Differential geometry methods for constructing manifold-targeted recurrent neural networks.

Neural computations can be framed as dynamical processes, whereby the structure of the dynamics within a neural network are a direct reflection of the computations that the network performs. A key step in generating mechanistic interpretations within this computation through dynamics framework is to establish the link between network connectivity, dynamics and computation. This link is only partly understood. Recent work has focused on producing algorithms for engineering artificial recurrent neural networks (RNN) with dynamics targeted to a specific goal manifold. Some of these algorithms only require a set of vectors tangent to the target manifold to be computed, and thus provide a general method that can be applied to a diverse set of problems. Nevertheless, computing such vectors for an arbitrary manifold in a high dimensional state space remains highly challenging, which in practice limits the applicability of this approach. Here we demonstrate how topology and differential geometry can be leveraged to simplify this task, by first computing tangent vectors on a low-dimensional topological manifold and then embedding these in state space. The simplicity of this procedure greatly facilitates the creation of manifold-targeted RNNs, as well as the process of designing task-solving on-manifold dynamics. This new method should enable the application of network engineering-based approaches to a wide set of problems in neuroscience and machine learning. Furthermore, our description of how fundamental concepts from differential geometry can be mapped onto different aspects of neural dynamics is a further demonstration of how the language of differential geometry can enrich the conceptual framework for describing neural dynamics and computation.

Implicit–explicit gradient of nondual awareness or consciousness as such

Consciousness is multi-dimensional but is most often portrayed with a two-dimensional (2D) map that has global levels or states on one axis and phenomenal contents on the other. On this map, awareness is conflated either with general alertness or with phenomenal content. This contributes to ongoing difficulties in the scientific understanding of consciousness. Previously, I have proposed that consciousness as such or nondual awareness—a basic non-conceptual, non-propositional awareness in itself free of subject-object fragmentation—is a unique kind that cannot be adequately specified by this 2D map of states and contents. Here, I propose an implicit–explicit gradient of nondual awareness to be added as the z-axis to the existing 2D map of consciousness. This gradient informs about the degree to which nondual awareness is manifest in any experience, independent of the specifics of global state or local content. Alternatively, within the multi-dimensional state space model of consciousness, nondual awareness can be specified by several vectors, each representing one of its properties. In the first part, I outline nondual awareness or consciousness as such in terms of its phenomenal description, its function and its neural correlates. In the second part, I explore the implicit–explicit gradient of nondual awareness and how including it as an additional axis clarifies certain features of everyday dualistic experiences and is especially relevant for understanding the unitary and nondual experiences accessed via different contemplative methods, mind-altering substances or spontaneously.

Distributed Policy Evaluation with Fractional Order Dynamics in Multiagent Reinforcement Learning

The main objective of multiagent reinforcement learning is to achieve a global optimal policy. It is difficult to evaluate the value function with high-dimensional state space. Therefore, we transfer the problem of multiagent reinforcement learning into a distributed optimization problem with constraint terms. In this problem, all agents share the space of states and actions, but each agent only obtains its own local reward. Then, we propose a distributed optimization with fractional order dynamics to solve this problem. Moreover, we prove the convergence of the proposed algorithm and illustrate its effectiveness with a numerical example.

Spatiotemporal blocking of the bouncy particle sampler for efficient inference in state-space models

We propose a novel blocked version of the continuous-time bouncy particle sampler of Bouchard-Côté et al. (J Am Stat Assoc 113(522):855–867, 2018) which is applicable to any differentiable probability density. This alternative implementation is motivated by blocked Gibbs sampling for state-space models (Singh et al. in Biometrika 104(4):953–969, 2017) and leads to significant improvement in terms of effective sample size per second, and furthermore, allows for significant parallelization of the resulting algorithm. The new algorithms are particularly efficient for latent state inference in high-dimensional state-space models, where blocking in both space and time is necessary to avoid degeneracy of MCMC. The efficiency of our blocked bouncy particle sampler, in comparison with both the standard implementation of the bouncy particle sampler and the particle Gibbs algorithm of Andrieu et al. (J R Stat Soc Ser B Stat Methodol 72(3):269–342, 2010), is illustrated numerically for both simulated data and a challenging real-world financial dataset.

On the Nonlinear Observability of Polynomial Dynamical Systems

Controllability and observability are important system properties in control theory. These properties cannot be easily checked for general nonlinear systems. This paper addresses the local and global observability as well as the decomposition with respect to observability of polynomial dynamical systems embedded in a higher-dimensional state-space. These criteria are applied on some example system.

The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.

The metastable brain associated with autistic-like traits of typically developing individuals

Metastability in the brain is thought to be a mechanism involved in dynamic organization of cognitive and behavioral functions across multiple spatiotemporal scales. However, it is not clear how such organization is realized in underlying neural oscillations in a high-dimensional state space. It was shown that macroscopic oscillations often form phase-phase coupling (PPC) and phase-amplitude coupling (PAC) which result in synchronization and amplitude modulation, respectively, even without external stimuli. These oscillations can also make spontaneous transitions across synchronous states at rest. Using resting-state electroencephalographic signals and the autism-spectrum quotient scores acquired from healthy humans, we show experimental evidence that the PAC combined with PPC allows amplitude modulation to be transient, and that the metastable dynamics with this transient modulation is associated with autistic-like traits. In individuals with a longer attention span, such dynamics tended to show fewer transitions between states by forming delta-alpha PAC. We identified these states as two-dimensional metastable states that could share consistent patterns across individuals. Our findings suggest that the human brain dynamically organizes inter-individual differences in a hierarchy of macroscopic oscillations with multiple timescales by utilizing metastability.

Implicit-explicit gradient of nondual awareness or consciousness as such

Consciousness is multi-dimensional but is most often portrayed with a 2-D map that has global levels or states on one axis, and phenomenal contents on the other. On this map, phenomenal content is conflated with awareness itself, which contributes to ongoing difficulties in the scientific understanding of consciousness. Previously (Josipovic 2014, 2019; Josipovic and Miskovic, 2020) I have proposed that consciousness as such, or nondual awareness - a basic non-conceptual, non-propositional awareness in itself free of subject-object fragmentation, is phenomenally, functionally and neurobiologically, a unique kind that cannot be adequately specified by a 2-D map of levels/modes and contents. Here, I propose an implicit-explicit gradient of nondual awareness to be added as the third dimension on z-axis. an axis to the 2D map of consciousness. Alternatively, within the multi-dimensional state space model of consciousness, nondual awareness can be specified by several vectors, each representing one of its properties.I explore how including the implicit-explicit gradient of nondual awareness as an additional axis clarifies certain features of everyday dualistic experiences and is especially relevant for understanding the unitary and nondual experiences accessed via different contemplative methods, mind altering substances, or spontaneously. I discuss the relevance of this for current theories of consciousness.

Towards Analysing multivariate weather/climate extremes

<p>Extreme analysis, via e.g., GEV, was developed to deal with univariate time series, and is very difficult to extend beyond that dimension. Here we explore a different method, the archetypal analysis, which focuses on multivariate extremes. The method seeks to approximate the convex hull in high-dimensional state space, by identifying corners representing "pure" types, i.e. archetypes. The method, encompasses, in particular, the virtues of EOFs and clustering. The method is presented with a new manifold-based optimization algorithm, and applied to a number of atmospheric fields, including SST and SLP gridded data. The application to SST, in particular, reveals important features related to SST extremes. The strengths and weaknesses of the method and possible future perspectives will be discussed.</p>

Quantum Aitchison geometry

Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper, we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation, it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.