Solutions of symbolic regression problems are expressions that are composed of input variables and operators from a finite set of function symbols. One measure for evaluating symbolic regression algorithms is their ability to recover formulae, up to symbolic equivalence, from finite samples. Not unexpectedly, the recovery problem becomes harder when the formula gets more complex, that is, when the number of variables and operators gets larger. Variables in naturally occurring symbolic formulas often appear only in fixed combinations. This can be exploited in symbolic regression by substituting one new variable for the combination, effectively reducing the number of variables. However, finding valid substitutions is challenging. Here, we address this challenge by searching over the expression space of small substitutions and testing for validity. The validity test is reduced to a test of functional dependence. The resulting iterative dimension reduction procedure can be used with any symbolic regression approach. We show that it reliably identifies valid substitutions and significantly boosts the performance of different types of state-of-the-art symbolic regression algorithms.

Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra systems (CASs) provide support for solving ODEs by first simplifying them, in particular through the use of Lie point symmetries. Finding these symmetries is, however, itself a difficult problem for CASs. Recent works in symbolic regression have shown promising results for recovering symbolic expressions from data. Here, we adapt search-based symbolic regression to the task of finding generators of Lie point symmetries. With this approach, we can find symmetries of ODEs that existing CASs cannot find.

Deep learning models for different NLP tasks often rely on pre-trained word embeddings, that is, vector representations of words. Therefore, it is crucial to evaluate pre-trained word embeddings independently of downstream tasks. Such evaluations try to assess whether the geometry induced by a word embedding captures connections made in natural language, such as, analogies, clustering of words, or word similarities. Here, traditionally, similarity is measured by comparison to human judgment. However, explicitly annotating word pairs with similarity scores by surveying humans is expensive. We tackle this problem by formulating a similarity measure that is based on an agent for routing the Wikipedia hyperlink graph. In this graph, word similarities are implicitly encoded by edges between articles. We show on the English Wikipedia that our measure correlates well with a large group of traditional similarity measures, while covering a much larger proportion of words and avoiding explicit human labeling. Moreover, since Wikipedia is available in more than 300 languages, our measure can easily be adapted to other languages, in contrast to traditional similarity measures.

Topic models are characterized by a latent class variable that represents the different topics. Traditionally, their observable variables are modeled as discrete variables like, for instance, in the prototypical latent Dirichlet allocation (LDA) topic model. In LDA, words in text documents are encoded by discrete count vectors with respect to some dictionary. The classical approach for learning topic models optimizes a likelihood function that is non-concave due to the presence of the latent variable. Hence, this approach mostly boils down to using search heuristics like the EM algorithm for parameter estimation. Recently, it was shown that topic models can be learned with strong algorithmic and statistical guarantees through Pearson's method of moments. Here, we extend this line of work to topic models that feature discrete as well as continuous observable variables (features). Moving beyond discrete variables as in LDA allows for more sophisticated features and a natural extension of topic models to other modalities than text, like, for instance, images. We provide algorithmic and statistical guarantees for the method of moments applied to the extended topic model that we corroborate experimentally on synthetic data. We also demonstrate the applicability of our model on real-world document data with embedded images that we preprocess into continuous state-of-the-art feature vectors.