Optimal fuzzy classification aggregation functions are weighted means.
problem Characterizing optimal fuzzy classification aggregation functions.
method Proving optimality of weighted arithmetic means for fuzzy classification.
result Optimal fuzzy classification aggregation functions are weighted means.
We propose a system for calculating a "scaling constant" for layers and weights of neural networks. We relate this scaling constant to two important quantities that relate to the optimizability of neural networks, and argue that a network that is "preconditioned" via scaling, in the sense that all weights have the same…
This note justifies approximations of arithmetic forwards using weighted averages of overnight forwards.
problem Theoretical justification for approximations of arithmetic forwards.
method Presentation of a central equation and computationally cheaper methods to approximate Fa. result Theoretical bounds and closed-form expressions for arithmetic factors in Gaussian HJM models.
New analysis of annealing paths in sampling and estimation.
problem Sampling from complex distributions and estimating normalization constants.
method Extending known results on Bregman divergence to quasi-arithmetic means under monotonic embedding.
result Analogous result for quasi-arithmetic means, highlighting the interplay between means, parametric families, and divergence functionals.
The study finds arithmetic groups often in square-tiled surface monodromies.
problem Understanding arithmetic properties of square-tiled surfaces.
method Analyzing variations of Hodge structures and Kontsevich-Zorich monodromies.
result Arithmetic groups are frequent in low genus square-tiled surfaces.
The recently introduced dropout training criterion for neural networks has been the subject of much attention due to its simplicity and remarkable effectiveness as a regularizer, as well as its interpretation as a training procedure for an exponentially large ensemble of networks that share parameters. In this work we …
New market makers improve on existing models in DeFi.
problem Improving liquidity and efficiency in decentralized finance.
method Developed a new family of market makers based on generalized means.
result G3Ms offer properties preferable to existing models.
New arithmetic phenomenon 'murmurations' detected using AI.
problem Detecting new arithmetic patterns in large datasets.
method Machine learning interpretability tools (PCA, saliency, convolutional filters).
result Murmurations encode Frobenius traces and connect to number theory.
This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.
problem Understanding which mean to use for symmetrizing Bregman divergences on positive definite matrices.
method Axiomatic definition of mean functionals and variational principles over the cone of positive definite matrices.
result The arithmetic mean is canonical for forward symmetrization, and the arithmetic, log-Euclidean, and harmonic means for reverse symmetrization.
The study finds infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
problem Existence and properties of semi-arithmetic Riemann surfaces.
method Combining number theory and hyperbolic geometry to prove existence and properties of semi-arithmetic Riemann surfaces.
result Existence of infinitely many semi-arithmetic Riemann surfaces with dense systoles and distinct invariant trace fields.
MC-LSTM extends LSTM to conserve mass in neural networks.
problem Conservation laws in real-world systems.
method Extending LSTM's inductive bias to conserve mass.
result MC-LSTM sets new state-of-the-art for predicting peak flows.
A new metric framework for weighted projective spaces improves clustering and analysis.
problem Proximity measurement in weighted projective spaces with intrinsic scaling and topology.
method Hierarchical clustering framework based on Finsler geometry, quotienting weighted scaling action.
result The constructed metric dF satisfies the triangle inequality, making it a genuine metric. Löbell polyhedra have small systoles and are quasi-arithmetic.
problem Finding compact hyperbolic polyhedra with small systoles.
method Elementary and conceptual means to observe systole behavior, number theoretic invariants to refine results.
result Löbell polyhedra give examples of closed hyperbolic 3-manifolds with arbitrarily small systole and are quasi-arithmetic.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines cQ1/2+O(Q1/4) square-rootable Salem numbers of degree 4 which are…
The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in …
New research disproves a key conjecture in optimization.
problem Comparison of sampling methods in stochastic optimization.
method Reduction to noncommutative arithmetic-geometric mean inequality and application of noncommutative Positivstellensatz.
result The Recht-Ré conjecture is false for general n.
WrapNet optimizes inference for low-resolution neural networks by using 8-bit additions.
problem Reducing multiplication complexity in low-resolution neural networks.
method Adapting neural networks to use low-resolution (8-bit) additions in accumulators, with a cyclic activation layer and overflow penalty regularizer.
result Achieves comparable classification accuracy to 32-bit counterparts using low-resolution additions.
Study on multiplicities in length spectrum of Salem numbers.
problem Understanding multiplicities in the length spectrum of Salem numbers.
method Analysis of square-rootable Salem numbers and their growth rate.
result Proved exponential growth rate for mean multiplicities in length spectrum.
A new method for averaging data on manifolds is proposed, offering simplicity and efficiency.
problem The difficulty of computing Fréchet means on manifolds, especially Stiefel and Grassmann.
method Proposed RL-barycenters, simpler arithmetic means projected onto the manifold.
result RL-barycenters yield simple yet effective means on Stiefel and Grassmann manifolds.
This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, R…
The study finds a diameter bound for graphs with positive entropic Ricci curvature, with optimal bounds for arithmetic mean.
problem Finding diameter bounds for graphs with positive entropic Ricci curvature.
method Using a localized gradient estimate and an equivalent definition of entropic Ricci curvature, the study derives a Bonnet-Myers type diameter bound.
result The derived diameter bound is optimal for arithmetic mean, but not for logarithmic mean.
Unified learning-rate scale for CNNs and ResNets, avoiding depth imbalance.
problem Challenges in choosing an appropriate learning rate for deep networks, especially as depth increases.
method Introduces Arithmetic-Mean μP (AM-μP), constraining network-wide average pre-activation second moment to a constant scale, combined with residual-aware He fan-in initialization. result Demonstrates a −3/2 scaling law for learning rates across depths, enabling zero-shot learning-rate transfer. Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has in…
Improved error feedback method reduces communication complexity in distributed training.
problem Improving convergence in distributed training methods with compression techniques.
method Introduced and improved a modern form of error feedback (EF21) by reducing its theoretical communication complexity.
result Communication complexity depends on the arithmetic mean of smoothness parameters, leading to significant improvements.
Neural networks learn modular arithmetic but not all, extending known solutions to generalize.
problem Neural networks struggle with modular arithmetic, especially for polynomials.
method Developed analytical solutions for MLP networks to learn modular addition and multiplication, then combined these solutions to generalize on arbitrary modular polynomials.
result Neural networks can learn and generalize solutions to modular polynomials, supporting the hypothesis that some polynomials are learnable.
Efficient Winograd convolution for INT8 networks using RNS.
problem Difficulty in applying Winograd algorithm to low-precision quantized networks.
method Extends Winograd algorithm to Residue Number System (RNS) for efficient INT8 convolution.
result Arithmetic complexity reduction up to 7.03x with performance improvement up to 2.30x-4.69x.
Quantized neural networks can represent all fixed-point functions under certain conditions.
problem Expressive power of quantized neural networks under fixed-point arithmetic.
method Analyzing necessary and sufficient conditions for quantized networks to represent all fixed-point functions.
result Various popular activation functions satisfy the sufficient condition for representing all fixed-point functions.
CUTS removes corruption from models without clean data, improving utility and security.
problem Removing corruption from models without access to clean training data.
method CUTS uses a proxy set to amplify corruption and subtract it from model weights.
result CUTS recovers a large fraction of lost utility and nearly eliminates attacks with minimal damage.
Bayesian optimisation's mean function choice affects convergence speed.
problem The choice of mean function in Bayesian optimisation impacts convergence speed.
method Empirical investigation of 8 mean functions on 10 synthetic and 2 real-world problems.
result Using a constant mean function equal to the worst observed quality value promotes faster convergence.
Randomized algorithms that base iteration-level decisions on samples from some pool are ubiquitous in machine learning and optimization. Examples include stochastic gradient descent and randomized coordinate descent. This paper makes progress at theoretically evaluating the difference in performance between sampling wi…
A new deep learning method using Boolean logic reduces training and inference energy.
problem High computational and energy costs in deep learning training and inference.
method Introduces Boolean weights and inputs for efficient training using Boolean logic.
result Achieves full-precision accuracy in ImageNet classification and surpasses state-of-the-art results in semantic segmentation.
New meta-learning method improves domain generalization by balancing parameters closer to domain centroids.
problem Improving domain generalization by reducing overfitting to specific domains.
method Arithmetic meta-learning with arithmetic-weighted gradients to balance parameters closer to domain centroids.
result Experimental validation of improved domain generalization performance.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
Consider oriented surfaces immersed in R3. Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k1,k2 is positive. The leaves of the foliations …
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsific…
Associated to oriented surfaces immersed in R^3 here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k_1, k_2 of the immersion, is positive. The leaves of the foliations are the lines of M- m…
We develop a Vector Quantized Spectral Clustering (VQSC) algorithm that is a combination of Spectral Clustering (SC) and Vector Quantization (VQ) sampling for grouping Soybean genomes. The inspiration here is to use SC for its accuracy and VQ to make the algorithm computationally cheap (the complexity of SC is cubic in…
Develops arithmetic PDE geometry concepts like curvature and cohomology.
problem Creating a geometry framework for arithmetic PDEs.
method Introducing arithmetic analogues of Levi-Civita and Chern connections, then developing curvature and characteristic classes.
result Arithmetic analogues of curvature and characteristic classes have been developed.
New geometric invariant limits the number of semi-arithmetic groups.
problem Understanding the structure of semi-arithmetic Fuchsian groups.
method Introducing a new geometric invariant called stretch and using the arithmetic Margulis lemma.
result There exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch, and coarea.
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
New findings show mapping class groups of certain high-dimensional manifolds are not residually finite.
problem Understanding the mapping class groups of simply connected high-dimensional manifolds.
method Provided a counterexample showing mapping class groups are not residually finite.
result Mapping class groups of some high-dimensional manifolds are not residually finite.
We introduce a novel scheme to train binary convolutional neural networks (CNNs) -- CNNs with weights and activations constrained to {-1,+1} at run-time. It has been known that using binary weights and activations drastically reduce memory size and accesses, and can replace arithmetic operations with more efficient bit…
Paper shows non-arithmetic surface with unique geometric property.
problem Non-arithmetic surfaces with unique geometric properties.
method Example of a non-arithmetic surface with marked length variety rigidity.
result Found a non-arithmetic surface with marked length variety rigidity.
The study analyzes numerical stability in large language models using mixed-precision arithmetic.
problem Numerical stability of large language models using low-precision arithmetic.
method Developed a mixed-precision analysis of transformer inference, deriving bounds for condition numbers and forward error.
result Established that numerical stability is determined by the interplay between weight magnitude and the growth of the residual stream.
New classification of hyperbolic Coxeter prisms.
problem Classifying hyperbolic Coxeter prisms.
method Determine which prisms are quasi-arithmetic or arithmetic.
result New insights into commensurability and systoles of associated orbifolds.
The paper extends the market price of risk for electricity swap contracts, incorporating jump risk.
problem Pricing electricity swap contracts with consideration of jump risk.
method Introducing a Merton type model with jumps and transferring to the physical measure, comparing arithmetic and geometric averaging.
result A decomposition of swap's market price of risk into classical and market price of risk components.
Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.
problem Developing arithmetic analogues in Chern-Simons TQFT.
method Constructing arithmetic analogues of Chern-Simons 1-cocycle, prequantization bundle, and Chern-Simons functional.
result Decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.