observing that the mask obtained by  $M_{[\tau_1,\tau_2]}(x) - IMS_{[\tau_1,\tau_2]}(x,L\rightarrow)$  contains  $IMS_{[\tau_1,\tau_2]}(x,L\leftarrow)$  since  $L\leftarrow$  and  $L\rightarrow$  are assumed to be non-overlapping and thus the minimum operator used in the heuristics selects  $|IMS_{[\tau_1,\tau_2]}(x,L\leftarrow)|$ . Finally,  $IMS_{[\tau_1,\tau_2]}(x,L\leftarrow)\subseteq M_{[\tau_1,\tau_2]}(x)$  and  $IMS_{[\tau_1,\tau_2]}(x,L\rightarrow)\subseteq M_{[\tau_1,\tau_2]}(x)$  by definition of  $IMS_{[\tau_1,\tau_2]}(x,L)$ . Since the two masks are not overlapping and  $|M_{[\tau_1,\tau_2]}(x)|\geq IMS_{[\tau_1,\tau_2]}(x,L\leftarrow) + IMS_{[\tau_1,\tau_2]}(x,L\rightarrow)$  then eq. (24)  $\leq$  eq. (21) and thus  $I(L) = \widehat{I} (L)$ .
Let's begin with fully overlapping masks for the case of overlapping masks. In this case
$$
I _ {x} = \left\{\begin{array}{l l}\max  \left(\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right|, \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|\right)&o p = O R\\\min  \left(\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right|, \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|\right)&o p = A N D\\\min  \left(\left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\leftarrow}\right)\right|, \left| M _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x\right)\right| - \left| I M S _ {\left[ \tau_ {1}, \tau_ {2} \right]} \left(x, L _ {\rightarrow}\right)\right|\right)&o p = A N D N O T\end{array}\right. \tag {27}
$$
Comparing eq. (22-24) to eq. (280), it is easy to see that eq. (19) holds for AND and AND NOT operators since eq. (22) = eq. (28) and eq. (23) = eq. (29). When  $op = OR$ , one can observe that eq. (27) ≤ eq. (21) since the maximum between two cardinalities is lower than their sum and  $|M_{[\tau_1,\tau_2]}(x)| \geq \max(|IMS_{[\tau_1,\tau_2]}(x,L_{\leftarrow})|, |IMS_{[\tau_1,\tau_2]}(x,L_{\rightarrow})|)$  since the masks are fully overlapping.
The cases described above (fully overlapping and non-overlapping masks) for the estimation of  $\widehat{I_x}$  cover the best-case scenarios for all the involved operators. Therefore, when the masks are partially overlapping, the real intersection is smaller than the case of non-overlapping for the OR operator and smaller than the case of fully overlapping for the AND and AND NOT operator. Thus,  $|\widehat{I_x} | \geq |I_x|$  holds also for partially overlapping masks.
Equation (20) Recall that the heuristic computes  $\widehat{S(x, L)}$  as:
$$
\widehat {S (x , L)} = \left\{\begin{array}{l l}\max  \left(\left| S (x, L _ {\leftarrow}) \right|, \left| S (x, L _ {\rightarrow}) \right|, \left| S \left(x, L _ {\leftarrow} \cup L _ {\rightarrow}\right)\right|\right.\left. \right.&o p = O R\\\max  \left(M i n O v e r (L), \left| \widehat {I _ {x}} \right|\right)&o p = A N D\\\max  \left(\left| S (x, L _ {\leftarrow}) \right| - M a x O v e r (L), \left| \widehat {I _ {x}} \right|\right)&o p = A N D N O T\end{array}\right. \tag {30}
$$
where  $\widehat{S(x, L_{\leftarrow} \cup L_{\rightarrow})} = |S(x, L_{\rightarrow})| + |S(x, L_{\rightarrow})| - \text{MaxOver}(L)$ . We can distinguish again between the case of non-overlapping and fully overlapping masks.
We proceed by proving for each case that (i)  $\widehat{S(x,L)}\leq S(x,L)$ , (ii)  $\widehat{S(x,L)} -\widehat{I_x}\leq S(x,L) - I_x$  and (iii)  $\widehat{S(x,L)} -\widehat{I_x}\geq 0$
In the case of non-overlapping masks, the real joint label is
$$
S (x, L) = \left\{\begin{array}{l l}| S (x, L _ {\leftarrow}) | + | S (x, L _ {\rightarrow}) |&o p = O R\\0&o p = A N D\\| S (x, L _ {\leftarrow}) |&o p = A N D N O T\end{array}\right. \tag {35}
$$
Let us begin by comparing eq. (30) and eq. (33) for the OR operator.
When the max operator selects  $|S(x, L_{\leftarrow})|$  or  $|S(x, L_{\leftarrow})|$  (i) is verified since  $|S(x, L_{\leftarrow})| \leq |S(x, L_{\leftarrow})| + |S(x, L_{\rightarrow})|$  and  $|S(x, L_{\rightarrow})| \leq |\widehat{S(x, L_{\leftarrow})}| + |S(x, L_{\rightarrow})|$  due to the non-negativity of the cardinality. When the maximum is  $|\widehat{S(x, L_{\leftarrow} \cup L_{\rightarrow})}|}$  (i) is verified since  $MaxOver(L)$  by definition returns a cardinality (the one of the overlapping between the largest possible extensions), then  $MaxOver(L) \geq 0$  and thus  $|\widehat{S(x, L_{\leftarrow} \cup L_{\rightarrow})}|} \leq |\widehat{S(x, L_{\leftarrow})}| + |S(x, L_{\rightarrow})|$ . (ii) is proved by observing that  $|\widehat{I_x}|$  is an overestimation of  $I$  and thus  $|\widehat{S(x, L)}| - |\widehat{I_x}| \leq |S(x, L)| - |I|$ .
(iii) can be proved by showing that  $|\widehat{S}(x, \overline{L})| \geq |\widehat{L_x}|$ . The proof follows by noting that  $IMS_{[\tau_1, \tau_2]}(x, L_{\leftarrow}) \subseteq S(x, L_{\leftarrow})$ ,  $IMS_{[\tau_1, \tau_2]}(x, L_{\rightarrow}) \subseteq S(x, L_{\rightarrow})$ , and the max operator selects the highest cardinality.