Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
The stability and robustness of compact schemes for parabolic PDEs are analyzed.
problem Stability and robustness of compact schemes for solving parabolic PDEs.
method Compact spatial discretization, Crank-Nicolson temporal discretization, eigenvalue analysis of amplification matrix.
result An upper bound on the condition number of the amplification matrix is derived, showing stability.
New NMF algorithm uses Toeplitz matrix for facial recognition.
problem Facial recognition performance improvement.
method Proposes TNMF algorithm with Toeplitz penalty for NMF.
result TNMF outperforms ZNMF and other constrained NMF algorithms.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
Study asymptotics of unitary matrix elements in quantum mechanics.
problem Asymptotic behavior of unitary matrix elements in quantum mechanics.
method Uses Berezin-Toeplitz quantization and symplectic geometry.
result Recover asymptotics of Wigner's d-matrix elements for spin representations.
Solves problem of describing transformations for upper triangular Toeplitz operators.
problem Describing coordinate transformations preserving upper triangular Toeplitz form of operator fields.
method Implicit formulas involving matrix-valued functions for describing transformations and Nijenhuis operators.
result Formulas for coordinate transformations and Nijenhuis operators in upper triangular Toeplitz form.
Paper solves a key problem in learning from high-dimensional covariance matrices.
problem Computing normalizing factors for Riemannian Gaussian distributions on high-dimensional covariance matrices.
method Equivalence with random matrix theory and log-normal matrix ensembles to approximate normalizing factors.
result Efficient approximation of normalizing factors with decreasing error as dimension increases.
Enhanced EEG classification improves motor imagery detection with less computation.
problem Improving classification accuracy of motor imagery EEG signals.
method Integrates Block-Toeplitz structure into augmented covariance matrices and uses Siegel metric.
result Significantly reduces computational time without compromising classification accuracy.
Paper speeds up GP inference by reducing precision matrix computation.
problem High computational complexity in computing kernel precision matrices.
method Splitting precision matrix into Hankel-Toeplitz matrices and computing only unique entries.
result Precision matrix computation reduced from O(NM2) to O(NM). New findings on optimization landscape of Toeplitz covariance estimation.
problem Understanding the geometry of the Gaussian maximum-likelihood objective for Toeplitz covariance estimation.
method Overparameterized Carathéodory representation of positive definite Toeplitz covariance matrices, focusing on both amplitudes and frequencies.
result Joint optimization of amplitudes and frequencies leads to a benign population landscape, allowing for global recovery of the true Toeplitz covariance.
Improved singular value approximation for convolutional layers.
problem Improving accuracy of singular value approximation for linear convolutional layers.
method Developed a new spectral density matrix method for singular value approximation with improved accuracy and reduced computational complexity.
result Obtained moderate improvement in singular value distribution compared to circular approximation.
New method tightens Lipschitz bounds for CNNs efficiently.
problem Lipschitz regularization of Convolutional Neural Networks (CNNs).
method Using Toeplitz matrix theory, introduces a tight and computationally efficient upper bound for convolutional layers.
result Developed an algorithm to train Lipschitz regularized CNNs.
Proposes a method for forecasting large-scale interval-valued time series.
problem Modeling and forecasting large-scale interval-valued time series.
method Feature extraction procedure involving auto-segmentation, clustering, and precision matrix estimation.
result The method enhances forecasting performance for large-scale interval-valued time series.
SKI speeds up Toeplitz Neural Networks by avoiding explicit decay bias and using frequency response.
problem Efficiently compute and update Toeplitz matrices in neural networks.
method Sparse plus low-rank decomposition, asymmetric SKI, frequency response modeling.
result Achieved significant speedup with minimal performance loss.
The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow define…
This paper considers the problem of robustly estimating a structured covariance matrix with an elliptical underlying distribution with known mean. In applications where the covariance matrix naturally possesses a certain structure, taking the prior structure information into account in the estimation procedure is benef…
New star-product defined on Poisson manifolds using Toeplitz operators.
problem Defining star-products on Poisson manifolds induced by symplectic Lie algebroids.
method Using Toeplitz operators on groupoids with Heisenberg group structure.
result Generalization of Guillemin and Melrose's symplectic approach.
We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for no…
Quantizes symplectic manifolds with toric singularities using Toeplitz operators.
problem Quantize symplectic manifolds with toric singularities.
method Establishes quantization for compact toric symplectic manifolds with transversal singular real polarizations using Toeplitz operators.
result Toeplitz operators determine a star product on compact toric symplectic manifolds with toric singularities as ℏo0+. Study small eigenvalues of Toeplitz operators and their relation to Mabuchi geodesics.
problem Analyzing small eigenvalues of Toeplitz operators on complex projective manifolds.
method Proving the existence of exponentially decaying eigenvalues for Toeplitz operators with specific symbols, and establishing a connection to Mabuchi geodesics.
result Logarithmic distribution of small eigenvalues correlates with Mabuchi geodesics between polarizations.
We study the Berezin-Toeplitz quantization on Kaehler manifolds. We explain first how to compute various associated asymptotic expansions, then we compute explicitly the first terms of the expansion of the kernel of the Berezin-Toeplitz operators, and of the composition of two Berezin-Toeplitz operators. As application…
Constructs families of Toeplitz operators for symplectic fibrations.
problem Quantization of symplectic fibrations.
method Smooth families of Szegö projections and Toeplitz operators.
result Deformation quantization of prequantizable symplectic fibrations.
Formula for Toeplitz operator kernel on CR manifolds.
problem Analyzing Toeplitz operators on CR manifolds.
method Formula for the symbol of the kernel, asymptotic expansions.
result Formula for the values at the diagonal of the second coefficient in the expansion of the symbol of the kernel.
We prove an off-diagonal expansion for a Toeplitz operator with an indicator function.
problem Asymptotics of Toeplitz operators with indicator function
method Off-diagonal expansion
result We extend two results to the non-compact setting.
We survey recent results about the asymptotic expansion of Toeplitz operators and their kernels, as well as Berezin-Toeplitz quantization. We deal in particular with calculation of the first coefficients of these expansions.
Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.
problem Quantization of symplectic manifolds with bounded geometry.
method Berezin-Toeplitz quantization theory.
result Correct semiclassical limit achieved.
Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.
problem Analyzing the asymptotics of weighted Bergman kernels for submultiplicative filtrations.
method Demonstrated that weight operator is a Toeplitz operator; analyzed asymptotics of weighted Bergman kernels.
result Local refinement of convergence of jumping measures towards geodesic ray pushforward measure.
Paper extends index theorem to odd-dimensional manifolds with even-dimensional boundaries.
problem Index theorem for odd-dimensional manifolds with boundaries.
method Equivariant Toeplitz index theory.
result Established equivariant version of Dai-Zhang's theorem.
We obtain the semi-classical expansion of the kernels and traces of Toeplitz operators with $\cC^k$--\,symbol on a symplectic manifold. We also give a semi-classical estimate of the distance of a Toeplitz operator to the space of self-adjoint and multiplication operators.
New algorithm optimizes matrix reordering for noisy disordered matrices.
problem Optimizing matrix reordering for noisy disordered matrices in single-cell biology and metagenomics.
method Proposed a polynomial-time adaptive sorting algorithm to improve upon spectral seriation.
result Our algorithm achieves superior performance compared to existing methods in real datasets.
We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin-Toeplitz quantization obtained recently by Ma-Marinescu, and of the composition of two Berezin-Toeplitz quantizations. Our main tool is the stationary phase formula of Melin-Sjöstrand.
Study recovers C*-algebra from fields of Toeplitz algebras on specific groups.
problem Recovering C*-algebra from fields of Toeplitz algebras on specific groups.
method Using continuous fields of Toeplitz algebras and a crossed product.
result Algebra of principal symbols can be recovered from fields of Toeplitz algebras.
Isomorphic algebra connects Toeplitz to Heisenberg group.
problem Connecting Toeplitz algebra to Heisenberg group.
method Isomorphism between Toeplitz algebra and Heisenberg group ideal.
result Found isomorphism between algebra and Heisenberg group ideal.
For phase-space manifolds which are compact Kaehler manifolds relations between the Berezin-Toeplitz quantization and the quantization with the help of Berezin's coherent states and symbols are studied. First the results on the Berezin-Toeplitz quantization of arbitrary compact Kaehler manifolds due to Bordemann, Meinr…
A new method solves complex financial equations efficiently.
problem Solving worst-case and best-case prices for two-factor uncertain volatility models.
method Decompose and integrate, then optimize; piecewise constant control; closed-form Green's functions; 2D convolution integrals; monotone numerical integration; Fast Fourier Transforms.
result The method efficiently computes the value function and optimal control, converging to the viscosity solution of the HJB equation.
A new model uses Toeplitz matrices to analyze time-series data transitions.
problem Analyzing transitions in time-series data from nonautonomous systems.
method Deep Koopman-layered models with learnable Toeplitz matrices, leveraging Toeplitz matrices' universal property.
result The model demonstrates universality and generalization, outperforming existing methods.
The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.
problem Quantization of compact symplectic manifolds with higher Landau levels.
method Develops Berezin-Toeplitz quantization using a Bochner Laplacian with specific spectral properties.
result The quantization provides a formal star-product for the lowest Landau level.
Using geometric quantization, we represent curve operators in the TQFT of Witten-Reshetikhin-Turaev with jauge group SU_2 as Toeplitz operators with symbols corresponding to trace functions. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, an…
Quantum propagation studied for Berezin-Toeplitz operators.
problem Asymptotic behavior of quantum propagators and spectral projectors.
method Geometric analysis of Hamiltonian flows and Maslov indices.
result Introduction of quantum states associated with Lagrangian submanifolds.
Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin-Toeplitz quantization of the open set of M where the curvature on L is non-degenerate. The quantum spaces are the spectral spaces corresponding to [0,k−N] (N>1 fixed), of the Kodaira…
Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global Toeplitz operator…
Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.
We establish some subprincipal estimates for Berezin-Toeplitz operators on symplectic compact manifolds. From this, we construct a family of subprincipal symbol maps and we prove that these maps are the only ones satisfying some expected conditions.
We consider Toeplitz operators associated with the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigenfunctions of a self-adjoint Toeplitz opera…
We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of…
Study of spectral flow in symmetric Toeplitz operator families.
problem Understanding spectral flow in families of symmetric Toeplitz operators.
method Analog of Atiyah-Singer-Robbin-Salamon theorem for Z2-valued spectral flow. result Graded secondary spectral flow equals secondary index of a Callias-type operator.
We compute the second coefficient of the composition of two Berezin-Toeplitz operators associated with the spinc Dirac operator on a symplectic manifold, making use of the full-off diagonal expansion of the Bergman kernel.
Researchers calculate the second coefficient in the expansion of a Toeplitz operator.
problem Analyzing the second coefficient in the semi-classical expansion of Toeplitz operators.
method Functional calculus of Toeplitz operators with Reeb vector fields and asymptotic analysis.
result The second coefficient of the expansion is calculated.