Type Ia supernovae (SNe Ia) provide us with a unique tool for measuring extragalactic distances and determining cosmological parameters. As a result, the precise and effective calibration for peak luminosities of SNe Ia becomes extremely crucial and thus is critically scrutinized for cosmological explorations. In this Letter, we reveal clear evidence for a tight linear correlation between peak luminosities of SNe Ia and their B − V colors ∼ 12 days after the B maximum denoted by ∆C 12 . By introducing such a novel color parameter, ∆C 12 , this empirical correlation allows us to uniformly standardize SNe Ia with decline rates ∆m 15 in the range of 0.8 < ∆m 15 < 2.0 and to reduce scatters in estimating their peak luminosities from ∼ 0.5 mag to the levels of 0.18 and 0.12 mag in the V and I bands, respectively. For a sample of SNe Ia with insignificant reddenings of host galaxies [e.g., E(B − V ) host < ∼ 0.06 mag], the scatter drops further to only 0.07 mag (or 3−4% in distance), which is comparable to observational accuracies and is better than other calibrations for SNe Ia. This would impact observational and theoretical studies of SNe Ia and cosmological scales and parameters.
Recognizing facial action units (AUs) during spontaneous facial displays is a challenging problem. Most recently, Convolutional Neural Networks (CNNs) have shown promise for facial AU recognition, where predefined and fixed convolution filter sizes are employed. In order to achieve the best performance, the optimal filter size is often empirically found by conducting extensive experimental validation. Such a training process suffers from expensive training cost, especially as the network becomes deeper.This paper proposes a novel Optimized Filter Size CNN (OFS-CNN), where the filter sizes and weights of all convolutional layers are learned simultaneously from the training data along with learning convolution filters. Specifically, the filter size is defined as a continuous variable, which is optimized by minimizing the training loss. Experimental results on two AU-coded spontaneous databases have shown that the proposed OFS-CNN is capable of estimating optimal filter size for varying image resolution and outperforms traditional CNNs with the best filter size obtained by exhaustive search. The OFS-CNN also beats the CNN using multiple filter sizes and more importantly, is much more efficient during testing with the proposed forward-backward propagation algorithm.
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