This study was motivated by two image discrimination examples: handwritten digit recognition and COVID-19 lung CT scanning image recognition. These two problems have a significant difference. Handwritten ones, for example, have a slash in the middle of all images, whereas locations of lung damage vary from one person to another. Linear classifiers excel at handling the former due to the consistent patterns, but they struggle with the latter due to the varying lung damage locations. To tackle the latter discrimination problem, we propose a novel approach called convolutional multiple-instance logistic regression (CMILR) that combines convolutional neural network (CNN) and multiple-instance learning. In the case of COVID-19 lung CT scans, CMILR resulted in an accuracy of 0.81 with only 169 parameters. In contrast, a fine-tuned CNN model resulted in an accuracy of 0.88 and 377,858 parameters. Additionally, CMILR provides a probability map indicating the likelihood of lung damage, offering valuable insights for medical diagnosis and making the learning algorithm explainable.