arXiv research

Guillemin trace formula adapted for group actions.

Introduces trace operator for quasi-plurisubharmonic functions on Kähler manifolds.

Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.

The Ray-Singer analytic torsion is the zeta-function trace of a certain sum of logarithm operators on the de Rham complex. In this note we examine the residue analytic torsion, defined using the residue-trace instead of the spectral zeta function quasi-trace.

Non-injectivity proven for trace map on character varieties.

We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier de…

We study the heat trace for both the drifting Laplacian as well as Schrödinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder…

A semicontinuous semifinite trace is constructed on the C*-algebra generated by the finite propagation operators acting on the L^2-sections of a hermitian vector bundle on an amenable open manifold of bounded geometry. This trace is the semicontinuous regularization of a functional already considered by J. Roe. As an a…

Sharp inequalities in unit ball with constraints on moments.

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and $n-2$ polynomial functions defined recursively.

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Trace and inverse trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3, 2026--2040.). Namely, we show that equality of the estimate holds if and only if the manifold …

This work proposes an adaptive trace lasso regularized L1-norm based graph cut method for dimensionality reduction of Hyperspectral images, called as `Trace Lasso-L1 Graph Cut' (TL-L1GC). The underlying idea of this method is to generate the optimal projection matrix by considering both the sparsity as well as the corr…

We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trac…

We present a new trace estimator of the matrix whose explicit form is not given but its matrix multiplication to a vector is available. The form of the estimator is similar to the Hutchison stochastic trace estimator, but instead of the random noise vectors in Hutchison estimator, we use small number of probing vectors…

Ray tracing sampler improves neural network sampling efficiency and resilience.

Using Laurent expansions of the Kontsevich-Vishik canonical trace of holomorphic families of classical pseudodifferential operators, we define functionals on the space of Riemannian metrics and investigate their conformal properties, thereby giving a unified description of several conformal invariants and anomalies.

The trace norm is widely used in multi-task learning as it can discover low-rank structures among tasks in terms of model parameters. Nowadays, with the emerging of big datasets and the popularity of deep learning techniques, tensor trace norms have been used for deep multi-task models. However, existing tensor trace n…

Develops interpolation methods for matrix functions in statistics and machine learning.

Develops a new nonparametric trace regression model for high-dimensional data.

Study of Hamiltonian flows on character varieties for self-intersecting curves.

New principle for supersymmetric localization on Lie groups.

This paper proposes a methodology for host-based anomaly detection using a semi-supervised algorithm namely one-class classifier combined with a PCA-based feature extraction technique called Eigentraces on system call trace data. The one-class classification is based on generating a set of artificial data using a refer…

Formula derived for zeta functions of 3D foliated systems.

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B…

Simplified construction recovers Todd class using algebraic methods.

Unified formula for higher traces of linear maps on finite-dimensional normed spaces.

This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is $n-1$, then the metric is a warped product where t…

Using the $\ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, ca…

We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this …

Quantization of the Teichmüller space of a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $γ$ in $S$ gives rise to a natural trace-of-monodromy function $\mathbb{I}(γ)$ on the Teichmüller space. For any…

We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry …

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…

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifol…

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a uniq…

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obt…

This is a continuation of our previous work arXiv:1601.05617 on trace and inverse trace of Steklov eigenvalues. More new inequalities for the trace and inverse trace of Steklov eigenvalues are obtained.

The flat trace of geodesic Koopman operators varies with negatively curved surfaces.

Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic space…

Study the Bochner-Schrödinger operator's trace in semiclassical limit.

Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.

New Finsler metrics describe trace function growth rates in convex projective surfaces.

Study of torus surgeries on knot traces, finding exotic surfaces and traces.

Policy evaluation with linear function approximation is an important problem in reinforcement learning. When facing high-dimensional feature spaces, such a problem becomes extremely hard considering the computation efficiency and quality of approximations. We propose a new algorithm, LSTD($λ$)-RP, which leverages rando…

Classifies knot traces with specific trisection genus limits.

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…

Study trace systoles on surfaces, finding optimal bounds and implications.

RKT model improves knowledge tracing by considering exercise relations and student forget behavior.

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein …