We propose a statistical model for natural language that begins by considering language as a monoid, then representing it in complex matrices with a compatible translation invariant probability measure. We interpret the probability measure as arising via the Born rule from a translation invariant matrix product state.
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48 results for “Compatibility Matrix”
A new GNN framework for graphs with heterophily.
problem Graphs with heterophily (nodes from different classes often connected).
method CPGNN framework with an interpretable compatibility matrix.
result CPGNN achieves state-of-the-art results in heterophily settings.
Improved neural network training with ADMM for hardware compatibility.
problem Training scalable and parallelizable neural networks.
method Alternating Direction Method of Multipliers (ADMM) combined with iterative least-square methods.
result Achieved significant accuracy improvements over SGD and Adam.
The Random Parameters model was proposed to explain the structure of the covariance matrix in problems where most, but not all, of the eigenvalues of the covariance matrix can be explained by Random Matrix Theory. In this article, we explore other properties of the model, like the scaling of its PDF as one take larger …
New optimizer designs respect symmetry, improving deep learning models.
problem Optimizers lack respect for symmetry in neural networks.
method Introduce symmetry-compatible principle for optimizer design.
result Symmetry-compatible optimizers improve model performance.
New method estimates graph compatibility from sparse labels.
problem Estimating graph compatibility from sparse labeled data.
method Factorized graph representations and algebraic amplification.
result End-to-end classification accuracy comparable to gold standard.
In this paper we consider the Poisson algebraic structure associated with a classical -matrix, i.e. with a solution of the modified classical Yang--Baxter equation. In Section 1 we recall the concept and basic facts of the -matrix type Poisson orbits. Then we describe the -matrix Poisson pencil (i.e the pair o…
We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there ex…
As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current notes: (1) basic -algebrogeometric foundations of …
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.
Derives a Hamiltonian model for 3D axially symmetric magnetohydrodynamics.
problem Modeling of 3D axially symmetric magnetohydrodynamics.
method Hamiltonian formulation and matrix discretization.
result First discrete model for 3D magnetohydrodynamics compatible with underlying Lie-Poisson structure.
Paper presents a deep learning method for estimating asset return precision matrices in noisy financial markets.
problem Estimating precision matrices of asset returns in low signal-to-noise ratio environments.
method Non-linear factor model within deep learning framework, consistent estimator with error covariance estimator.
result Superior accuracy in simulations and empirical data.
Parity on based matricesmath.GT
Parity defined for based matrices, a new example of virtual knot parity.
problem Defining parity for a new algebraic structure.
method Introduced parity for based matrices, defined reduced stable parity.
result New example of parity for virtual knots.
Improved covariance matrix estimation for portfolio optimization with guaranteed PSD and controlled conditioning.
problem Guaranteeing positive semidefinite ness and controlling spectral conditioning in IQ estimators.
method Introducing squeezing identity and atomic-IQ parameterization to construct structured channel matrices with PSD guarantees and analytic eigen floor for conditioning control.
result Atomic-IQ improves Sharpe ratios and delivers a more stable risk profile compared to standard estimators.
Notions of compatible and almost compatible pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics, are introduced and studied.
Framework controls uncertainty in LLMs without labels or probabilities.
problem Managing uncertainty in black-box LLMs without token-level probability or true labels.
method Integrates generative models, UCP, and conformal alignment to control uncertainty.
result Achieves close-to-nominal coverage and tighter thresholds than split UCP.
Defines compatibility between Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
problem Generalizing compatibility between Poisson and pseudo-Riemannian metrics to Jacobi structures.
method Introduces and studies compatibility conditions for Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
result Compatibility conditions are preserved under Poissonization and equivalent to Sasakian structures for contact pseudo-metrics.
The paper characterizes compatible linear connections on 3D Finsler manifolds.
problem Characterizing compatible linear connections on Finsler manifolds of dimension three.
method Intrinsic method to characterize compatible linear connections, focusing on indicatrices and Euclidean symmetries.
result If a compatible linear connection is not unique, indicatrices must be Euclidean surfaces of revolution.
Deep neural networks are vulnerable to adversarial attacks. The literature is rich with algorithms that can easily craft successful adversarial examples. In contrast, the performance of defense techniques still lags behind. This paper proposes ME-Net, a defense method that leverages matrix estimation (ME). In ME-Net, i…
The paper proposes reusable network components by making them compatible across tasks.
problem Training networks for different tasks independently leads to incompatible components.
method The paper splits a network into a features extractor and a target task head, and proposes various approaches to make them compatible.
result The proposed methods produce components that are directly compatible without compromising accuracy on original tasks.
Proves compatibility of light cones and projective structures.
problem Clarifying different concepts of compatibility between conformal and projective structures.
method Analyzes compatibility criteria introduced by Ehlers-Pirani-Schild and Trautman-Scholz.
result Proves that the compatibility criterion introduced by Ehlers-Pirani-Schild is correct.
The paper investigates compatible linear connections on Randers spaces and finds a unique extremal connection.
problem Investigating compatible linear connections on Randers spaces.
method Transformed compatibility equations by taking torsion components as variables and determined when these equations have solutions.
result Characterized Randers spaces as non-Riemannian generalized Berwald spaces with a positive constant norm of perturbing term.
Paper speeds up Gaussian process inference using Matérn kernels.
problem Efficiently performing Gaussian process inference for large datasets.
method Exact Matérn kernel decomposition into empirical cumulative distribution functions, combined with divide-and-conquer approach.
result The proposed algorithm significantly speeds up Gaussian process inference for low-dimensional problems with hundreds of thousands of data points.
The paper studies properties of group relations induced by compatible coarse structures.
problem Properties of asymptotic resemblance relations on groups.
method Generalization of asymptotic dimension and introduction of set theoretic coupling.
result Groups with compatible coarse structures that admit a set theoretic coupling are asymptotic equivalent.
Riemann compatible tensorsmath.DG
Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …
The causal compatibility question asks whether a given causal structure graph -- possibly involving latent variables -- constitutes a genuinely plausible causal explanation for a given probability distribution over the graph's observed variables. Algorithms predicated on merely necessary constraints for causal compatib…
Compatibility equations adapted to magnetic geometry.
problem No specific problem stated; magnetic geometry is the setting.
method Established compatibility equations in magnetic geometry.
result Analogues of Gauss, Ricci, and Codazzi-Mainardi equations in magnetic geometry.
Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additi…
Compatible director fields in cond-mat.soft
New methods reveal compatible liquid crystal phases in 3D.
problem Understanding compatible director fields in 3D liquid crystals.
method Re-derived compatibility conditions using vector calculus.
result Characterized a wide range of compatible liquid crystal phases.
We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact -manifolds. More precisely, we show that a contact -manifold admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb fl…
Study of spheres and circles on a manifold with a specific metric structure.
problem Understanding geometric objects on a manifold with a skew-circulant structure.
method Analyzing hyper-spheres, spheres, and circles in a tangent space of a 4D manifold with a skew-circulant tensor structure.
result Characterization of geometric objects under an indefinite metric.
Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (1) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor; (2) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; (3)…
The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilitie…
We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.
New proof for unique semi-symmetric compatible linear connection on Finsler manifolds.
problem Existence and uniqueness of semi-symmetric compatible linear connections on Finsler manifolds.
method New linear algebra proof without integration, using convex body properties and intrinsic equations.
result Uniqueness of semi-symmetric compatible linear connection proved.
New method improves compatibility of risk stratification models without sacrificing accuracy.
problem Compatibility issues arise when updating clinical machine learning models.
method Proposes rank-based compatibility measure and new loss function.
result Increased compatibility of models by 0.019 with no loss in discriminative performance.
Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel et al., 2009) or the…
Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor is named `compatible' with the curvature tensor if . Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetriz…
The study examines backward compatibility issues in ML systems, especially with noisy data.
problem Backward compatibility challenges in ML systems, especially with noisy data.
method Empirical analysis of ML systems across different architectures and datasets, focusing on data shifts and noise.
result Backward compatibility issues arise even without data shift due to optimization stochasticity and training on large-scale noisy datasets can significantly decrease compatibility.
Generalized Lagrange-Weyl structures and compatible connections are introduced as a natural generalization of similar notions from Riemannian geometry. Exactly as in Riemannian case, the compatible connection is unique if certain symmetry conditions with respect to vertical and horizontal Christoffel symbols are impose…
We will introduce two notions of compatibility bettwen pseudo-Riemannian metric and Poisson structure using the notion of contravariant connection introduced by Fernandes R. L., we will study some proprities of manifold endowed with such compatible structures an we will give some examples.
AI systems are being deployed to support human decision making in high-stakes domains. In many cases, the human and AI form a team, in which the human makes decisions after reviewing the AI's inferences. A successful partnership requires that the human develops insights into the performance of the AI system, including …
The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demo…
Researchers create compatibility complexes for Einstein metrics.
problem Constructing solutions to the conformal-to-Einstein operator.
method Using a method that leverages symmetries and geometric properties.
result At most one independent solution exists under genericity assumptions.
In this Note, we will characterize the Poisson structures compatible with the canonical metric of $\reel^3$. We will also give some relvant examples of such structures. The notion of compatibility used in this Note was introduced and studied by the author in previous papers.
Study star products on Poisson manifolds compatible with reduction.
problem Finding star products compatible with coisotropic reduction.
method Compute second constraint Hochschild cohomology of constraint algebra.
result Determine infinitesimal star products on Poisson manifolds.
In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solv…
Perfectly Fitting CDO Prices Across Tranches: A Theoretical Framework with Efficient Algorithmsq-fin.RM
Paper proposes a perfect-fit model for CDO tranches.
problem Achieving a perfect fit to market prices across all CDO tranches.
method Introduces compatibility levels, derives conditions, constructs copula models.
result Demonstrates efficient verification and construction of perfect-fit models.