In the spirit of Ray and Singer we define a complex valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex valued Reidemeister torsion, including its phase. …
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48 results for “complex-valued torsion”
We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bi…
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.
In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian structures, to non-selfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we verify, up to sign, the conjecture about the comparison of the Milnor-Turaev torsi…
We extend the main result in the previous paper of Zhang and the author relating the Milnor-Turaev torsion with the complex valued analytic torsion to the equivariant case.
Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.
problem Addressing surfaces in Riemann-Cartan geometry with nontrivial torsion.
method Introducing a complex-valued 2-form associated with the torsion, which interacts with other geometric concepts.
result Complex-valued mean curvature quantity interacts with Hopf differential and Gauss map.
Complex Valued Analytic Torsion for Flat Bundles and for Holomorphic Bundles with (1,1) Connectionsmath.DG
The work of Ray and Singer which introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting t…
Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.
problem Behavior of dynamical zeta functions at the origin for compact hyperbolic manifolds.
method Uses complex-valued torsion instead of Ray-Singer analytic torsion.
result Holomorphicity and value at s=0 for twisted Ruelle zeta function for arbitrary representations.
Survey on twisted dynamical zeta functions and Fried's conjecture.
problem Analyzing twisted dynamical zeta functions and their relation to Fried's conjecture.
method Review of existing literature and mini-course presentation.
result Discussion and validation of Fried's conjecture.
Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.
problem Analyzing the twisted Ruelle zeta function on hyperbolic manifolds.
method Investigating the twisted Ruelle zeta function associated with geodesic flow and acyclic representations.
result The twisted Ruelle zeta function equals the square of the refined analytic torsion multiplied by an exponential involving the eta invariant.
We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
problem Calculating the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
method Renormalization of the Chern-Simons invariant using asymptotics along an equidistance foliation.
result The leading coefficient introduces a complex-valued quantity consisting of mean curvature and torsion 2-form.
Co-Euler structures were studied by Burghelea and Haller on closed manifolds as dual objects to Euler structures. We extend the notion of co-Euler structures to the situation of compact manifolds with boundary. As an application, by studying their variation with respect to smooth changes of the Riemannian metric, co-Eu…
Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray-Singer torsion, the Milnor-Turaev torsion, and the dynamical torsion. They are associated …
New method creates minimal submanifolds using complex-valued eigenfunctions.
problem Creating minimal submanifolds in compact Riemannian manifolds.
method Employing complex-valued eigenfunctions.
result Manufactured minimal submanifolds in compact Riemannian manifolds.
Complex-valued neural networks avoid spurious local minima.
problem Finding spurious local minima in neural networks.
method Proved no spurious local minima for shallow complex neural networks with quadratic activations.
result Complex-valued weights eliminate spurious local minima in neural networks.
A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors followed by (2) taking the absolute value of every entry…
In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…
Complex-valued (p,q)-harmonic morphisms defined and studied.
problem Defining and studying (p,q)-harmonic morphisms between Riemannian manifolds.
method Introducing and characterizing complex-valued (p,q)-harmonic morphisms.
result Characterization and new non-trivial examples of complex-valued (p,q)-harmonic morphisms.
The paper uses complex-valued functions to simplify plane differential geometry and kinematics.
problem Simplifying complex problems in plane differential geometry and kinematics.
method Consistent use of complex-valued functions of a real variable.
result Derives results in a particularly simple, uniform, and transparent way.
Bayesian sparsification improves complex-valued neural networks by 50-100x with minimal performance loss.
problem Efficiently compressing complex-valued neural networks for embedded systems.
method Extending Sparse Variational Dropout to complex-valued networks and conducting a numerical study.
result Achieved state-of-the-art performance on MusicNet with 50-100x compression.
Latent-space metrics for Complex-Valued VAE out-of-distribution detection under radar cluttereess.SP
Study complex-valued VAEs for radar OOD detection.
problem Detecting out-of-distribution signals in complex radar environments.
method Proposed and compared several detection metrics for CVAE.
result CVAE-MSE and latent-based scores outperform ANMF-Tyler.
The construction of synthetic complex-valued signals from real-valued observations is an important step in many time series analysis techniques. The most widely used approach is based on the Hilbert transform, which maps the real-valued signal into its quadrature component. In this paper, we define a probabilistic gene…
Complex-valued neural networks are not a new concept, however, the use of real-valued models has often been favoured over complex-valued models due to difficulties in training and performance. When comparing real-valued versus complex-valued neural networks, existing literature often ignores the number of parameters, r…
We study the curvature of a manifold on which there can be defined a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form. We also give a necessary curvature condition for the exis…
Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudo kernel, is needed to learn any function in complex-valued fields. Accordingly, we…
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
Complex-Valued Restricted Boltzmann Machine for Direct Speech Parameterization from Complex Spectraeess.AS
This paper describes a novel energy-based probabilistic distribution that represents complex-valued data and explains how to apply it to direct feature extraction from complex-valued spectra. The proposed model, the complex-valued restricted Boltzmann machine (CRBM), is designed to deal with complex-valued visible unit…
C-SURE improves complex-valued deep learning models by shrinking estimates, outperforming MLE and SurReal.
problem Improving accuracy and robustness of complex-valued deep learning models.
method Proposes a Stein's unbiased risk estimate (SURE) for complex-valued data and integrates it into a prototype CNN classifier.
result C-SURE outperforms SurReal and MLE in accuracy and robustness on complex-valued datasets.
Over the last decade, both the neural network and kernel adaptive filter have successfully been used for nonlinear signal processing. However, they suffer from high computational cost caused by their complex/growing network structures. In this paper, we propose two random Euler filters for complex-valued nonlinear filt…
This paper proves a generalization bound for complex-valued neural networks scaling with spectral complexity.
problem Ensuring the performance of complex-valued neural networks on unseen data.
method Theoretical derivation using Maurey Sparsification Lemma and Dudley Entropy Integral, empirical validation on various datasets.
result The spectral complexity of weight matrices is a significant factor in the generalization ability of complex-valued neural networks.
This research explores complex-valued neural networks and their implementation.
problem The challenges of implementing complex-valued neural networks and their potential for non-complex data.
method Detailed theory and implementation of CVNN, including Wirtinger calculus, complex backpropagation, and modules like complex layers and activation functions. Python implementation using cvnn toolbox.
result Demonstrates the potential of CVNN for non-complex data through simulations.
We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complex-valued kernel least-mean-square (gCKLMS). We borrow from the new results on widely linear reproducing kernel Hilbert space (WL-RKHS) for nonlinear regression and complex-valued signals, recently proposed by th…
New submersion proves complex-valued harmonic map existence.
problem Existence of non-constant harmonic morphisms.
method Constructing harmonic Riemannian submersions from symmetric spaces.
result Existence of non-constant, globally defined complex-valued harmonic morphism.
CVNN outperforms RVNN on non-circular data.
problem Classifying complex-valued data with statistical dependence.
method Comparison of CVNN and RVNN on non-circular data.
result CVNN outperforms RVNN in accuracy and generalization.
CVNNs improve performance in tasks with complex-valued inputs.
problem Improving performance in tasks with complex-valued inputs.
method Analyze the approximation properties of complex-valued neural networks (CVNNs).
result Quantitative approximation bounds for CVNNs, showing error scales as .
In this note, we report the back propagation formula for complex valued singular value decompositions (SVD). This formula is an important ingredient for a complete automatic differentiation(AD) infrastructure in terms of complex numbers, and it is also the key to understand and utilize AD in tensor networks.
We describe the relationship between complex-valued harmonic morphisms from Minkowski 4-space} and the shear-free ray congruences of mathematical physics. Then we show how a horizontally conformal submersion on a domain of Euclidean 3-space gives the boundary values at infinity of a complex-valued harmonic morphism on …
Complex-valued signals are used in the modeling of many systems in engineering and science, hence being of fundamental interest. Often, random complex-valued signals are considered to be proper. A proper complex random variable or process is uncorrelated with its complex conjugate. This assumption is a good model of th…
CAP-BM learns complex-valued data's amplitude and phase distributions.
problem Learning from complex-valued data with amplitude variation.
method Complex Amplitude-Phase Boltzmann machine (CAP-BM) with Gibbs sampling.
result Necessity of amplitude-amplitude coupling term in CAP-BM.
PolarBM models complex-valued audio signals in polar coordinates, improving over conventional methods.
problem Discarding structural information in complex-valued problems simplifies models but loses important amplitude-phase relationships.
method Proposes PolarBM, a novel Boltzmann machine for complex-valued variables in polar coordinates, and LogPolarBM for logarithmic amplitude.
result PolarBM and LogPolarBM achieve superior modeling accuracy compared to conventional models, including deep neural networks.
Complex-valued neural networks can approximate any continuous function with bounded widths and depths.
problem Approximating continuous functions with complex-valued neural networks of bounded widths and depths.
method Analyzing activation functions and proving universality for complex-valued networks.
result Deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor -affine.
Survey of complex-valued neural networks for improved performance.
problem Lack of complex-valued neural networks in machine learning frameworks.
method Literature review of CVNNs.
result Advantages of CVNNs over real-valued neural networks.
There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. I…
Risk diversification is one of the dominant concerns for portfolio managers. Various portfolio constructions have been proposed to minimize the risk of the portfolio under some constrains including expected returns. We propose a portfolio construction method that incorporates the complex valued principal component anal…
We construct new complex-valued harmonic morphisms from Euclidean spaces from functions which are holomorphic with respect to Hermitian structures. In particular, we give the first global examples of complex-valued harmonic morphisms from for each which do not arise from a Kähler structure; it is know…
The paper studies geometric representations of submanifolds using complex-valued functions.
problem Exploring the geometry of codimension-2 submanifolds.
method Implicitly representing submanifolds by complex-valued functions and showing a prequantum bundle structure.
result The space of implicit representations admits a prequantum bundle structure over the space of submanifolds.
In this paper, we generalize Huber's criterion to multichannel sparse recovery problem of complex-valued measurements where the objective is to find good recovery of jointly sparse unknown signal vectors from the given multiple measurement vectors which are different linear combinations of the same known elementary vec…
New complex-valued maps found on complex geometries.
problem Finding new maps on complex geometries.
method Solving non-linear PDEs based on manifold geometry.
result Constructed new proper biharmonic and (2,1)-harmonic maps.