Motivated by the problem of domain formation in chromosomes, we studied a co-polymer model where only a subset of the monomers feel attractive interactions. These monomers are displaced randomly from a regularly-spaced pattern, thus introducing some quenched disorder in the system. Previous work has shown that in the case of regularly-spaced interacting monomers this chain can fold into structures characterized by multiple distinct domains of consecutive segments. In each domain, attractive interactions are balanced by the entropy cost of forming loops. We show by advanced replica-exchange simulations that adding disorder in the position of the interacting monomers further stabilizes these domains. The model suggests that the partitioning of the chain into well-defined domains of consecutive monomers is a spontaneous property of heteropolymers. In the case of chromosomes, evolution could have acted on the spacing of interacting monomers to modulate in a simple way the underlying domains for functional reasons.
We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations that achieve the Bayes-optimal generalization error, even in the case of unbalanced clusters. We explore analytically the error-counting loss landscape in the vicinity of a Bayes-optimal solution, and show that the closer we get to such configurations, the higher the local entropy, implying that the Bayes-optimal solution lays inside a wide flat region. We also consider the algorithmically relevant case of targeting wide flat minima of the (differentiable) mean squared error loss. Our analytical and numerical results show not only that in the balanced case the dependence on the norm of the weights is mild, but also, in the unbalanced case, that the performances can be improved.
The differing ability of polypeptide conformations to act as the native state of proteins has long been rationalized in terms of differing kinetic accessibility or thermodynamic stability. Building on the successful applications of physical concepts and sampling algorithms recently introduced in the study of disordered systems, in particular artificial neural networks, we quantitatively explore how well a quantity known as the local entropy describes the native state of model proteins. In lattice models and all-atom representations of proteins, we are able to efficiently sample high local entropy states and to provide a proof of concept of enhanced stability and folding rate. Our methods are based on simple and general statistical-mechanics arguments, and thus we expect that they are of very general use.
Generative processes in biology and other fields often produce data that can be regarded as resulting from a composition of basic features. Here we present an unsupervised method based on autoencoders for inferring these basic features of data. The main novelty in our approach is that the training is based on the optimization of the 'local entropy' rather than the standard loss, resulting in a more robust inference, and enhancing the performance on this type of data considerably. Algorithmically, this is realized by training an interacting system of replicated autoencoders. We apply this method to synthetic and protein sequence data, and show that it is able to infer a hidden representation that correlates well with the underlying generative process, without requiring any prior knowledge. AUTHOR SUMMARYExtracting compositional features from noisy data and identifying the corresponding generative models is a fundamental challenge across sciences. The composition of elementary features can have highly non-linear effects which makes them very hard to identify from experimental data. In biology, for instance, one challenge is to identify the key steps or components of molecular and cellular processes. Representative examples are the modeling of protein sequences as the composition of patterns influenced by phylogeny or the identification of gene clusters in which the presence of specific genes depends on the evolutionary history of the cell. Here we present an unsupervised machine learning technique for the analysis of compositional data which is based on entropic neural autoencoders. Our approach aims at finding deep autoencoders that are highly invariant with respect to perturbations in the inputs and in the parameters. The procedure is efficient to implement and we have validated it both on synthetic and protein sequence data, where it can be shown that the latent variables of the autoencoders are non trivially correlated with the true underlying generative processes. Our results suggests that the local entropy approach represents a general valuable tool for the extraction of compositional features in hard unsupervised learning problems.
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