This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. Wh…
On-device research index
arXiv research
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.
168,695 papers · 148 categories
Trend · papers per month
48 results for “nonlinear superposition rule”
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …
Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, char…
The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.
problem Finding general solutions for Lie systems.
method Develops mixed superposition rules for Lie systems with imprimitive Lie algebras and semidirect sums.
result Extends coalgebra method to Lie systems of partial differential equations.
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study…
Geometric methods integrate Lie systems for optimal control problems.
problem Integrating Lie systems for optimal control problems.
method Geometric numerical methods based on Magnus expansions and Runge-Kutta-Munthe-Kaas.
result Accurate numerical solutions for Lie systems in optimal control problems.
We consider the demixing problem of two (or more) structured high-dimensional vectors from a limited number of nonlinear observations where this nonlinearity is due to either a periodic or an aperiodic function. We study certain families of structured superposition models, and propose a method which provably recovers t…
Study new -contact distributions and Lie systems.
problem Characterizing and understanding -contact distributions.
method Analyzing Goursat distributions and Lie systems.
result Characterized new types of -contact distributions.
Quantum computing speeds up asset pricing models exponentially.
problem Solving dynamic nonlinear asset pricing models efficiently.
method Utilizes quantum superposition and entanglement to solve models exponentially faster than classical methods.
result Exponential computational speed-up for solving asset pricing models.
This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are i…
It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also allows us to characterize Riccati chain equations geometrically in terms of the pro…
In this paper we develop some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and we discuss some of its integrability conditions from a group theoretical perspective. The nonlinear superposition principle also arises in a simple way.
We consider the demixing problem of two (or more) high-dimensional vectors from nonlinear observations when the number of such observations is far less than the ambient dimension of the underlying vectors. Specifically, we demonstrate an algorithm that stably estimate the underlying components under general \emph{struc…
Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
We study the problem of demixing a pair of sparse signals from noisy, nonlinear observations of their superposition. Mathematically, we consider a nonlinear signal observation model, , where denotes the superposition signal, and are orthonormal bases in $\mathb…
MIMONets: Multiple-Input-Multiple-Output Neural Networks Exploiting Computation in Superpositioncs.LG
MIMONets speed up neural network inference by processing multiple inputs in parallel.
problem Reducing computational cost in neural network inference for large datasets.
method Proposes MIMONets, which augment neural network architectures with variable binding mechanisms to handle multiple inputs in superposition.
result Achieves significant speedups (2-4x) with minimal accuracy loss, demonstrating adaptability across different architectures.
Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularizationstat.ML
New method recovers sparse signals from nonlinear observations with robust error bounds.
problem Recovering two sparse vectors from nonlinearly mixed observations with limited data.
method Regularization-based framework combining Huberized data fidelity and generalized folded-concave penalties with a proximal alternating algorithm.
result Estimation error bounds of order at every localized stationary point, with oracle rate under beta-min condition.
The -symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the -symplectic structures to investigate a type of systems of first-order ordinary differential equations, the -symplectic Lie systems. In part…
Resonator Networks outperform optimization methods at solving high-dimensional vector factorizationcs.NE
We develop theoretical foundations of Resonator Networks, a new type of recurrent neural network introduced in Frady et al. (2020) to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dim…
Biological neurons learn tensor decompositions of higher-order correlations using nonlinear Hebbian plasticity.
problem Learning higher-order correlations in biological neurons.
method Introduce and study generalized nonlinear Hebbian learning rules.
result Neurons can learn tensor eigenvectors of higher-order input correlation tensors.
Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformati…
This paper proposes a brain-inspired approach to quantum machine learning with the goal of circumventing many of the complications of other approaches. The fact that quantum processes are unitary presents both opportunities and challenges. A principal opportunity is that a large number of computations can be carried ou…
Study harmonic surfaces in 3D space, proving superposition principle.
problem Understanding harmonic surfaces in .
method Using harmonic Enneper immersions and superposition principle.
result Minimal and maximal surfaces can be decomposed into harmonic components.
Quasi-Lie schemes for PDEsmath.AP
The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining -dependent superposition rules and integrability conditions are analysed.…
We introduce a nonlinear aggregation type classifier for functional data defined on a separable and complete metric space. The new rule is built up from a collection of arbitrary training classifiers. If the classifiers are consistent, then so is the aggregation rule. Moreover, asymptotically the aggregation rule b…
The present paper proposes generalized Gaussian kernel adaptive filtering, where the kernel parameters are adaptive and data-driven. The Gaussian kernel is parametrized by a center vector and a symmetric positive definite (SPD) precision matrix, which is regarded as a generalization of the scalar width parameter. These…
New Lie systems derived from Goursat distributions with applications to differential equations.
problem Analyzing Lie systems associated with Goursat distributions and their applications.
method Analyzing bracket-generating distributions and their relation to Lie systems, focusing on reductions and reconstructions.
result Lie systems associated with Goursat distributions can be reduced and solutions reconstructed from reduced systems.
In many learning tasks, structural models usually lead to better interpretability and higher generalization performance. In recent years, however, the simple structural models such as lasso are frequently proved to be insufficient. Accordingly, there has been a lot of work on "superposition-structured" models where mul…
Quantum machine learning uses superposition to create a large ensemble of classifiers.
problem Improving machine learning efficiency on quantum computers.
method Using superposition to create an exponentially large ensemble of classifiers, trained with an optimization-free learning algorithm.
result Adding an optimization step improves the performance of quantum ensembles of classifiers.
This paper improves fund net value prediction using ARIMA-LSTM hybrid model.
problem Inaccurate prediction of fund net value due to linear characteristics of models.
method ARIMA-LSTM hybrid model: ARIMA for linear characteristics, LSTM for nonlinear.
result Hybrid model shows higher accuracy in predicting fund net value.
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
New algorithm extracts features from superpositions in machine learning models.
problem Challenges in extracting interpretable features from complex models in superposition.
method An efficient query algorithm that identifies non-degenerate feature directions and reconstructs the function.
result Identifies all feature directions whose responses are non-degenerate and reconstructs the function \( f \) in a general superposition setting.
Observing a linear superposition principle, a family of new minimal hypersurfaces in Euclidean space is found, as well as that linear combinations of generalized helicoids induce new algebraic minimal cones of arbitrarily high degree.
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property…
High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models …
Superposition in autoencoders leads to loss in simple models.
problem Mechanistic interpretability of neural networks
method Analyzing mathematical basis for superposition and providing bounds for reconstruction loss
result Upper and lower bounds for L2 reconstruction loss in the sparse regime
This work analyses types of group actions on families of -dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to…
Paper introduces Manifold Probe for discovering representation manifolds in superposition.
problem Discovering representation manifolds in complex superposition representations.
method Generalizes linear regression probes to learn feature spaces and directions in superposition representations.
result Demonstrates Manifold Probe on Llama 2-7b representations, finding causally involved manifolds in model behaviour.
This paper uses NLDT to find interpretable control rules from complex DRL policies.
problem Complex, non-interpretable policies from black-box AI methods.
method Evolutionary optimization of NLDT for hierarchical control rules.
result Interpretable control rules with similar performance to black-box DRL.
New gravitational instantons and non-holomorphic minimal spheres discovered.
problem Existence of non-holomorphic minimal spheres in ALF spaces.
method Gluing construction of ALF gravitational instantons using specific metrics.
result Existence of non-holomorphic minimal spheres in a general class of ALF spaces.
New methods combine model predictions to avoid linear mixtures' limitations.
problem Combining predictions from different models to avoid linear mixtures' limitations.
method Log-linear pooling (locking) and quantum superposition (quacking) to optimise model weights.
result Demonstrated locking method with illustrative example and practical application.
Optimal Volt/VAR control rules designed using deep learning.
problem Designing optimal Volt/VAR control rules for DERs to regulate voltage fluctuations.
method Formulated as a deep learning problem, where a DNN emulates Volt/VAR dynamics and optimizes rule parameters.
result DNN-based optimization outperforms MINLP in efficiency and accuracy.
A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot--Guldberg …
We consider the learning of multi-agent Hawkes processes, a model containing multiple Hawkes processes with shared endogenous impact functions and different exogenous intensities. In the framework of stochastic maximum likelihood estimation, we explore the associated risk bound. Further, we consider the superposition o…
This paper considers generalized linear models using rule-based features, also referred to as rule ensembles, for regression and probabilistic classification. Rules facilitate model interpretation while also capturing nonlinear dependences and interactions. Our problem formulation accordingly trades off rule set comple…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a -dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…