Hard phase in inference problems is glassy and hard to reconstruct.
problem Hard phase in inference problems that are hard to solve algorithmically.
method Study of metastable states and their entropy in low-rank matrix factorization.
result AMP algorithm performance is not improved by considering glassy states.
Deep neural networks and glassy systems share dynamics but differ in landscape properties.
problem Comparing training dynamics of DNNs and glassy systems.
method Statistical physics methods applied to DNN training.
result DNN dynamics slow down due to many flat directions, diffusing at the loss minimum.
Review of gradient-based algorithms in statistical inference problems.
problem Understanding the dynamics of gradient-based algorithms in statistical inference.
method Insights from physics of glassy systems.
result Quantitative and qualitative understanding of algorithm performance.
Experimental fractal landscape dynamics observed in emulsions.
problem Understanding anomalous motions in soft glassy materials.
method Quantitative analysis of oil droplet trajectories in dense emulsions.
result Experimental fractal geometry matches computational model of soft glassy dynamics.
Combining ML and physics for understanding glassy systems.
problem Understanding supercooled liquids and glasses due to disorder and non-equilibrium effects.
method Data-driven approach using machine learning with physical intuition.
result Building a phenomenological theory of disordered materials.
Study dynamics of alternating minimization for bilinear regression under large system limits.
problem Understanding the time evolution of alternating minimization for bilinear regression.
method Replica method applied to a multi-temperature glassy system.
result Dynamics of alternating minimization can be described by a two-dimensional discrete stochastic process.
Study of Langevin algorithm in noisy high-dimensional inference.
problem Analyzing the Langevin algorithm's performance in noisy high-dimensional inference.
method Analytic study of Langevin algorithm's performances using the spiked matrix-tensor model.
result The algorithmic threshold of the Langevin algorithm is sub-optimal compared to AMP.
Paper introduces supervised and unsupervised TAM models for binary neurons.
problem Learning and retrieval of structured triplets of patterns in neural networks.
method Extends Hebbian paradigm to supervised and unsupervised protocols, using glassy statistical mechanical techniques.
result Obtained self-consistency equations for critical dataset sizes and retrieval performance.
We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive because any network can be embedded on a surface with sufficiently high genus. The…
The dynamical behavior of the currency exchange rate after its large-scale catastrophe is discussed through a case study of the rate of Russian rubles to US dollars after its crash in 2014. It is shown that, similarly to the case of the stock market crash, the relaxation is characterized by a power law, which is in ana…
Paper introduces ASP, a variant of AMP for low-rank matrix estimation, showing improved performance under model mismatch.
problem Statistical inference for low-rank matrix estimation problems.
method Introduces approximate survey propagation (ASP) algorithm for low-rank matrix estimation problems.
result ASP converges in a larger regime and can reach lower errors compared to AMP when there is a model mismatch.
In our model, n traders interact with each other and with a central bank; they are taxed on the money they make, some of which is dissipated away by corruption. A generic feature of our model is that the richest trader always wins by 'consuming' all the others: another is the existence of a threshold wealth, below wh…
Study energy landscapes in glass models, focusing on Gaussian and spiked-tensor functions.
problem Characterize statistical properties and phase transitions of high-dimensional energy landscapes.
method Developed a Kac-Rice method framework to compute landscape complexity and analyze phase transitions rigorously.
result Characterized the ruggedness and arrangements of local minima in energy landscapes.
Empirical data of supermarket sales show stylised facts that are similar to stock markets, with a broad (truncated) Levy distribution of weekly sales differences in the baseline sales [R.D. Groot, Physica A 353 (2005) 501]. To investigate the cause of this, the influence of social interactions and advertisements are st…
Model explains asset price fluctuations using interacting neurons.
problem Understanding volatility and price dynamics in financial markets.
method Structural model based on interacting neurons, generating functional analysis, simulations.
result Model predicts fat-tailed and broader distributions of asset returns.
Deep neural networks undergo hierarchical free-energy landscape transitions with increasing data size.
problem Understanding the design space and dynamics of deep neural networks.
method Statistical mechanical approach based on replica method.
result Hierarchical free-energy landscape transitions with ultrametricity, leading to simpler configurations in deeper layers.
Classifies 7- and 8-dimensional naturally reductive spaces.
problem Classifying naturally reductive spaces in 7 and 8 dimensions.
method Combines structure theory and new construction methods.
result Complete classification of 7- and 8-dimensional naturally reductive spaces.
Study quantifies firm risks from nature decline, showing significant equity losses.
problem Estimating the financial impact of nature deterioration on companies.
method Developed metrics (Country Degradation Index, Nature Risk Score) and assessed five environmental hazards.
result Global equities lose 26.8% in a nature decline scenario, with worst firms losing 75%.
When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…
Integration procedure for Lie groupoid natural transformations.
problem Infinitesimal counterpart of natural transformations in Lie groupoids.
method Integration procedure for Lie groupoid morphisms.
result Provides smooth natural transformations between Lie groupoid morphisms.
Generates natural product-like compounds using GPT models.
problem Challenges in generating and evaluating natural product-like compounds.
method Trained GPT-based chemical language models on natural product dataset.
result Generated compounds have similar distribution to natural products.
New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
Defines and proves the uniqueness of a second natural connection on Riemannian Π-manifolds.
problem Characterizing and proving uniqueness of a natural connection on Riemannian Π-manifolds.
method Definition and proof of the second natural connection, proving its uniqueness and necessary/sufficient condition for coincidence with the first natural connection.
result Proves the uniqueness of the second natural connection on Riemannian Π-manifolds.
Paper constructs naturally reductive spaces with a general formula.
problem Understanding naturally reductive spaces.
method Explicit construction from \cite{Storm2018} and general formula derivation.
result Proves reducibility and isomorphism criteria.
Study characterizes naturally reductive metrics on homogeneous manifolds.
problem Characterizing naturally reductive (α1,α2) metrics on homogeneous manifolds. method Characterization through local f-products and equivalence of properties. result Explicit flag curvature formula for naturally reductive metrics.
Natural gradients boost performance in non-conjugate Gaussian process models.
problem Improving inference in non-conjugate Gaussian process models.
method Use of natural gradients in non-conjugate stochastic settings with hyperparameter learning.
result Natural gradients significantly improve performance, especially for ill-conditioned posteriors.
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A concrete transitive group of isometries is given for the new spaces and also the na…
Natural gradient simplification for deep learning networks.
problem Efficiency in training deep Bayesian networks.
method Analysis of two geometries of Fisher information matrix and development of a method to simplify natural gradient for the second geometry.
result A method to simplify natural gradient for deep networks using an auxiliary recognition model.
Study finds real-world datasets contain natural experiments that can improve model performance.
problem Detecting natural experiments in real-world datasets for causal inference.
method Synthetic graph simulation and feature selection based on causal links.
result Real-world datasets contain natural experiments that can be exploited for improved model performance.
We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fra…
TANGO optimizes models with small learning rates, converging to natural gradient.
problem Optimizing models with small learning rates to converge to natural gradient.
method TANGO, a simple algorithm that converges to natural gradient descent.
result TANGO achieves natural gradient descent with small learning rates.
Paper shows spectra can't distinguish naturally reductive manifolds.
problem Cannot distinguish naturally reductive manifolds using Laplace-Beltrami spectrum.
method Characterized naturally reductive 2-step nilpotent Lie groups via Ambrose-Singer's structures; constructed isospectral pairs of 9-dimensional nilmanifolds.
result Spectra of Laplace-Beltrami operator can't distinguish naturally reductive manifolds from non-naturally reductive ones.
A new natural gradient accounts for correlated variational parameters in variational inference.
problem Traditional natural gradients fail to correct for correlations in variational inference.
method Construct a new natural gradient called the Variational Predictive Natural Gradient (VPNG).
result VPNG accounts for the relationship between model parameters and variational parameters.
In the present paper we study naturally reductive homogeneous (α,β)-metric spaces. Under some conditions, we give some necessary and sufficient conditions for a homogeneous (α,β)-metric space to be naturally reductive. Then we show that for such spaces the two definitions of naturally reductive homogeneous Finsler …
I.A.B. Strachan introduced the notion of a natural Frobenius submanifold of a Frobenius manifold and gave a sufficient but not necessary condition for a submanifold to be a natural Frobenius submanifold. This paper will give a necessary and sufficient condition and classify the natural Frobenius hypersurfaces.
Introduces a natural parallel translation for navigation data.
problem Navigation data geometric representation and parallelism.
method Introduces a natural parallel translation using Riemannian parallelism.
result The natural parallel translation preserves the Randers norm and has a finite-dimensional holonomy group.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
Symmetry distribution in naturally reductive nilpotent Lie groups matches invariant induced by fixed vectors.
problem Understanding symmetry distribution in naturally reductive nilpotent Lie groups.
method Analyzing the invariant distribution induced by fixed vectors of isotropy.
result Symmetry distribution matches the invariant induced by fixed vectors.
Real Heegaard Floer theory shown to be natural and invariant.
problem Defining and proving naturality of real Heegaard Floer homology.
method Defined and proved naturality of real Heegaard Floer homology and other related theories.
result Real Heegaard Floer homology is shown to be natural and admits an action of the equivariant mapping class group.
A framework for natural gradient with arbitrary similarity measures.
problem Unclear metric for natural gradient in non-Euclidean spaces.
method Derive a metric for natural gradient given an arbitrary similarity measure.
result General framework for natural gradient in non-Euclidean spaces.
A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-ad…
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
Square-root natural-gradient improves variational inference convergence.
problem Challenges in establishing theoretical convergence guarantees for natural-gradient descent.
method Square-root parameterization for Gaussian covariance.
result Establishes novel convergence guarantees for natural-gradient Gaussian inference.
Paper proposes a natural hedging framework with graphical assessment for longevity risk management.
problem Lack of a unified framework for natural hedging and graphical risk assessment.
method Structured natural hedging framework integrated with a graphical risk metric.
result Demonstrates flexibility, interpretability, and practical value for longevity risk management.
We extend natural-gradient methods to mixtures of exponential-family distributions, improving inference speed.
problem Complex, multimodal posterior distributions are difficult to approximate with simple exponential-family distributions.
method We use minimal conditional-EF representations and derive simple natural-gradient updates.
result Our natural-gradient method converges faster than black-box methods with reparameterization gradients.
Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.
problem Understanding natural quasiconvexity and its implications in risk measures.
method Relates natural quasiconvexity to decomposable sums, proposes a general treatment of convexity index, and proves equivalence for certain spaces.
result Natural quasiconvexity and convexity are equivalent for conditional risk measures on Lp spaces under mild conditions. Study natural invariants for differential operators, simplifying their equivalence problem.
problem Equivalence problem of nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to simplify differential operator equivalence.
Study natural operators transforming tensor fields, proving all bilinear ones are of order one.
problem Understanding natural differential operators on tensor fields.
method Proved all bilinear operators are of order one, then classified operators in specific cases.
result Full classification of natural differential operators on tensor fields.