This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations and metric realization of one-parameter filtrations of simplicial complexes. Inferring topological information from combinatorial models which have been built according to metric relations (Vietoris-Rips complexes) has proven useful in topological data analysis and general machine learning applications. This encourages the use of such objects for geometric inference. We extend this research direction by proposing a clear theoretical pipeline that not only provides a comprehensive guide for assigning a (triangulated) metric space to every admissible one- parameter filtration of simplicial complexes but also offers a method for merging these objects. With this, our method presents 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 with known geometries and in applications to data, in particular from computational biology.

In spatial cognition, the Successor Representation (SR) from reinforcement learning provides a compelling candidate of how predictive representations are used to encode space. In particular, hippocampal place cells are hypothesized to encode the SR. Here, we investigate how varying the temporal symmetry in learning rules influences those representations. To this end, we use a simple local learning rule which can be made insensitive to the temporal order. We analytically find that a symmetric learning rule results in a successor representation under a symmetrized version of the experienced transition structure. We then apply this rule to a two-layer neural network model loosely resembling hippocampal subfields CA3 - with a symmetric learning rule and recurrent weights - and CA1 - with an asymmetric learning rule and no recurrent weights. Here, when exposed repeatedly to a linear track, neurons in our model in CA3 show less shift of the centre of mass than those in CA1, in line with existing empirical findings. Investigating the functional benefits of such symmetry, we employ a simple reinforcement learning agent which may learn symmetric or classical successor representations. Here, we find that using a symmetric learning rule yields representations which afford better generalization, when the agent is probed to navigate to a new target without relearning the SR. This effect is reversed when the state space is not symmetric anymore. Thus, our results hint at a potential benefit of the inductive bias afforded by symmetric learning rules in areas employed in spatial navigation, where there naturally is a symmetry in the state space.

Abstract This is the first of a series of papers that develop a systematic bridge between constructions in discrete mathematics and the corresponding continuous analogs. In this paper, we establish an equivalence between Forman’s discrete Morse theory on a simplicial complex and the continuous Morse theory (in the sense of any known non-smooth Morse theory) on the associated order complex via the Lovász extension. Furthermore, we propose a new version of the Lusternik–Schnirelman category on abstract simplicial complexes to bridge the classical Lusternik–Schnirelman theorem and its discrete analog on finite complexes. More generally, we can suggest a discrete Morse theory on hypergraphs by employing piecewise-linear (PL) Morse theory and Lovász extension, hoping to provide new tools for exploring the structure of hypergraphs.

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)."I do not need a friend who changes when I change and who nods when I nod; my shadow does that much better."(Plutarch, Quomodo adulator ab amico internoscatur 53b 10)

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.

Structural inhomogeneities in synaptic efficacies have a strong impact on population response dynamics of cortical networks and are believed to play an important role in their functioning. However, little is known about how such inhomogeneities could evolve by means of synaptic plasticity. Here we present an adaptive model of a balanced neuronal network that combines two different types of plasticity, STDP and synaptic scaling. The plasticity rules yield both long-tailed distributions of synaptic weights and firing rates. Simultaneously, a highly connected subnetwork of driver neurons with strong synapses emerges. Coincident spiking activity of several driver cells can evoke population bursts and driver cells have similar dynamical properties as leader neurons found experimentally. Our model allows us to observe the delicate interplay between structural and dynamical properties of the emergent inhomogeneities. It is simple, robust to parameter changes and able to explain a multitude of different experimental findings in one basic network.

The information that two random variables Y, Z contain about a third random variable X can have aspects of shared information (contained in both Y and Z), of complementary information (only available from (Y, Z) together) and of unique information (contained exclusively in either Y or Z). Here, we study measures SĨ of shared, UĨ unique and CĨ complementary information introduced by Bertschinger et al. [1] which are motivated from a decision theoretic perspective. We find that in most cases the intuitive rule that more variables contain more information applies, with the exception that SĨ and CĨ information are not monotone in the target variable X. Additionally, we show that it is not possible to extend the bivariate information decomposition into SĨ, UĨ and CĨ to a non-negative decomposition on the partial information lattice of Williams and Beer [2]. Nevertheless, the quantities UĨ, SĨ and CĨ have a well-defined interpretation, even in the multivariate setting.

Many neurons have epochs in which they fire action potentials in an approximately periodic fashion. To see what effects noise of relatively small amplitude has on such repetitive activity we recently examined the response of the Hodgkin-Huxley (HH) space-clamped system to such noise as the mean and variance of the applied current vary, near the bifurcation to periodic firing. This article is concerned with a more realistic neuron model which includes spatial extent. Employing the Hodgkin-Huxley partial differential equation system, the deterministic component of the input current is restricted to a small segment whereas the stochastic component extends over a region which may or may not overlap the deterministic component. For mean values below, near and above the critical values for repetitive spiking, the effects of weak noise of increasing strength is ascertained by simulation. As in the point model, small amplitude noise near the critical value dampens the spiking activity and leads to a minimum as noise level increases. This was the case for both additive noise and conductance-based noise. Uniform noise along the whole neuron is only marginally more effective in silencing the cell than noise which occurs near the region of excitation. In fact it is found that if signal and noise overlap in spatial extent, then weak noise may inhibit spiking. If, however, signal and noise are applied on disjoint intervals, then the noise has no effect on the spiking activity, no matter how large its region of application, though the trajectories are naturally altered slightly by noise. Such effects could not be discerned in a point model and are important for real neuron behavior. Interference with the spike train does nevertheless occur when the noise amplitude is larger, even when noise and signal do not overlap, being due to the instigation of secondary noise-induced wave phenomena rather than switching the system from one attractor (firing regularly) to another (a stable point).