Generative Adversarial Networks (GANs) excel at creating realistic images with complex models for which maximum likelihood is infeasible. However, the convergence of GAN training has still not been proved. We propose a two time-scale update rule (TTUR) for training GANs with stochastic gradient descent on arbitrary GAN loss functions. TTUR has an individual learning rate for both the discriminator and the generator. Using the theory of stochastic approximation, we prove that the TTUR converges under mild assumptions to a stationary local Nash equilibrium. The convergence carries over to the popular Adam optimization, for which we prove that it follows the dynamics of a heavy ball with friction and thus prefers flat minima in the objective landscape. For the evaluation of the performance of GANs at image generation, we introduce the `Fréchet Inception Distance'' (FID) which captures the similarity of generated images to real ones better than the Inception Score. In experiments, TTUR improves learning for DCGANs and Improved Wasserstein GANs (WGAN-GP) outperforming conventional GAN training on CelebA, CIFAR-10, SVHN, LSUN Bedrooms, and the One Billion Word Benchmark.

MOTIVATION: Biclustering of transcriptomic data groups genes and samples simultaneously. It is emerging as a standard tool for extracting knowledge from gene expression measurements. We propose a novel generative approach for biclustering called 'FABIA: Factor Analysis for Bicluster Acquisition'. FABIA is based on a multiplicative model, which accounts for linear dependencies between gene expression and conditions, and also captures heavy-tailed distributions as observed in real-world transcriptomic data. The generative framework allows to utilize well-founded model selection methods and to apply Bayesian techniques. RESULTS: On 100 simulated datasets with known true, artificially implanted biclusters, FABIA clearly outperformed all 11 competitors. On these datasets, FABIA was able to separate spurious biclusters from true biclusters by ranking biclusters according to their information content. FABIA was tested on three microarray datasets with known subclusters, where it was two times the best and once the second best method among the compared biclustering approaches. AVAILABILITY: FABIA is available as an R package on Bioconductor (http://www.bioconductor.org). All datasets, results and software are available at http://www.bioinf.jku.at/software/fabia/fabia.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: As more genomes are sequenced, the demand for fast gene classification techniques is increasing. To analyze a newly sequenced genome, first the genes are identified and translated into amino acid sequences which are then classified into structural or functional classes. The best-performing protein classification methods are based on protein homology detection using sequence alignment methods. Alignment methods have recently been enhanced by discriminative methods like support vector machines (SVMs) as well as by position-specific scoring matrices (PSSM) as obtained from PSI-BLAST. However, alignment methods are time consuming if a new sequence must be compared to many known sequences-the same holds for SVMs. Even more time consuming is to construct a PSSM for the new sequence. The best-performing methods would take about 25 days on present-day computers to classify the sequences of a new genome (20,000 genes) as belonging to just one specific class--however, there are hundreds of classes. Another shortcoming of alignment algorithms is that they do not build a model of the positive class but measure the mutual distance between sequences or profiles. Only multiple alignments and hidden Markov models are popular classification methods which build a model of the positive class but they show low classification performance. The advantage of a model is that it can be analyzed for chemical properties common to the class members to obtain new insights into protein function and structure. We propose a fast model-based recurrent neural network for protein homology detection, the 'Long Short-Term Memory' (LSTM). LSTM automatically extracts indicative patterns for the positive class, but in contrast to profile methods it also extracts negative patterns and uses correlations between all detected patterns for classification. LSTM is capable to automatically extract useful local and global sequence statistics like hydrophobicity, polarity, volume, polarizability and combine them with a pattern. These properties make LSTM complementary to alignment-based approaches as it does not use predefined similarity measures like BLOSUM or PAM matrices. RESULTS: We have applied LSTM to a well known benchmark for remote protein homology detection, where a protein must be classified as belonging to a SCOP superfamily. LSTM reaches state-of-the-art classification performance but is considerably faster for classification than other approaches with comparable classification performance. LSTM is five orders of magnitude faster than methods which perform slightly better in classification and two orders of magnitude faster than the fastest SVM-based approaches (which, however, have lower classification performance than LSTM). Only PSI-BLAST and HMM-based methods show comparable time complexity as LSTM, but they cannot compete with LSTM in classification performance. To test the modeling capabilities of LSTM, we applied LSTM to PROSITE classes and interpreted the extracted patterns. In 8 out of 15 classes, LSTM automatically extracted the PROSITE motif. In the remaining 7 cases alternative motifs are generated which give better classification results on average than the PROSITE motifs. AVAILABILITY: The LSTM algorithm is available from http://www.bioinf.jku.at/software/LSTM_protein/.