A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We study the logarithmic L(α)-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent t…
We introduce a temperature into the exponential function and replace the softmax output layer of neural nets by a high temperature generalization. Similarly, the logarithm in the log loss we use for training is replaced by a low temperature logarithm. By tuning the two temperatures we create loss functions that are non…
Method identifies low-dimensional structure in high-dimensional probability measures.
problem Identifying low-dimensional structure in high-dimensional probability measures.
method Extends prior work on minimizing majorizations of the Kullback-Leibler divergence to identify optimal approximations within a specific class of measures.
result Connection between dimensional logarithmic Sobolev inequality and approximations with the ansatz.
We prove the logarithmic divergence of equivariant analytic torsion for one-parameter degenerations of projective algebraic manifolds, when the coefficient vector bundle is given by a Nakano semi-positive vector bundle twisted by the relative canonical bundle.
Given for instance a finite volume negatively curved Riemannian manifold M, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of M and their linear divergence rates under the geodesic flow. As…
We study the geometry of probability distributions with respect to a generalized family of Csiszár f-divergences. A member of this family is the relative α-entropy which is also a Rényi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply E…
Paper analyzes risk bounds for in-context learning in multiclass classification.
problem Risk bounds for in-context learning in multiclass classification.
method Formalizes tasks as sequences of labeled examples and queries, estimates conditional class probabilities, establishes oracle inequality for KL divergence.
result ICL achieves minimax optimal rate for conditional probability estimation.
We construct new families of two-ended O(m)×O(n)-invariant solutions to the Allen- Cahn equation Δu+u-u3=0 in RN+1, with N≥7, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi-Toda system on a given $O(m)…
We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weig…
A new approach to L2-consistent estimation of a general density functional using k-nearest neighbor distances is proposed, where the functional under consideration is in the form of the expectation of some function f of the densities at each point. The estimator is designed to be asymptotically unbiased, using t…
A method for learning skeleton of Bayesian networks robust to outliers and corruption.
problem Learning the exact skeleton of discrete Bayesian networks from corrupted data.
method Distributionally robust optimization and regression approach, optimizing worst-case risk over distributions within bounded Wasserstein distance or KL divergence.
result Logarithmic sample complexities for successful structure learning of bounded-degree graphs.
In their papers published in 1993 and 1994, by expressing certain physical quantity in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in a paper published in 2008, …
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…
The paper tightens bounds for estimating Schrödinger potentials in unpaired data translation.
problem Estimating Schrödinger potentials in unpaired data translation.
method Using stochastic optimal control and Ornstein-Uhlenbeck process, the paper derives tight bounds on the generalization ability of an empirical risk minimizer.
result The approach achieves almost optimal convergence rates for Gaussian mixtures.
Study scaling of optimal solutions for reliability constraints in resource provisioning.
problem Achieving high reliability in resource provisioning under stringent requirements.
method Chance-constrained optimization, distributionally robust optimization, f-divergence balls, line search.
result Correct scaling properties of optimal decisions are preserved by using appropriate f-divergence balls, leading to conservative yet near-optimal solutions.