Jürgen Jost

Ivan P. Yamshchikov, Alexey Tikhonov, Yorgos Pantis et al.

The extensive surviving corpus of the ancient scholar Plutarch of Chaeronea (ca. 45-120 CE) also contains several texts which, according to current scholarly opinion, did not originate with him and are therefore attributed to an anonymous author Pseudo-Plutarch. These include, in particular, the work Placita Philosophorum (Quotations and Opinions of the Ancient Philosophers), which is extremely important for the history of ancient philosophy. Little is known about the identity of that anonymous author and its relation to other authors from the same period. This paper presents a BERT language model for Ancient Greek. The model discovers previously unknown statistical properties relevant to these literary, philosophical, and historical problems and can shed new light on this authorship question. In particular, the Placita Philosophorum, together with one of the other Pseudo-Plutarch texts, shows similarities with the texts written by authors from an Alexandrian context (2nd/3rd century CE).

Parvaneh Joharinad, Jürgen Jost

Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric concept of curvature. This enables us, on one hand, to reconceptualize curvature and link it to the geometric notion of hyperconvexity. On the other hand, we can then also understand methods of topological data analysis from a geometric perspective.

Lukas Silvester Barth, Fatemeh, Fahimi et al.

This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We present a systematic and detailed construction of a metric representation for locally distorted metric spaces that captures complex data structures more accurately than the previous schemes. Our approach addresses limitations in existing methods by accommodating non-uniform data distributions and intricate local geometries. We validate its performance through extensive experiments on examples of various geometric objects and benchmark real-world datasets, demonstrating significant improvements in representation quality.

Janis Keck, Lukas Silvester Barth, Fatemeh et al.

Fuzzy simplicial sets have become an object of interest in dimensionality reduction and manifold learning, most prominently through their role in UMAP. However, their definition through tools from algebraic topology without a clear probabilistic interpretation detaches them from commonly used theoretical frameworks in those areas. In this work we introduce a framework that explains fuzzy simplicial sets as marginals of probability measures on simplicial sets. In particular, this perspective shows that the fuzzy weights of UMAP arise from a generative model that samples Vietoris-Rips filtrations at random scales, yielding cumulative distribution functions of pairwise distances. More generally, the framework connects fuzzy simplicial sets to probabilistic models on the face poset, clarifies the relation between Kullback-Leibler divergence and fuzzy cross-entropy in this setting, and recovers standard t-norms and t-conorms via Boolean operations on the underlying simplicial sets. We then show how new embedding methods may be derived from this framework and illustrate this on an example where we generalize UMAP using Čech filtrations with triplet sampling. In summary, this probabilistic viewpoint provides a unified probabilistic theoretical foundation for fuzzy simplicial sets, clarifies the role of UMAP within this framework, and enables the systematic derivation of new dimensionality reduction methods.

Charlotte Beylier, Parvaneh Joharinad, Jürgen Jost et al.

Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations between triples of points and other points. More significantly, based on this curvature profile, we introduce a quantitative measure to evaluate the effectiveness of data representations, such as those produced by dimensionality reduction techniques. Furthermore, Our experiments demonstrate that this curvature-based analysis can be employed to estimate the intrinsic dimensionality of datasets. We use this to explore the large-scale geometry of empirical networks and to evaluate the effectiveness of dimensionality reduction techniques.

Lukas Silvester Barth, Hannaneh Fahimi, Parvaneh Joharinad et al.

Many machine learning algorithms try to visualize high dimensional metric data in 2D in such a way that the essential geometric and topological features of the data are highlighted. In this paper, we introduce a framework for aggregating dissimilarity functions that arise from locally adjusting a metric through density-aware normalization, as employed in the IsUMap method. We formalize these approaches as m-schemes, a class of methods closely related to t-norms and t-conorms in probabilistic metrics, as well as to composition laws in information theory. These m-schemes provide a flexible and theoretically grounded approach to refining distance-based embeddings.

Ivan P. Yamshchikov, Cyrille Merleau Nono Saha, Igor Samenko et al.

This position paper looks into the formation of language and shows ties between structural properties of the words in the English language and their polysemy. Using Ollivier-Ricci curvature over a large graph of synonyms to estimate polysemy it shows empirically that the words that arguably are easier to pronounce also tend to have multiple meanings.

Ivan P. Yamshchikov, Viacheslav Shibaev, Aleksander Nagaev et al.

This paper focuses on latent representations that could effectively decompose different aspects of textual information. Using a framework of style transfer for texts, we propose several empirical methods to assess information decomposition quality. We validate these methods with several state-of-the-art textual style transfer methods. Higher quality of information decomposition corresponds to higher performance in terms of bilingual evaluation understudy (BLEU) between output and human-written reformulations.

Benjamin Inden, Jürgen Jost

Many experiments have been performed that use evolutionary algorithms for learning the topology and connection weights of a neural network that controls a robot or virtual agent. These experiments are not only performed to better understand basic biological principles, but also with the hope that with further progress of the methods, they will become competitive for automatically creating robot behaviors of interest. However, current methods are limited with respect to the (Kolmogorov) complexity of evolved behavior. Using the evolution of robot trajectories as an example, we show that by adding four features, namely (1) freezing of previously evolved structure, (2) temporal scaffolding, (3) a homogeneous transfer function for output nodes, and (4) mutations that create new pathways to outputs, to standard methods for the evolution of neural networks, we can achieve an approximately linear growth of the complexity of behavior over thousands of generations. Overall, evolved complexity is up to two orders of magnitude over that achieved by standard methods in the experiments reported here, with the major limiting factor for further growth being the available run time. Thus, the set of methods proposed here promises to be a useful addition to various current neuroevolution methods.

Johannes Rauh, Pradeep Kr. Banerjee, Eckehard Olbrich et al.

The unique information ($UI$) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the $UI$, which implies that the $UI$ is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the $UI$ lower bounds the two-way rate and discuss implications of the conjecture.

Pradeep Kr. Banerjee, Eckehard Olbrich, Jürgen Jost et al.

Given two channels that convey information about the same random variable, we introduce two measures of the unique information of one channel with respect to the other. The two quantities are based on the notion of generalized weighted Le Cam deficiencies and differ on whether one channel can approximate the other by a randomization at either its input or output. We relate the proposed quantities to an existing measure of unique information which we call the minimum-synergy unique information. We give an operational interpretation of the latter in terms of an upper bound on the one-way secret key rate and discuss the role of the unique informations in the context of nonnegative mutual information decompositions into unique, redundant and synergistic components.

Wiktor Młynarski, Jürgen Jost

Binaural sound localization is usually considered a discrimination task, where interaural time (ITD) and level (ILD) disparities at pure frequency channels are utilized to identify a position of a sound source. In natural conditions binaural circuits are exposed to a stimulation by sound waves originating from multiple, often moving and overlapping sources. Therefore statistics of binaural cues depend on acoustic properties and the spatial configuration of the environment. In order to process binaural sounds efficiently, the auditory system should be adapted to naturally encountered cue distributions. Statistics of cues encountered naturally and their dependence on the physical properties of an auditory scene have not been studied before. Here, we performed binaural recordings of three auditory scenes with varying spatial properties. We have analyzed empirical cue distributions from each scene by fitting them with parametric probability density functions which allowed for an easy comparison of different scenes. Higher order statistics of binaural waveforms were analyzed by performing Independent Component Analysis (ICA) and studying properties of learned basis functions. Obtained results can be related to known neuronal mechanisms and suggest how binaural hearing can be understood in terms of adaptation to the natural signal statistics.