Abstract: A computer-implemented method for training a normalizing flow. The normalizing flow predicts a first density value based on a first input image. The first density value characterizes a likelihood of the first input image to occur. The first density value is predicted based on an intermediate output of a first convolutional layer of the normalizing flow. The intermediate output is determined based on a plurality of weights of the first convolutional layer. The method for training includes: determining a second input image; determining an output, wherein the output is determined by providing the second input image to the normalizing flow and providing an output of the normalizing flow as output; determining a second density value based on the output tensor and on the plurality of weights; determining a natural gradient of the plurality of weights with respect to the second density value; adapting the weights according to the natural gradient.

Abstract: A computer-implemented method of training a machine learnable model for controlling and/or monitoring a computer-controlled system. The machine learnable model is configured to make inferences based on a probability distribution of sensor data of the computer-controlled system. The machine learnable model is configured to account for symmetries in the probability distribution imposed by the system and/or its environment. The training involves sampling multiple samples of the sensor data according to the probability distribution. Initial values are sampled from a source probability distribution invariant to the one or more symmetries. The samples are iteratively evolved according to a kernel function equivariant to the one or more symmetries. The evolution uses an attraction term and a repulsion term that are defined for a selected sample in terms of gradient directions of the probability distribution and of the kernel function for the multiple samples.

Abstract: A computer-implemented method of training an image generation model. The image generation model comprises an argmax transformation configured to compute a discrete index feature indicating an index of a feature of the continuous feature vector with an extreme value. The image generation model is trained using a log-likelihood optimization. This involves obtaining a value of the index feature for the training image, sampling values of the continuous feature vector given the value of the index feature according to a stochastic inverse transformation of the argmax transformation, and determining a likelihood contribution of the argmax transformation for the log-likelihood based on a probability that the stochastic inverse transformation generates the values of the continuous feature vector given the value of the index feature.

Abstract: A computer-implemented method for training a normalizing flow. The normalizing flow is configured to determine a first output signal characterizing a likelihood or a log-likelihood of an input signal. The normalizing flow includes at least one first layer which includes trainable parameters. A layer input to the first layer is based on the input signal and the first output signal is based on a layer output of the first layer. The training includes: determining at least one training input signal; determining a training output signal for each training input signal using the normalizing flow; determining a first loss value which is based on a likelihood or a log-likelihood of the at least one determined training output signal with respect to a predefined probability distribution; determining an approximation of a gradient of the trainable parameters; updating the trainable parameters of the first layer based on the approximation of the gradient.

Abstract: A computer-implemented method for training a normalizing flow. The normalizing flow predicts a first density value based on a first input image. The first density value characterizes a likelihood of the first input image to occur. The first density value is predicted based on an intermediate output of a first convolutional layer of the normalizing flow. The intermediate output is determined based on a plurality of weights of the first convolutional layer. The method for training includes: determining a second input image; determining an output, wherein the output is determined by providing the second input image to the normalizing flow and providing an output of the normalizing flow as output; determining a second density value based on the output tensor and on the plurality of weights; determining a natural gradient of the plurality of weights with respect to the second density value; adapting the weights according to the natural gradient.

Abstract: A computer-implemented method of training an image classifier which uses any combination of labelled and/or unlabelled training images. The image classifier comprises a set of transformations between respective transformation inputs and transformation outputs. An inverse model is defined in which for a deterministic, non-injective transformation of the image classifier, its inverse is approximated by a stochastic inverse transformation. During training, for a given training image, a likelihood contribution for this transformation is determined based on a probability of its transformation inputs being generated by the stochastic inverse transformation given its transformation outputs. This likelihood contribution is used to determine a log-likelihood for the training image to be maximized (and its label, if the training image is labelled), based on which the model parameters are optimized.

Abstract: A method for training a neural network. The neural network comprises a first layer which includes a plurality of filters to provide a first layer output comprising a plurality of feature maps. Training of the classifier includes: receiving, by a preceding layer, a first layer input in the first layer, wherein the first layer input is based on the input signal; determining the first layer output based on the first layer input and a plurality of parameters of the first layer; determining a first layer loss value based on the first layer output, wherein the first layer loss value characterizes a degree of dependency between the feature maps, the first layer loss value being obtained in an unsupervised fashion; and training the neural network. The training includes an adaption of the parameters of the first layer, the adaption being based on the first layer loss value.