The paper introduces closed-form expressions for interpreting Tsetlin Machines.
problem Interpreting complex Tsetlin Machines with a large number of clauses.
method Developed closed-form expressions for local and global interpretability of Tsetlin Machines.
result The expressions enable real-time feature importance assessment and data clustering.
Survey on closed-form Fisher-Rao distance expressions.
problem Finding closed-form expressions for Fisher-Rao distance.
method Collect and present examples of closed-form expressions for Fisher-Rao distance of discrete and continuous distributions.
result Presentation of closed-form expressions for Fisher-Rao distance of various distributions.
Summary of FastICA algorithm performance as data size grows.
problem Understanding the asymptotic performance of FastICA algorithms.
method Presentation of new closed-form expressions for several FastICA variants.
result New insights into the performance of FastICA algorithms as data size increases.
This paper provides a dictionary of closed-form kernel mean embeddings.
problem Challenges in deriving closed-form kernel mean embeddings.
method Comprehensive dictionary and practical tools for deriving new embeddings.
result Provides a Python library with minimal implementations of embeddings.
New model captures complex relationships from experimental data.
problem Capturing intricate feature interactions in empirical data.
method Shape Arithmetic Expressions (SHAREs) combining GAMs and mathematical expressions.
result SHAREs model captures complex feature interactions.
Conjectures closed-form expressions and cyclotomic expansions for knot invariants.
problem Calculating HOMFLY-PT invariants of knots colored by rectangular diagrams.
method Interpolation Macdonald polynomials and cyclotomic expansions.
result Conjectured closed-form expressions and cyclotomic expansions for knot invariants.
In this paper, we detail the main simulation methods used in practice to measure one-year reserve risk, and describe the bootstrap method providing an empirical distribution of the Claims Development Result (CDR) whose variance is identical to the closed-form expression of the prediction error proposed by Wüthrich et a…
Improved portfolio optimization using VaR and CVaR with NMVM models.
problem Optimizing portfolios with VaR and CVaR under NMVM distributions.
method Transformed mean-CVaR-skewness problems into quadratic optimization with closed-form solutions for NMVM models.
result Approximate closed-form expressions for VaR and CVaR of NMVM portfolios.
The paper derives risk measures for metalog distributions.
problem Deriving risk measures for metalog distributions.
method Closed-form expressions for Conditional Value at Risk and first-order partial moments.
result First-order partial moments are convex with respect to metalog parameters.
For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.
DCFSC uses a simple auto-encoder for subspace clustering without parameters.
problem Subspace clustering for large-scale high-dimensional datasets.
method Closed-form shallow auto-encoder for data-driven self-expressive layer, no parameters or optimization.
result Significant memory benefits over existing methods on large datasets.
This paper addresses the problem of inferring a regular expression from a given set of strings that resembles, as closely as possible, the regular expression that a human expert would have written to identify the language. This is motivated by our goal of automating the task of postmasters of an email service who use r…
We define and calculate the HOMFLY polynomial for a specific type of quiver.
problem Calculating the HOMFLY polynomial for forest quivers.
method Recursive definition and closed-form expression for forest quivers.
result Closed-form expression for the HOMFLY polynomial of a forest quiver.
Yau proved an existence theorem for Ricci-flat Kähler metrics in the 1970's, but we still have no closed form expressions for them. Nevertheless there are several ways to get approximate expressions, both numerical and analytical. We survey some of this work and explain how it can be used to obtain physical predictions…
The holomorphic torsion of a hermitian locally symmetric space is expressed as a special value of a geometric zeta function.
The paper uses moment matching method for pricing spread options under Lévy models.
problem Pricing spread options under Lévy models with mean-variance mixture.
method Moment matching method applied to Lévy models with mean-variance mixture.
result Obtains semi-closed form formulas for spread option prices.
We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.
Researchers derive expressions for metric perturbations of extremal surfaces.
problem Understanding changes in extremal surfaces under metric perturbations.
method Derived explicit expressions for position and surface area changes.
result Found an expansion of surface area involving multiple integrals of geometric quantities.
Paper shows leafwise cohomological expression for dynamical zeta functions.
problem Analyzing dynamical zeta functions on foliated dynamical systems.
method Leafwise cohomological approach.
result Leafwise cohomological expression of dynamical zeta functions.
Closed loop solitons in a plane, whose curvatures obey the modified Korteweg-de Vries equation, were investigated. It was shown that their tangential vectors are expressed by ratio of Weierstrass sigma functions for genus one case and ratio of Baker's sigma functions for the genus two case. This study is closely relate…
We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We intr…
We prove a McShane-type identity - a series, expressed in terms of geodesic lengths, that sums to 2πfor any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman-Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.
Tree-based regularization improves latent variable inference from related datasets.
problem Inferring latent variables from multiple related datasets in causal systems.
method Tree-Based Regularization (TBR) for sparse changes across environments.
result TBR identifies true latent variables up to simple transformations under sparse changes.
Extracts interpretable potential energy from Hamiltonian systems.
problem Learning an interpretable potential energy function from Hamiltonian systems.
method Constructs a neural network model of the potential and applies equation discovery to extract a closed-form algebraic expression.
result Close agreement between learned neural potentials and ground truth potentials, including correct effective potential for a central force problem.
The paper develops flows for tori and spheres, addressing complex geometries.
problem Learning flows on tori and spheres for complex geometries.
method Recursive flows starting from circles, intervals, or spheres.
result Expressive and numerically stable flows on tori and spheres.
The paper finds the explicit expression of Graham-Witten's invariant for 4D submanifolds.
problem Finding conformal invariants of submanifolds.
method Volume renormalization of minimal surfaces in conformally compact Einstein manifolds.
result Explicit expression of Graham-Witten's conformal invariant for 4D submanifolds.
Expressive quantum circuits are harder to train due to flatter cost landscapes.
problem Designing quantum circuits that are both expressive and trainable.
method Deriving a relationship between expressibility and gradient magnitude, extending barren plateau phenomenon.
result Highly expressive ansätze exhibit flatter cost landscapes, making them harder to train.
The holomorphic torsion of a compact locally symmetric manifold is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.
LCD improves causal discovery in high-dimensional gene data.
problem Predicting causal effects in large-scale gene expression data.
method Local Causal Discovery (LCD) with practical estimators, ICP algorithm inspiration, preselection method, and statistical tests.
result LCD estimator closely matches ICP's accuracy but is simpler and faster.
We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formul…
The equivariant holomorphic torsion of a compact locally symmetric manifold and an automorphism is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.
Optimal investment strategy under demand and cost uncertainties with shocks.
problem Optimal investment decision in a project with stochastic demand and costs.
method Extended Dixit and Pindyck's approach to jump diffusion processes, derived closed expression for value of the firm.
result Closed-form solution to the optimization problem for isoelastic functions.
Formula for integrating random variables on hyperbolic surfaces.
problem Integrating random variables on the moduli space of hyperbolic surfaces.
method Integration formula for lengths of closed geodesics.
result Integral of geometric random variables can be expressed as an integral over R.
In this paper, we address the aggregation of dependent stop loss reinsurance risks where the dependence among the ceding insurer(s) risks is governed by the Sarmanov distribution and each individual risk belongs to the class of Erlang mixtures. We investigate the effects of the ceding insurer(s) risk dependencies on th…
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
Unified framework explains geometric properties of CNNs.
problem Understanding why encoder-decoder CNNs perform well.
method Unified mathematical framework based on recent neural network theories.
result Encoder-decoder CNNs are related to nonlinear basis representation using combinatorial convolution frames.
We solve the ANOVA decomposition for categorical inputs.
problem Lack of a closed-form expression for ANOVA decomposition with categorical dependent variables.
method Bridge functional analysis with discrete Fourier analysis to derive a closed-form decomposition.
result Closed-form decomposition for categorical inputs without assumptions.
Study examines pricing of target volatility options in fractional SABR model.
problem Pricing target volatility options in the lognormal fractional SABR model.
method Used Ito's calculus for a theoretical replicating strategy and derived approximations and closed-form expressions.
result Accuracy of approximations for target volatility option pricing in various parameter ranges.
Unified framework for pricing various debt securities.
problem Pricing of different types of debt securities under general short-rate processes.
method Unifying framework using continuous-time Markov chain approximations and bi-dimensional diffusion processes.
result Closed-form matrix expressions and efficient algorithms for pricing various debt securities.
We find the maximum mutual information for neural networks and its key determinants.
problem Understanding the maximum mutual information in neural architectures.
method Derived closed-form expression for maximum mutual information across neural network families.
result Maximum mutual information stems from a generalized formula and is influenced by network width and statistical invariances.
Naz and Chaudhry found multiple solutions, while Boucekkine and Ruiz-Tamarit found unique solutions for the Lucas-Uzawa model.
problem Determining the uniqueness of closed-form solutions for the Lucas-Uzawa model.
method Equating expressions for h(t) and u(t) to find conditions for unique solutions.
result Proposed a condition for unique closed-form solutions and an open question for integral evaluation.
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
The paper connects knot homology, quantum 6j-symbols, and complements of knots.
problem Investigating the relationship between knot homology, quantum 6j-symbols, and knot complements.
method Developed a grading rule for HOMFLY-PT and Kauffman homology, found relationships between A-polynomials, and conjectured closed-form expressions for quantum 6j-symbols and knot complements.
result Closed-form expressions for SO(N) quantum 6j-symbols and conjectured expressions for (a,t)-deformed F_K for knot complements.
Bayes-optimal learning of deep random networks with Gaussian weights is studied.
problem Learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights.
method Closed-form expressions for Bayes-optimal test error, ridge regression, kernel and random features regression are computed.
result Optimally regularized ridge regression and kernel regression achieve Bayes-optimal performances, while logistic loss yields a near-optimal test error for classification.
Paper finds explicit expressions for Jenkins-Strebel differentials on a sphere with four poles.
problem Finding explicit Jenkins-Strebel differentials on a Riemann sphere with four poles.
method Using the Weierstrass ℘ function and simple closed curves. result Explicit expressions and algorithm for Jenkins-Strebel differentials on the sphere with four poles.
This work studies Gaussian geometry under entropy-regularized 2-Wasserstein distance.
problem Understanding Gaussian distributions in uncertainty quantification and diffusivity.
method Entropy-regularized 2-Wasserstein distance, closed-form solutions, fixed-point characterization.
result Closed-form expressions for the 2-Sinkhorn divergence and fixed-point barycenter.
This research connects 4-manifold trisections to their underlying surfaces and their intersection forms.
problem Understanding the intersection forms of 4-manifolds from their trisection diagrams.
method Expressing the twisted homology and Reidemeister torsion of a 4-manifold in terms of the first homology of its trisection surface and the intersection form.
result The intersection form of a 4-manifold can be derived from the intersection form of its trisection surface.
Unified method for deriving ridgelet transforms for various neural network architectures.
problem Deriving closed-form expressions for ridgelet transforms in modern neural network architectures.
method Unified Fourier slice method to derive ridgelet transforms for diverse neural network types.
result Systematic method to derive ridgelet transforms for various neural network architectures.