| accuracy | time |
| Perraudin etal. (2019),2D CNNbaseline | 54.2 | 104 ms |
| Perraudin et al. (2019), CNN variant, k = 8 | 62.1 | 185ms |
| Perraudin etal. (2019),FCN variant, k = 8 | 83.8 | 185 ms |
| k = 8 neighbors,t from section 3.2 | 87.1 | 185 ms |
| k = 2O neighbors,t from section 3.2 | 91.3 | 250 ms |
| k = 40 neighbors,t from section 3.2 | 92.5 | 363 ms |
+
+
+Figure 7: Tradeoff between cost and accuracy.
+
+# 4.2 COSMOLOGICAL MODEL CLASSIFICATION
+
+Given observations, cosmologists estimate the posterior probability of cosmological parameters, such as the matter density $\Omega _ { m }$ and the normalization of the matter power spectrum $\sigma _ { 8 }$ . Those parameters are typically estimated by likelihood-free inference, which requires a function to predict the parameters from simulations. As that is complicated to setup, prediction methods are typically benchmarked on the classification of spherical maps instead (Schmelzle et al., 2017). We used the same task, data, and setup as Perraudin et al. (2019): the classification of 720 partial convergence maps made of $n \approx 1 0 ^ { 6 }$ pixels $( 1 / 1 2 \approx 8 \%$ of a sphere at $N _ { s i d e } = 1 0 2 4 )$ from two $\Lambda { \bf C D M }$ cosmological models, $\Omega _ { m } = 0 . 3 1$ , $\sigma _ { 8 } = 0 . 8 2 ,$ ) and $\Omega _ { m } = 0 . 2 6$ , $\sigma _ { 8 } = 0 . 9 1$ ), at a relative noise level of 3.5 (i.e., the signal is hidden in noise of 3.5 times higher standard deviation). Convergence maps represent the distribution of over- and under-densities of mass in the universe (see Bartelmann, 2010, for a review of gravitational lensing).
+
+Graphs are built with (5), $k = 8 , 2 0 , 4 0$ neighbors, and the corresponding optimal kernel widths $t$ given in section 3.2. Following Perraudin et al. (2019), the NN is made of 5 graph convolutional layers, each followed by a max pooling layer which down-samples by 4. A GAP and a fully connected layer with softmax follow. The polynomials are all of order $P = 4$ and the number of channels per layer is 16, 32, 64, 64, 64, respectively. The cross-entropy loss is optimized with Adam for 80 epochs. The learning rate is $2 \cdot 1 \dot { 0 } ^ { - 4 } \cdot 0 . 9 9 9 ^ { \mathrm { s t e p } }$ and the batch size is 8.
+
+Unlike on SHREC’17, results (table 2) show that a lower equivariance error on the convolutions translates to higher performance. That is probably due to the high frequency content of those maps (figure 5). There is a clear cost-accuracy tradeoff, controlled by the number of neighbors $k$ (figure 7). This experiment moreover demonstrates DeepSphere’s flexibility (using partial spherical maps) and scalability (competing spherical CNNs were tested on maps of at most 10, 000 pixels).
+
+# 4.3 CLIMATE EVENT SEGMENTATION
+
+We evaluate our method on a task proposed by (Mudigonda et al., 2017): the segmentation of extreme climate events, Tropical Cyclones (TC) and Atmospheric Rivers (AR), in global climate simulations (figure 1c). The data was produced by a 20-year run of the Community Atmospheric Model v5 (CAM5) and consists of 16 channels such as temperature, wind, humidity, and pressure at multiple altitudes. We used the pre-processed dataset from (Jiang et al., 2019).11 There is 1,072,805 spherical maps, down-sampled to a level-5 icosahedral sampling $( n = 1 0 \cdot 4 ^ { l } + 2 = 1 0$ , 242 pixels). The labels are heavily unbalanced with $0 . 1 \%$ TC, $2 . 2 \%$ AR, and $9 7 . 7 \%$ background (BG) pixels.
+
+The graph is built with (5), $k = 6$ neighbors, and a kernel width $t$ set to the average of the distances. Following Jiang et al. (2019), the NN is an encoder-decoder with skip connections. Details in section C.3. The polynomials are all of order $P = 3$ . The cross-entropy loss (weighted or nonweighted) is optimized with Adam for 30 epochs. The learning rate is $1 \cdot 1 0 { - 3 }$ and the batch size is 64.
+
+Results are shown in table 3 (details in tables 6, 7 and 8). The mean and standard deviation are computed over 5 runs. Note that while Jiang et al. (2019) and Cohen et al. (2019) use a weighted cross-entropy loss, that is a suboptimal proxy for the mAP metric. DeepSphere achieves state-of
+
+Table 3: Results on climate event segmentation: mean accuracy (over TC, AR, BG) and mean average precision (over TC and AR). DeepSphere achieves state-of-the-art performance.
+
+ | TC | AR | BG | mean |
| Mudigonda et al. (2017) | 74 | 65 | 97 | 78.67 |
| Jiang et al. (2019) (paper) | 94 | 93 | 97 | 94.67 |
| Jiang et al. (2019) (rerun) | 93.9 | 95.7 | 95.2 | 94.95 |
| Cohen et al. (2019) (S2R) | 97.8 | 97.3 | 97.3 | 97.5 |
| Cohen et al. (2019) (R2R) | 97.9 | 97.8 | 97.4 | 97.7 |
| DS (Jiang architecture, weighted loss) | 97.1 | 97.6 | 96.5 | 97.1 |
| DS (weighted loss) | 97.4 ± 1.1 | 97.7± 0.7 | 98.2 ± 0.5 | 97.8 ± 0.3 |
| DS (wider architecture, weighted loss) | 91.5 | 93.4 | 99.0 | 94.6 |
| DS (Jiang architecture, non-weighted loss) | 33.6 | 93.6 | 99.3 | 75.5 |
| DS (non-weighted loss) | 69.2 ± 3.7 | 94.5 ± 2.9 | 99.7± 0.1 | 87.8 ± 0.5 |
| DS (wider architecture, non-weighted loss) | 73.4 | 92.7 | 99.8 | 88.7 |
+
+Table 7: Results on climate event segmentation: average precision. Tropical cyclones (TC) and atmospheric rivers (AR) are the two positive classes. Note that a weighted cross-entropy loss is not optimal for the average precision metric.
+
+ | accuracy | time |
| Perraudin etal. (2019),2D CNNbaseline | 54.2 | 104 ms |
| Perraudin et al. (2019), CNN variant, k = 8 | 62.1 | 185ms |
| Perraudin etal. (2019),FCN variant, k = 8 | 83.8 | 185 ms |
| k = 8 neighbors,t from section 3.2 | 87.1 | 185 ms |
| k = 2O neighbors,t from section 3.2 | 91.3 | 250 ms |
| k = 40 neighbors,t from section 3.2 | 92.5 | 363 ms |
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+ "table_body": " | TC | AR | BG | mean |
| Mudigonda et al. (2017) | 74 | 65 | 97 | 78.67 |
| Jiang et al. (2019) (paper) | 94 | 93 | 97 | 94.67 |
| Jiang et al. (2019) (rerun) | 93.9 | 95.7 | 95.2 | 94.95 |
| Cohen et al. (2019) (S2R) | 97.8 | 97.3 | 97.3 | 97.5 |
| Cohen et al. (2019) (R2R) | 97.9 | 97.8 | 97.4 | 97.7 |
| DS (Jiang architecture, weighted loss) | 97.1 | 97.6 | 96.5 | 97.1 |
| DS (weighted loss) | 97.4 ± 1.1 | 97.7± 0.7 | 98.2 ± 0.5 | 97.8 ± 0.3 |
| DS (wider architecture, weighted loss) | 91.5 | 93.4 | 99.0 | 94.6 |
| DS (Jiang architecture, non-weighted loss) | 33.6 | 93.6 | 99.3 | 75.5 |
| DS (non-weighted loss) | 69.2 ± 3.7 | 94.5 ± 2.9 | 99.7± 0.1 | 87.8 ± 0.5 |
| DS (wider architecture, non-weighted loss) | 73.4 | 92.7 | 99.8 | 88.7 |
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| k = 8 neighbors,t from section 3.2 | 87.1 | 185 ms |
| k = 2O neighbors,t from section 3.2 | 91.3 | 250 ms |
| k = 40 neighbors,t from section 3.2 | 92.5 | 363 ms |
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| Mudigonda et al. (2017) | 74 | 65 | 97 | 78.67 |
| Jiang et al. (2019) (paper) | 94 | 93 | 97 | 94.67 |
| Jiang et al. (2019) (rerun) | 93.9 | 95.7 | 95.2 | 94.95 |
| Cohen et al. (2019) (S2R) | 97.8 | 97.3 | 97.3 | 97.5 |
| Cohen et al. (2019) (R2R) | 97.9 | 97.8 | 97.4 | 97.7 |
| DS (Jiang architecture, weighted loss) | 97.1 | 97.6 | 96.5 | 97.1 |
| DS (weighted loss) | 97.4 ± 1.1 | 97.7± 0.7 | 98.2 ± 0.5 | 97.8 ± 0.3 |
| DS (wider architecture, weighted loss) | 91.5 | 93.4 | 99.0 | 94.6 |
| DS (Jiang architecture, non-weighted loss) | 33.6 | 93.6 | 99.3 | 75.5 |
| DS (non-weighted loss) | 69.2 ± 3.7 | 94.5 ± 2.9 | 99.7± 0.1 | 87.8 ± 0.5 |
| DS (wider architecture, non-weighted loss) | 73.4 | 92.7 | 99.8 | 88.7 |
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| 1:Wo ←wo;t←0 | /*Set running average to wo*/ /*Set momentum value*/ |
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