After giving a general introduction to the main known results on the anisotropic Calder{ó}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{ó}n problem at fixed frequency, in dimension n $…
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48 results for “Calderón reproducing formula”
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
Derives properties of heat kernel for Rumin complex on Heisenberg groups.
problem Analyzing heat kernel properties for Rumin complex.
method Derives properties of heat equation with Hodge operator on Heisenberg groups.
result Constructs Calderón reproducing formula using heat kernel for Rumin forms.
We establish new Calderón reproducing formulas for self-adjoint operators that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with through holomorphic functional calculus whilst the synthesising function interacts with through funct…
We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We …
The article analyzes LCE in Hilbert space, deriving new formulas and regularisation methods.
problem Analyzing conditional expectation in infinite-dimensional Hilbert space.
method Establishing analytical properties and regularisation for LCE in Hilbert space, deriving new formulas.
result Simple derivation and intuitive justification of conditional mean embedding formula.
The paper examines how adversarial training and noise affect neural network performance.
problem Overfitting in adversarial training and data augmentation.
method Adversarial training and data augmentation with noise in the context of regularized regression in RKHS.
result Appropriate regularization can prevent overfitting and improve performance.
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …
Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answ…
Proposes estimators for complex dose-response curves using kernel methods.
problem Estimating complex dose-response curves with continuous treatments, mediators, and covariates.
method Kernel ridge regression with sequential kernel embedding technique.
result Simple estimators for mediated and time-varying dose response curves with nonasymptotic uniform rates.
Study on deep neural networks using branching processes and Mehler's formula.
problem Understanding the mathematical role of activation functions in compositional neural networks.
method Connection between compositional kernels and branching processes via Mehler's formula; new random features algorithm.
result Explicit formulas for eigenvalues of compositional kernels quantify complexity.
We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier integral operators of a particular kind. They act on oscillatory wave functions, …
We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call "homotopic curve shortening" (HCS), starts with a closed curve (which might self-intersect) in the presence of a set of point obstacles, and evolves in discrete…
In previous papers, the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e., is a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [ ] of knots such that [K]=K. It turns out that …
The holographic description in the presence of gravitational Chern-Simons term is studied. The modified gravitational equations are integrated by using the Fefferman-Graham expansion and the holographic stress-energy tensor is identified. The stress-energy tensor has both conformal anomaly and gravitational or, if re-f…
A Theory of Non_Gaussian Option Pricingcond-mat.stat-mech
Option pricing formulas are derived from a non-Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker-Planck equation which maximizes the Tsallis nonextensive entropy of index . A generalized form of the Black-Scholes differential equation is found, and we derive a martin…
AlphaEval evaluates alpha mining models efficiently and comprehensively.
problem Lack of systematic evaluation for alpha mining models.
method Unified, parallelizable evaluation framework assessing predictive power, stability, robustness, financial logic, and diversity.
result AlphaEval achieves evaluation consistency comparable to comprehensive backtesting, providing more comprehensive insights and higher efficiency.
We extend the now classic structural credit modeling approach of Black and Cox to a class of "two-factor" models that unify equity securities such as options written on the stock price, and credit products like bonds and credit default swaps. In our approach, the two sides of the stylized balance sheet of a firm, namel…
We develop a modelling framework for multiple yield curves driven by continuous-state branching processes with immigration (CBI processes). Exploiting the self-exciting behavior of CBI jump processes, this approach can reproduce the relevant empirical features of spreads between different interbank rates. In particular…
Two formulae estimate sensitivity of random vectors to distributional parameters.
problem Estimating sensitivity of random vectors to distributional parameters.
method Two analytical formulae and four numerical algorithms.
result Validated numerical algorithms and demonstrated effectiveness.
We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollar…
We consider a standard symplectic dynamics on TM generated by a natural Lagrangian L. The Lagrangian is assumed to be invariant with respect to the action TR_g of a Lie group G lifted from the free and proper action R_g of G on M. It is shown that under these conditions a connection on principal bundle pi: M \rightarro…
Modeling risk and performance with Levy-stable distributions.
problem Understanding risk and performance in financial markets with non-Gaussian distributions.
method Developed a finite-horizon model using Levy-stable scaling, identified parameters from data, derived formulas for various financial ratios.
result Horizon-correct formulas for risk measures are derived and validated across different horizons.
Study topological correlators for SYM on four-manifolds, deriving explicit formulae and confirming S-duality.
problem Topological correlation functions of , SYM on four-manifolds.
method Coupling to a Spin^c structure, deriving explicit formulae, and confirming S-duality.
result Topological correlators are mock modular forms for .
We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential eq…
Study entropic regularization of Gaussian measures and processes on Hilbert space.
problem Regularizing 2-Wasserstein distance for infinite-dimensional Gaussian measures and processes.
method Minimum Mutual Information property, closed form formulas, Fréchet differentiability, Sinkhorn barycenter equation.
result Entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable in Hilbert space.
Study on reproducibility in optimization with bounds on limits.
problem Limits of reproducibility in noisy or error-prone optimization procedures.
method Defined a quantitative measure of reproducibility and analyzed convex optimization settings.
result Revealed a fundamental trade-off between computation and reproducibility.
Survival analysis models research reproducibility, offering new insights.
problem Reproducibility crisis in machine learning research.
method Survival analysis to model reproducibility as a continuous process.
result Survival analysis provides deeper insights into research reproducibility.
Clarifies the scope of 'reproducibility' in AI and ML.
problem Unclear meaning of 'reproducibility' in AI and ML.
method Analyzes the community's works on reproducibility.
result Identifies eight general topic areas of reproducibility.
This chapter introduces reproducibility in machine learning for medical imaging.
problem Lack of reproducibility in machine learning for medical imaging.
method Distinguishes and defines types of reproducibility, outlines requirements, and discusses utility.
result Discussion on benefits and a plea for a non-dogmatic approach to reproducibility.
Optimal Guarantees for Algorithmic Reproducibility and Gradient Complexity in Convex Optimizationcs.LG
New algorithms ensure reproducibility and optimal convergence in convex optimization.
problem Trade-off between reproducibility and convergence rate in convex optimization.
method Regularization-based algorithms for smooth convex minimization and minimax optimization.
result Achieves optimal reproducibility and near-optimal gradient complexity for various oracle settings.
Working on different aspects of algorithmic trading we empirically discovered a new market invariant. It links together the volatility of the instrument with its traded volume, the average spread and the volume in the order book. The invariant has been tested on different markets and different asset classes. In all cas…
FX Smile in the Heston Modelq-fin.CP
The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is non-negative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options.…
Proposes a method to improve neural architectures reproducibly.
problem Lack of reproducibility in Neural Architecture Transformer (NAT).
method Differentiable Neural Architecture Transformation (DNAT).
result DNAT outperforms NAT and is applicable to various models and datasets.
The paper aims to improve reproducibility of machine learning experiments.
problem Lack of reproducibility in machine learning experiments.
method Investigates factors affecting reproducibility and proposes FAIR data practices.
result Preliminary results show the role of ProvBook in capturing and comparing provenance.
Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functionsstat.ML
In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (nor…
NeurIPS 2019 program improves reproducibility in machine learning.
problem Ensuring machine learning research results are reproducible and reliable.
method Code submission policy, reproducibility challenge, and checklist integration.
result Improved reproducibility standards across the machine learning community.
Challenge encourages reproducible deep learning methods.
problem Inconsistent training behavior in deep learning competitions.
method Evaluate methods based on their training procedures, retrain in controlled settings.
result Guaranteed reproducibility and generalization of methods.
In this paper, we discuss the approaches we took and trade-offs involved in making a paper on a conceptual topic in pattern recognition research fully reproducible. We discuss our definition of reproducibility, the tools used, how the analysis was set up, show some examples of alternative analyses the code enables and …
Paper explores RKHS properties for derivative and integral operators.
problem Establishing sufficient conditions for reproducing property in RKHS.
method Establishing reproducing property for combinations of composition operators.
result Provides framework for regularized learning algorithms involving function values, gradients, or operators.
Replication confirms CGD's effectiveness in competitive games.
problem Reproducibility of a novel Nash equilibrium algorithm.
method Replicated experiments and provided Python implementation.
result CGD avoids oscillatory and divergent behaviours.
The reproducibility of scientific research has become a point of critical concern. We argue that openness and transparency are critical for reproducibility, and we outline an ecosystem for open and transparent science that has emerged within the human neuroimaging community. We discuss the range of open data sharing re…
What makes a paper independently reproducible? Debates on reproducibility center around intuition or assumptions but lack empirical results. Our field focuses on releasing code, which is important, but is not sufficient for determining reproducibility. We take the first step toward a quantifiable answer by manually att…
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with…
Diffusion models' consistency across splits explained by random matrix theory.
problem Consistency of diffusion models trained on non-overlapping subsets.
method Random matrix theory framework to quantify dataset effects on denoiser and sampling map.
result The theory explains and predicts cross-split disagreement in diffusion models.
Paper reproduces and enhances a method for cross-lingual word embeddings.
problem Creating robust cross-lingual mappings of word embeddings without supervision.
method Reproduces and enhances a self-learning method with grid search for hyperparameters.
result Model's robustness is demonstrated across four new languages.
ERICA assesses reproducibility in cluster analysis.
problem Lack of a unified framework for evaluating cluster analysis replicability.
method ERICA (iterative clustering assignments) method to quantify replicability.
result Demonstrates ERICA's ability to identify reproducible cluster structure.
This paper extends mirror descent to Banach spaces with reproducing kernels.
problem Optimizing in Banach spaces with reproducing kernels.
method Mirror descent algorithm adapted for Banach spaces with reproducing kernels.
result Mirror descent achieves linear convergence in certain conditions and standard convergence in a constrained setting.