Principal manifolds and graphs in practice: from molecular biology to dynamical systems

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.Comment: 12 pages, 9 figure

Complex Systems Science: Dreams of Universality, Reality of Interdisciplinarity

Using a large database (~ 215 000 records) of relevant articles, we empirically study the "complex systems" field and its claims to find universal principles applying to systems in general. The study of references shared by the papers allows us to obtain a global point of view on the structure of this highly interdisciplinary field. We show that its overall coherence does not arise from a universal theory but instead from computational techniques and fruitful adaptations of the idea of self-organization to specific systems. We also find that communication between different disciplines goes through specific "trading zones", ie sub-communities that create an interface around specific tools (a DNA microchip) or concepts (a network).Comment: Journal of the American Society for Information Science and Technology (2012) 10.1002/asi.2264

SOM-VAE: Interpretable Discrete Representation Learning on Time Series

High-dimensional time series are common in many domains. Since human cognition is not optimized to work well in high-dimensional spaces, these areas could benefit from interpretable low-dimensional representations. However, most representation learning algorithms for time series data are difficult to interpret. This is due to non-intuitive mappings from data features to salient properties of the representation and non-smoothness over time. To address this problem, we propose a new representation learning framework building on ideas from interpretable discrete dimensionality reduction and deep generative modeling. This framework allows us to learn discrete representations of time series, which give rise to smooth and interpretable embeddings with superior clustering performance. We introduce a new way to overcome the non-differentiability in discrete representation learning and present a gradient-based version of the traditional self-organizing map algorithm that is more performant than the original. Furthermore, to allow for a probabilistic interpretation of our method, we integrate a Markov model in the representation space. This model uncovers the temporal transition structure, improves clustering performance even further and provides additional explanatory insights as well as a natural representation of uncertainty. We evaluate our model in terms of clustering performance and interpretability on static (Fashion-)MNIST data, a time series of linearly interpolated (Fashion-)MNIST images, a chaotic Lorenz attractor system with two macro states, as well as on a challenging real world medical time series application on the eICU data set. Our learned representations compare favorably with competitor methods and facilitate downstream tasks on the real world data.Comment: Accepted for publication at the Seventh International Conference on Learning Representations (ICLR 2019

Computational physics of the mind

In the XIX century and earlier such physicists as Newton, Mayer, Hooke, Helmholtz and Mach were actively engaged in the research on psychophysics, trying to relate psychological sensations to intensities of physical stimuli. Computational physics allows to simulate complex neural processes giving a chance to answer not only the original psychophysical questions but also to create models of mind. In this paper several approaches relevant to modeling of mind are outlined. Since direct modeling of the brain functions is rather limited due to the complexity of such models a number of approximations is introduced. The path from the brain, or computational neurosciences, to the mind, or cognitive sciences, is sketched, with emphasis on higher cognitive functions such as memory and consciousness. No fundamental problems in understanding of the mind seem to arise. From computational point of view realistic models require massively parallel architectures

Magnification Control in Self-Organizing Maps and Neural Gas

We consider different ways to control the magnification in self-organizing maps (SOM) and neural gas (NG). Starting from early approaches of magnification control in vector quantization, we then concentrate on different approaches for SOM and NG. We show that three structurally similar approaches can be applied to both algorithms: localized learning, concave-convex learning, and winner relaxing learning. Thereby, the approach of concave-convex learning in SOM is extended to a more general description, whereas the concave-convex learning for NG is new. In general, the control mechanisms generate only slightly different behavior comparing both neural algorithms. However, we emphasize that the NG results are valid for any data dimension, whereas in the SOM case the results hold only for the one-dimensional case.Comment: 24 pages, 4 figure