Ambient Space Research Articles

Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator.

We propose a machine-learning approach to construct reduced-order models (ROMs) to predict the long-term out-of-sample dynamics of brain activity (and in general, high-dimensional time series), focusing mainly on task-dependent high-dimensional fMRI time series. Our approach is a three stage one. First, we exploit manifold learning and, in particular, diffusion maps (DMs) to discover a set of variables that parametrize the latent space on which the emergent high-dimensional fMRI time series evolve. Then, we construct ROMs on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem, i.e., the construction of a map from the low-dimensional manifold to the original high-dimensional (ambient) space by coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using two benchmark fMRI time series: (i) a simplistic five-dimensional model of stochastic discrete-time equations used just for a "transparent" illustration of the approach, thus knowing a priori what one expects to get, and (ii) a real fMRI dataset with recordings during a visuomotor task. We show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient space; one can use instead the low-frequency truncation of the DMs function space of L2-integrable functions to predict the entire list of coordinate functions in the ambient space and to solve the pre-image problem.

Versatile Curve Design by Level Set with Quadratic Convergence.

Many 3D mesh processing tasks revolve around generating and manipulating curves on surface meshes. While it is intuitive to explicitly model these curves using mesh edges or parametric curves in the ambient space, these methods often suffer from numerical instability or inaccuracy due to the projection operation. Another natural strategy is to adapt spline based tools, these methods are quite fast but are hard to be extended to more versatile constraints and need heavy manual interactions. In this paper, we present an efficient and versatile approach to curve design based on an implicit representation known as the level set. While previous works have explored the use of the level set to generate curves with minimal length, they typically have limitations in accommodating additional conditions for rich and robust control. To address these challenges, we formulate curve editing with constraints like smoothness, interpolation, tangent control, etc., via a level set based variational problem by constraining the values or derivatives of the level set function. However, the widely used gradient flow strategy converges very slowly for this complicated variational problem compared to the classical geodesic one. Thus, we propose to solve it via Newton's method enhanced by local Hessian correction and a trust-region strategy. As a result, our method not only enables versatile control, but also excels in terms of performance due to nearly quadratic convergence and almost linear complexity in each iteration via narrow band acceleration. In practice, these advantages effectively benefit various applications, such as interactive curve manipulation, boundary smoothing for surface segmentation and path planning with obstacles as demonstrated.

Least Squares Projection Onto the Behavior for SISO LTI Models

We consider the least squares projection onto the behavior for linear time-invariant (LTI) single-input single-output (SISO) models, in which the observed input-output data are modified in a least squares (LS) sense by subtracting so-called misfits, so that the modified data satisfy a given linear dynamic relation. We show that the LS-criterion of the projection problem induces an orthogonal decomposition of the ambient data space and we characterize this decomposition by means of banded block-Toeplitz matrices, the elements of which are the coefficients of the difference equation describing the SISO LTI dynamics. We thereby generalize earlier results in the literature on autonomous LTI models to the more complicated SISO case. Additionally, we illustrate that the novel characterization is equivalent (up to a change of model representation) to results derived using (isometric) state-space representations in the literature on behavioral systems theory.

A Note on Secant-Defective Varieties and Clifford Modules

We generalise a construction of Landsberg, which associates certain Clifford algebra representations to Severi varieties. We thus obtain a new proof of Russo’s Divisibility Property for LQEL varieties whose secant varieties do not fill the ambient space. We also make an observation about the question of whether there exists a smooth, irreducible, non-degenerate, projective variety with secant deficiency delta > 8.

WITHDRAWN: A post-quantum key exchange protocol from the intersection of conics

In this paper we present a key exchange protocol in which Alice and Bob have secret keys given by two conics embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded curves. Both of them construct some common invariants given by the intersection of two conics.

Relaxation and optimal finiteness domain for degenerate quadratic functionals. One-dimensional case

The aim of this paper is the study, in the one-dimensional case, of the relaxation of a quadratic functional admitting a very degenerate weight w, which may not satisfy both the doubling condition and the classical Poincaré inequality. The main result deals with the relaxation on the greatest ambient space L0(Ω) of measurable functions endowed with the topology of convergence in measure w dx. Here w is an auxiliary weight fitting the degenerations of the original weight w. Also the relaxation w.r.t. the L2(Ω, w˜)-convergence is studied. The crucial tool of the proof is a Poincaré type inequality, involving the weights w and w, on the greatest finiteness domain Dw of the relaxed functionals.

A fundamental inequality for 3-dimensional CAT(ε) hypersurfaces

By J.F. Nash’s Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash’s Theorem effectively is finding useful relationships between intrinsic and extrinsic elements that are characterizing immersions. After 1993, when a groundbreaking work written by B.-Y. Chen on this theme was published, many explorations pursued this important avenue of inquiry. Bearing in mind this historical context, in our present paper we point out a new relationship involving intrinsic and extrinsic curvature invariants, under natural geometric conditions. We conclude our work with an obstruction to minimality, in the spirit of S.-S. Chern’s reflection from 1968.

On the Robustness of Bayesian Neural Networks to Adversarial Attacks.

Vulnerability to adversarial attacks is one of the principal hurdles to the adoption of deep learning in safety-critical applications. Despite significant efforts, both practical and theoretical, training deep learning models robust to adversarial attacks is still an open problem. In this article, we analyse the geometry of adversarial attacks in the over-parameterized limit for Bayesian neural networks (BNNs). We show that, in the limit, vulnerability to gradient-based attacks arises as a result of degeneracy in the data distribution, i.e., when the data lie on a lower dimensional submanifold of the ambient space. As a direct consequence, we demonstrate that in this limit, BNN posteriors are robust to gradient-based adversarial attacks. Crucially, by relying on the convergence of infinitely-wide BNNs to Gaussian processes (GPs), we prove that, under certain relatively mild assumptions, the expected gradient of the loss with respect to the BNN posterior distribution is vanishing, even when each NN sampled from the BNN posterior does not have vanishing gradients. The experimental results on the MNIST, Fashion MNIST, and a synthetic dataset with BNNs trained with Hamiltonian Monte Carlo and variational inference support this line of arguments, empirically showing that BNNs can display both high accuracy on clean data and robustness to both gradient-based and gradient-free adversarial attacks.

The Role and Value of Architectural Art for Game Design

Architecture is an important element of game design, and all kinds of architectural art in the game scene provide players with unlimited reverie virtual places and spaces. The article focuses on the application of architectural design in the virtual space of the game, taking the functional attributes of the place element of architecture in the composition of the virtual space as the entry point, and comparing the different attributes of architecture in the real and the virtual space of the game. Architecture and games are often seen as two unrelated fields, but they are closely related. The similarity between game design and architectural design is that they both need to be designed based on certain rules and constraints and take the user’s experience into account. Based on this commonality, game designers need to integrate the virtual environment space composed of buildings with the architectural features of the real world in the context of the game, which is one of the most important factors determining whether players can understand and immerse themselves in the game. Game designers often take inspiration from architectural design to help them create realistic environments in their games. At the same time, in games, architecture is not only used as a background for a virtual city or a realistic rendering of an existing city, but more importantly, the well-structured ambient space constituted by architecture is utilised to convey psychological cues such as emotions and feelings.

A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows

We propose a novel modular inference approach combining two different generative models — generative adversarial networks (GAN) and normalizing flows — to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. GAN-Flow does not involve Markov chain Monte Carlo simulation, making it particularly suitable for solving large-scale inverse problems. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows. Notably, one of the applications we consider involves a 65,536-dimensional inverse problem of phase retrieval wherein an object is reconstructed from sparse noisy measurements of the magnitude of its Fourier transform.

Non-negative Tucker decomposition with graph regularization and smooth constraint for clustering

Non-negative Tucker decomposition (NTD) and its graph regularized extensions are the most popular techniques for representing high-dimensional non-negative data, which are typically found in a low-dimensional sub-manifold of ambient space, from a geometric perspective. Therefore, the performance of the graph-based NTD methods relies heavily on the low-dimensional representation of the original data. However, most existing approaches treat the last factor matrix in NTD as a low-dimensional representation of the original data. This treatment leads to the loss of the original data’s multi-linear structure in the low-dimensional subspace. To remedy this defect, we propose a novel graph regularized Lp smooth NTD (GSNTD) method for high-dimensional data representation by incorporating graph regularization and an Lp smoothing constraint into NTD. The new graph regularization term constructed by the product of the core tensor and the last factor matrix in NTD, and it is used to uncover hidden semantics while maintaining the intrinsic multi-linear geometric structure of the data. The addition of the Lp smoothing constraint to NTD may produce a more accurate and smoother solution to the optimization problem. The update rules and the convergence of the GSNTD method are proposed. In addition, a randomized variant of the GSNTD algorithm based on fiber sampling is proposed. Finally, the experimental results on four standard image databases show that the proposed method and its randomized variant have better performance than some other state-of-the-art graph-based regularization methods for image clustering.

Inevitable monokineticity of strongly singular alignment

We prove that certain types of measure-valued mappings are monokinetic i.e. the distribution of velocity is concentrated in a Dirac mass. These include weak measure-valued solutions to the strongly singular Cucker–Smale model with singularity of order greater or equal to the dimension of the ambient space. Consequently, we are able to answer a couple of open questions related to the singular Cucker–Smale model. First, we prove that weak measure-valued solutions to the strongly singular Cucker–Smale kinetic equation are monokinetic, under very mild assumptions that they are uniformly compactly supported and weakly continuous in time. This can be interpreted as a rigorous derivation of the macroscopic fractional Euler-alignment system from the kinetic Cucker–Smale equation without the need to perform any hydrodynamical limit. This suggests the superior suitability of the macroscopic framework to describe large-crowd limits of strongly singular Cucker–Smale dynamics. Second, we perform a direct micro- to macroscopic mean-field limit from the Cucker–Smale particle system to the fractional Euler-alignment model. This leads to the final result—the existence of weak solutions to the fractional Euler-alignment system with almost arbitrary initial data in 1D, including the possibility of a vacuum. Existence can be extended to 2D under the a priori assumption that the density of the mean-field limit has no atoms.

Estimation of sharp geometric inequality in \(D_{\alpha}\)-homothetically deformed Kenmotsu manifolds

In this article, we investigate the Kenmotsu manifold when applied to a \(D_{\alpha}\)-homothetic deformation. Then, given a submanifold in a \(D_{\alpha}\)-homothetically deformed Kenmotsu manifold, we derive the generalized Wintgen inequality. Additionally, we find this inequality for submanifolds such as slant, invariant, and anti-invariant in the same ambient space.

Modelling the air pollution induced health effects of energy consumption across varied spaces in OECD countries: An asymmetric analysis

The air pollution-induced health effects of energy consumption remain a grey area in the extant literature. As of yet, there are very limited studies on the subject matter for countries in the Organisation for Economic Co-operation and Development (OECD). To this end, we employ panel data from 1990 to 2019 for 37 OECD countries and panel estimation techniques that control for distributional asymmetry. We also utilize a composite variable of mortality and morbidity to capture the full spectrum of air pollution-induced health burdens. Our findings reveal that renewable energy ameliorates the health burden of air pollution in indoor spaces, evidence for the ameliorating effects of renewable energy in outdoor and occupational spaces was however weak. We also show that income has an ameliorating effect on air pollution-induced health burdens across all spaces and that the effect of non-renewable energy is asymmetric and disparate across all spaces. Furthermore, while technological innovation ameliorates the health burden of air pollution in indoor spaces, it exacerbates the health burden of air pollution in both occupational and outdoor spaces. These findings thus show that the positive health implications of renewable energy policy interventions have more traction in indoor spaces. Capacity needs to be built in the renewable energy sector to reduce the pollution-inducing health burdens emanating from ambient and occupational spaces.

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Possible ambient kinematics

In a seminal paper, Bacry and Lévy–Leblond classified kinematical algebras, a class of Lie algebras encoding the symmetries of spacetime. Homogeneous spacetimes (infinitesimally, Klein pairs) associated with these possible kinematics can be partitioned into four families—riemannian, lorentzian, galilean, and carrollian—based on the type of invariant metric structure they admit. In this work, we classify possible ambient kinematics—defined as extensions of kinematical algebras by a scalar ideal—as well as their associated Klein pairs. Kinematical Klein pairs arising as quotient space along the extra scalar ideal are said to admit a lift into the corresponding ambient Klein pair. While all non-galilean Klein pairs admit a unique—trivial and torsionfree—higher-dimensional lift, galilean Klein pairs are constructively shown to admit lifts into two distinct families of ambient Klein pairs. The first family includes the bargmann algebra as well as its curved/torsional avatars while the second family is novel and generically allows lifts into torsional ambient spaces. We further comment on the relation between these two families and the maximally symmetric family of leibnizian Klein pairs.

The Discrete Theorema Egregium

In 1827, Gauss proved that Gaussian curvature is actually an intrinsic quantity, meaning that it can be calculated just from measurements within the surface. Before, curvature of surfaces could only be computed extrinsically, meaning that an ambient space is needed. Gauss named this remarkable finding Theorema Egregium. In this paper, we discuss a discrete version of this theorem for polyhedral surfaces. We give an elementary proof that the common extrinsic and intrinsic definitions of discrete Gaussian curvature are equivalent.

Generalized Minkowski Type Integral Formulas for Compact Hypersurfaces in Pseudo-Riemannian Manifolds

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature.

Risk reduction of tank explosion based on passivation of unusable propellant residues

To reduce the risk of explosion of propellant tanks of expended spacecraft and launch vehicles with liquid rocket engines in orbit, as well as in case of emergency situation, for example, loss of orientation, the Inter-Agency Space Debris Coordination Committee recommends passivation measures, including the discharge of residual liquid propellant and pressurant gas. In ANSYS-Fluent program complex possible initial positions of liquid propellant residues in a spherical tank at its rotation under conditions of low gravitational fields are determined. The values of liquid propellant residues depending on their initial position in the spherical tank at opening of the drain line for discharge of gas–liquid mixture into the ambient space are determined. The concept of formation of two-phase flows of liquid propellant on the example of the spherical tank at tangential entry of compressed gas is offered. The relationship between the number of gas inlet points and the effectiveness of the developed method (expressed as the ratio of the mass of expelled liquid propellant to the mass of gas expended) is demonstrated. For instance, the use of 2 gas inlet points achieves an efficiency of up to 30 %, while employing 3 gas inlet points increases it to 89 %.

Determining homology of an unknown space from a sample

The homology of an unknown subspace of Euclidean space can be determined from the intrinsic Čech complex of a sample of points in the subspace, without reference to the ambient Euclidean space. More precisely, given a subspace X of Euclidean space and a sample A of points in X, we give conditions for the homology of X to be isomorphic to a certain persistent homology group of the intrinsic Čech complex.