In this paper, we introduce new classes of divergences by extending the definitions of the Bregman divergence and the skew Jensen divergence. These new divergence classes (g-Bregman divergence and skew g-Jensen divergence) satisfy some properties similar to the Bregman or skew Jensen divergence. We show these g-diverge…
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48 results for “Jensen's Trace Inequality”
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
problem Computing divergence between Gaussian measures in infinite-dimensional Hilbert space.
method Closed form expression and regularization for divergence calculation.
result Closed form expression and regularization for Geometric Jensen-Shannon divergence.
The paper proves a Jensen's inequality in spaces with lower bounded curvature.
problem Proving Jensen's inequality in geodesic spaces with curvature constraints.
method Using properties of tangent cones and gradients for semi-concave functions in spaces with lower bounded curvature.
result The inequality holds for geodesically convex functions in spaces with curvature lower bounded.
On Jensen's inequality for generalized Choquet integral with an application to risk aversionq-fin.RM
In the paper we give necessary and sufficient conditions for the Jensen inequality to hold for the generalized Choquet integral with respect to a pair of capacities. Next, we apply obtained result to the theory of risk aversion by providing the assumptions on utility function and capacities under which an agent is risk…
Improved MMWU algorithm achieves instance-optimal regret bound for matrix LEA.
problem Matrix Learning from Expert Advice problem.
method Developed a general potential-based framework for matrix LEA, using a new Jensen's trace inequality.
result Achieved instance-optimal regret bound of .
Mathematical study of excess growth rate connects info theory with finance.
problem Understanding the excess growth rate in portfolio theory.
method Axiomatic characterization theorems of excess growth rate in terms of relative entropy, Jensen's inequality gap, and logarithmic divergence.
result Established rich connections between information theory and finance.
Paper improves particle variational inference by optimizing generalization error bound.
problem Improving the diversity of models in particle variational inference to enhance generalization.
method Develops a new second-order Jensen inequality with a repulsion term based on the loss function, leading to a tighter generalization error bound.
result The proposed PVI optimizes the generalization error bound directly, improving performance compared to existing methods.
Extends online learning to metric spaces using exponential weights.
problem Online learning in metric spaces.
method Exponentially weighted average forecaster, barycenters, Jensen's inequality, measure contraction property.
result Results in a statistical learning framework.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.
problem Characterizing holomorphic automorphisms with high entropy on hyperkähler manifolds.
method Using Jensen's inequality and properties of stable and unstable distributions, the authors show uniform contraction and expansion, leading to the conclusion that the manifold is birational to a torus quotient.
result Holomorphic automorphisms with high entropy on hyperkähler manifolds are Kummer examples.
The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
Sharp inequalities in unit ball with constraints on moments.
problem Establishing Sobolev trace inequalities with constraints.
method Constructing smooth test functions for higher order moments.
result Almost optimal Sobolev trace inequalities for 2nd and 4th orders.
Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.
problem Classifying conformally flat hypersurfaces and characterizing rotational hypersurfaces.
method Derives a sharp inequality relating eigenvalues of trace-free matrices and applies it to hypersurfaces.
result New proof of the classification of conformally flat hypersurfaces and construction of a functional for rotational hypersurfaces.
We give a geometric interpretation of the linear trace Harnack inequality for the Ricci flow.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
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…
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 paper classifies minimizers for a specific inequality in Euclidean balls.
problem Classifying minimizers for a specific inequality in Euclidean balls.
method Adaptation of Frank-Lieb proof for the sharp Sobolev inequality.
result Implication of a fully nonlinear sharp Sobolev trace inequality.
Using Jeff Holman's comments in Quantitative Finance to illustrate 4 critical errors students should learn to avoid: 1) Mistaking tails (4th moment) for volatility (2nd moment), 2) Missing Jensen's Inequality, 3) Analyzing the hedging wihout the underlying, 4) The necessity of a numeraire in finance.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Paper resolves bias in ALFT training using generalized alignment games.
problem Systematic bias in estimating logarithmic rewards from small batches.
method Generalized Distributional Alignment Games, U-statistics, minimax polynomial estimators, Variance-Optimal Augmented Polynomial Optimization Program (AQP) Estimator.
result Proves optimal bias and accelerated convergence in ALFT training.
In this paper, we proved the mass angular momentum inequality\cite{D1}\cite{ChrusLiWe}\cite{SZ} for axisymmetric, asymptotically flat, vacuum constraint data sets with small trace. Given an initial data set with small trace, we construct a boost evolution spacetime of the Einstein vacuum equations as \cite{ChOM}. Then …
Bayesian Attention Networks compress data by focusing on key training samples.
problem Lossless data compression for efficiency.
method Bayesian Attention Networks with attention factors and latent space.
result Efficient prediction using a few correlated training samples.
The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
problem Establishing higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
method Introducing conformally covariant boundary operators, proving extension theorems, and establishing trace inequalities.
result Generalized CR Sobolev trace inequalities for all γ ∈ (0, n+1) \mathbb{N}.
In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization o…
The paper proves inequalities for isometries in loxodromic Kleinian groups.
problem Discreteness criteria for subgroups of PSL.
method Generalization of discreteness criteria, using trace inequalities and optimization problems.
result Inequalities involving traces and hyperbolic displacements for loxodromic Kleinian groups.
New bounds on homological eigenvalues relate to Weil-Petersson length.
problem Bounding growth of homological eigenvalues for pseudo-Anosov automorphisms.
method Established inequality linking homological Jensen square sum to Weil-Petersson translation length.
result Homological Jensen square sum grows at most linearly with covering degree compared to Weil-Petersson translation length.
Proposes a new divergence measure for probability distributions.
problem Challenges in estimating divergences from empirical samples.
method Embeds data into RKHS, computes Jensen-Shannon divergence between covariance operators.
result Establishes RJSD as a lower bound on Jensen-Shannon divergence, enabling variational estimation.
We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures on Wasserstein space, extending on one hand, results in the Euclidean case (for ba…
We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.
Jensen-Shannon Information Based Characterization of the Generalization Error of Learning Algorithmscs.IT
New bound on machine learning model performance using Jensen-Shannon information.
problem Understanding the performance of machine learning models.
method Proposes a new information-theoretic bound on generalization error.
result Shows that the new bound can be tighter than mutual information-based bounds under certain conditions.
New framework using Jensen-Shannon divergence improves domain adaptation theory.
problem Incoherence between empirical domain adversarial training and theoretical -divergence.
method Established new theoretical framework based on Jensen-Shannon divergence, derived bi-directional upper bounds.
result Framework exhibits flexibilities for various transfer learning problems.
Formula derived for sample complexity in binary hypothesis testing.
problem Determine the minimum number of samples to distinguish between two distributions.
method Developed a formula for sample complexity in both prior-free and Bayesian settings, using Jensen-Shannon and Hellinger divergences.
result Formula characterizes sample complexity for a wide range of error parameters, up to multiplicative constants.
Improved lower bound for first Dirichlet eigenvalue using variance refinement.
problem Finding a more precise lower bound for the first Dirichlet eigenvalue.
method Refined Jensen-Hölder averaging using variance term.
result Explicit closed-form in-diameter bound strictly stronger than previous estimates.
Study trace systoles on surfaces, finding optimal bounds and implications.
problem Optimal systolic inequalities on hyperbolic manifolds and non-Fuchsian representations.
method Defined trace systole, used Markoff maps correspondence, computed bounds.
result Explicit optimal bounds for one-holed torus, four-holed sphere, and non-orientable surface of genus 3.
Improved GANs estimate convergence rate for density estimation.
problem Improving the accuracy of density estimation with GANs.
method Proved an oracle inequality for JS divergence between GAN estimate and true density.
result JS-divergence rate of convergence is .
A new objective function using Jensen-Shannon divergence improves generative learning from multiple data types.
problem Learning from multiple data types efficiently and accurately.
method Proposes a novel objective function using Jensen-Shannon divergence to approximate multimodal posteriors directly.
result The mmJSD objective optimizes an ELBO and improves generative learning tasks.
This paper introduces Jensen, an easily extensible and scalable toolkit for production-level machine learning and convex optimization. Jensen implements a framework of convex (or loss) functions, convex optimization algorithms (including Gradient Descent, L-BFGS, Stochastic Gradient Descent, Conjugate Gradient, etc.), …
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
We prove certain generalization of Hardy's inequality where the "boundary defining function" is replaced by a polynomial defining a singular algebraic variety. An application is given on the existence of a small time heat trace expansion for a Schrödinger operator with mild singularities along this algebraic set.
Proposes a new loss function for learning with noisy labels.
problem Improving model learnability with noisy labels.
method Uses generalized Jensen-Shannon divergence as a noise-robust loss function.
result Shows state-of-the-art results on noisy data.
We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the one proved by Chow-Hamiltonm, we give another proof on the classification of the …
We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we ch…
In this report, we derive a non-negative series expansion for the Jensen-Shannon divergence (JSD) between two probability distributions. This series expansion is shown to be useful for numerical calculations of the JSD, when the probability distributions are nearly equal, and for which, consequently, small numerical er…
We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation on closed manifolds. We also derive a new interpolated Harnack inequality for the equation on closed surfaces under the -Ricci flow. Finally we prove…
We introduce a new class of lower bounds on the log partition function of a Markov random field which makes use of a reversed Jensen's inequality. In particular, our method approximates the intractable distribution using a linear combination of spanning trees with negative weights. This technique is a lower-bound count…