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.

$$\alpha $$-Dirac-harmonic maps from closed surfaces

Abstract$$\alpha $$ α -Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $$\alpha $$ α -harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For $$\alpha >1$$ α > 1 , the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing $$\alpha $$ α -harmonic maps for $$\alpha >1$$ α > 1 and then letting $$\alpha \rightarrow 1$$ α → 1 . The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $$\alpha $$ α -Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By $$\varepsilon $$ ε -regularity and suitable perturbations, we can then show that such a sequence of perturbed $$\alpha $$ α -Dirac-harmonic maps converges to a smooth coupled $$\alpha $$ α -Dirac-harmonic map.

Smooth functorial field theories from B-fields and D-branes

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.

Harmonic maps with torsion

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan (1924). The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.

On the p-pseudoharmonic map heat flow

In this paper, we consider the heat flow for [Formula: see text]-pseudoharmonic maps from a closed Sasakian manifold [Formula: see text] into a compact Riemannian manifold [Formula: see text]. We prove global existence and asymptotic convergence of the solution for the [Formula: see text]-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold [Formula: see text] is non-positive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the [Formula: see text]-pseudoharmonic map heat flow as well when its initial [Formula: see text]-energy is sufficiently small.

RoCGAN: Robust Conditional GAN

Conditional image generation lies at the heart of computer vision and conditional generative adversarial networks (cGAN) have recently become the method of choice for this task, owing to their superior performance. The focus so far has largely been on performance improvement, with little effort in making cGANs more robust to noise. However, the regression (of the generator) might lead to arbitrarily large errors in the output, which makes cGANs unreliable for real-world applications. In this work, we introduce a novel conditional GAN model, called RoCGAN, which leverages structure in the target space of the model to address the issue. Specifically, we augment the generator with an unsupervised pathway, which promotes the outputs of the generator to span the target manifold, even in the presence of intense noise. We prove that RoCGAN share similar theoretical properties as GAN and establish with both synthetic and real data the merits of our model. We perform a thorough experimental validation on large scale datasets for natural scenes and faces and observe that our model outperforms existing cGAN architectures by a large margin. We also empirically demonstrate the performance of our approach in the face of two types of noise (adversarial and Bernoulli).

Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

Discrete Stochastic Regulator on a Manifold, Minimizing Dispersion of the Output Macrovariable

A theoretical result is presented in the form of a new algorithm for the synthesis of a control system over a non-linear object, whose mathematical model represents a stochastic matrix difference equation having noise with a zero mean and finite dispersion in the righthand part. The new algorithm for synthesizing stochastic control for such an object is based on a three-stage procedure. In the first stage, the structure of the control system is formed in accordance with the classical method of analytical design of aggregated regulators (ADAR) in a fixed-noise assumption. In the second stage, the conditional mathematical expectation of the resulting expression for the first-stage control is determined. In the third stage, the control model is refined by excluding the noise variable from the control formula based on decomposing the initial control system affected by the new control. It is shown that the proposed control strategies minimize the target macro variable dispersion and ensure a stable, on average, achievement of the target manifold. A detailed example of an application of the algorithm for synthesizing control over the motion of an immobile center of mass is given, whose analog is represented by the objects such as by robot-manipulators, is given. The results of numerical modeling are presented, which confirm the operability of the constructed controller. Numerical simulations of the designed control system was performed using the authentic working equipment data.

Riemannian submersions whose total manifolds admit a Ricci soliton

In this paper, we study Riemannian submersions whose total manifolds admit a Ricci soliton. Here, we characterize any fiber of such a submersion is Ricci soliton or almost Ricci soliton. Indeed, we obtain necessary conditions for which the target manifold of Riemannian submersion is a Ricci soliton. Moreover, we study the harmonicity of Riemannian submersion from Ricci soliton and give a characterization for such a submersion to be harmonic.

Regularity of Dirac-harmonic maps with $$\lambda $$-curvature term in higher dimensions

In this paper, we will study the partial regularity for stationary Dirac-harmonic maps with $$\lambda $$λ-curvature term. For a weakly stationary Dirac-harmonic map with $$\lambda $$λ-curvature term $$(\phi ,\psi )$$(ϕ,ψ) from a smooth bounded open domain $$\Omega \subset {\mathbb {R}}^m$$Ω⊂Rm with $$m\ge 2$$m≥2 to a compact Riemannian manifold N, if $$\psi \in W^{1,p}(\Omega )$$ψ∈W1,p(Ω) for some $$p>\frac{2m}{3}$$p>2m3, we prove that $$(\phi , \psi )$$(ϕ,ψ) is smooth outside a closed singular set whose $$(m-2)$$(m-2)-dimensional Hausdorff measure is zero. Furthermore, if the target manifold N does not admit any harmonic sphere $$S^l$$Sl, $$l=2,\ldots , m-1$$l=2,…,m-1, then $$(\phi ,\psi )$$(ϕ,ψ) is smooth.