Study on the distribution of nodal sets on random superpositions of eigenfunctions.
problem Understanding the distribution of nodal sets on random superpositions of eigenfunctions.
method Analysis of the conormal cycle and integration current on nodal sets.
result Equidistribution of the conormal cycle on fibers of cotangent bundle in odd dimensions.
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.
Quantum algorithms for multi-armed bandits are explored with limited reward access.
problem Exploring quantum speed-ups in multi-armed bandit problems with limited reward information.
method Introduced new bandit models and showed query complexity equivalence with classical algorithms.
result No quadratic speed-up is possible for multi-armed bandits with limited reward access.
New minimal hypersurfaces and cones found through linear superposition.
problem Understanding minimal surfaces in Euclidean space.
method Linear superposition principle applied to minimal hypersurfaces and cones.
result New minimal hypersurfaces and cones discovered.
SupSup model learns thousands of tasks without forgetting, using randomly initialized subnetworks.
problem Sequentially learning many tasks without forgetting.
method Randomly initialized base network with task-specific subnetworks (supermasks).
result Gradient-based optimization can identify the correct subnetwork for new tasks.
Develops a scalable framework for optimizing superposition-structured models.
problem Insufficient interpretability and generalization performance of simple structural models.
method Proximal Newton-type method with smoothed conic dual approach and LBFGS updating formula.
result Achieves super-linear convergence rate for optimizing superposition-structured models.
We study random Morse functions on a Riemann manifold (Mm,g) defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric g. The randomness is determined by a fixed Schwartz function w and a small parameter ε>0. We first prove that as ε→0 the ex…
Paper develops a simple estimator for high-dimensional superposition models with various component structures.
problem Estimating high-dimensional superposition models with different component structures.
method Presented a simple estimator for general superposition models with any number of component parameters and any norm structure.
result Geometric condition and high probability non-asymptotic bounds for accurate component estimation.
Study harmonic surfaces in 3D space, proving superposition principle.
problem Understanding harmonic surfaces in R3. method Using harmonic Enneper immersions and superposition principle.
result Minimal and maximal surfaces can be decomposed into harmonic components.
Improved learning of Hawkes processes with superposition-assisted optimization.
problem Learning multi-agent Hawkes processes with shared and different intensities.
method Stochastic optimization with superposition-driven diversity strategy.
result Superposition improves risk bound and convergence properties.
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…
We develop dependent hierarchical normalized random measures and apply them to dynamic topic modeling. The dependency arises via superposition, subsampling and point transition on the underlying Poisson processes of these measures. The measures used include normalised generalised Gamma processes that demonstrate power …
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.
This paper explores robust recovery of a superposition of R distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of 2N−1 dimensional and R<<2N−1. This framework covers a large class of signals arising from real applications in biology, automation,…
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.
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.
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.
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
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.
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.
The study examines Euclid's Book I, focusing on area applications and construction methods.
problem Exploring Euclid's geometric constructions and proofs, particularly those involving area calculations.
method Summarizing medieval editions and ancient commentaries, comparing constructions and proofs.
result Medieval editions often avoid Euclid's use of superposition in area proofs, offering alternative constructions.
Paper proposes an algorithm to reconstruct optimal model structure from graph adjacency matrix.
problem Optimal model structure reconstruction from weighted colored graph adjacency matrix.
method Uses prize-collecting Steiner tree algorithm to reconstruct minimum spanning tree.
result Demonstrates the effectiveness of the prize-collecting Steiner tree algorithm for model structure reconstruction.
Sparse superposition codes were recently introduced by Barron and Joseph for reliable communication over the AWGN channel at rates approaching the channel capacity. The codebook is defined in terms of a Gaussian design matrix, and codewords are sparse linear combinations of columns of the matrix. In this paper, we prop…
In the information-based approach to asset pricing the market filtration is modelled explicitly as a superposition of signals concerning relevant market factors and independent noise. The rate at which the signal is revealed to the market then determines the overall magnitude of asset volatility. By letting this inform…
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.
Gravitational instantons are constructed as superpositions of Atiyah-Hitchin and Taub-NUT geometries.
problem Constructing gravitational instantons from Atiyah-Hitchin and Taub-NUT geometries.
method A gluing construction that captures the superposition of moduli spaces of centred SU(2) monopoles and Taub-NUT manifolds.
result Gravitational instantons are explicitly shown to be superpositions of Atiyah-Hitchin and Taub-NUT geometries.
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…
Resonator Networks solve high-dimensional vector factorization better than optimization methods.
problem High-dimensional vector factorization problem in Vector Symbolic Architectures.
method Recurrent neural network (Resonator Networks) that combines nonlinear dynamics and superposition search.
result Resonator Networks outperform optimization methods in solving high-dimensional vector factorization.
We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge t…
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.
New method robust to random distributional shifts in prediction.
problem Random distributional shifts in real-world settings.
method Hybrid approach combining long-term and proxy outcomes.
result Hybrid approach yields lower mean-squared error than current methods.
Explores tensor products in hyperdimensional computing.
problem Understanding tensor products in hyperdimensional computing.
method Generalized results from graph embeddings to vector symbolic architectures and hyperdimensional computing.
result Tensor product is the most general and expressive representation with errorless unbinding and detection.
Develops a method to interpolate data for efficient inversion in nonuniform geometries.
problem Efficient inversion in nonuniform geometries where not all sources see all receivers.
method Interpolates data to an ideal acquisition geometry while solving the inverse problem using simultaneous shots.
result Illustrates the flexibility and efficiency of the approach using synthetic experiments.
Curves in 3-sphere with constant curvature and torsion are described.
problem Understanding curves with constant curvature and torsion in a 3-sphere.
method Analyzing the motion of a point with superposition of circular motions.
result Behavior of these curves can be periodic or dense in a Clifford torus.
Word embeddings reveal multiple senses, which can be recovered using sparse coding.
problem Understanding word senses in polysemous words.
method Sparse coding of word embeddings to recover multiple senses.
result Sparse coding can approximate multiple word senses, with each sense associated with a discourse atom.
A theory of feature geometry using spectral analysis of weight matrices.
problem Current methods decompose neural network activations into sparse linear features, losing geometric structure.
method Develops a theory by analyzing the spectra of weight-derived matrices, introducing the frame operator.
result Features collapse onto single eigenspaces, organizing into tight frames, and admit discrete classification.
Lie systems method simplifies Riccati hierarchy study.
problem Simplifying study of Riccati hierarchy equations.
method Lie systems approach to projective Riccati equations.
result Characterization of Riccati chain equations geometrically.
Networks play a central role in modern data analysis, enabling us to reason about systems by studying the relationships between their parts. Most often in network analysis, the edges are given. However, in many systems it is difficult or impossible to measure the network directly. Examples of latent networks include ec…
Classical clients can verify quantum learning tasks efficiently.
problem Making quantum learning accessible to classical clients.
method Developed a framework for classical verification of quantum learning.
result Quantum learning tasks can be efficiently verified by classical verifiers.
Method recovers structured components from nonlinear observations.
problem Demixing structured vectors from nonlinear observations.
method Proposes a method to recover components from nearly m = O(s) samples.
result Strictly improves upon previous techniques and matches best sample complexity.
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…
Study new k-contact distributions and Lie systems.
problem Characterizing and understanding k-contact distributions. method Analyzing Goursat distributions and Lie systems.
result Characterized new types of k-contact distributions. Extends quasi-Lie systems to PDEs for integrability analysis.
problem Analyzing integrability conditions for PDEs.
method Develops a procedure to construct quasi-Lie systems for PDEs through quasi-Lie schemes.
result Obtains t-dependent superposition rules and integrability conditions. Superposed Hawkes processes improve risk bounds and solve cold-start issues.
problem Improving risk bounds in temporal point processes.
method Least squares estimation of superposed Hawkes processes.
result Superposed Hawkes processes tighten risk bounds under certain conditions.
New self-exciting random evolutions (SEREs) for modeling traffic and transport processes.
problem Modeling self-exciting and clustering effects in traffic and transport processes.
method Introducing a new process based on a superposition of a Markov chain and a Hawkes process, and constructing self-exciting random evolutions (SEREs).
result Developed new models and limit theorems for SEREs, including averaging and diffusion approximation.
We show that Scherk's first surface, a one-parameter family of solutions to the minimal surface equation, may be written as a linear superposition of other solutions with specific parametric values.