We rigorously prove statistical physics predictions for non-convex GLMs in high dimensions.
problem Analyzing high-dimensional optimization problems in non-convex Generalized Linear Models.
method Developed a systematic framework using the Gaussian Min-Max Theorem and AMP to rigorously prove replica-symmetric formulas.
result Validated statistical physics predictions for non-convex GLMs, aligning with physicist's conjectures.
PLS-SVD struggles with missing data in multimodal datasets, showing a phase transition in performance.
problem Missing data in PLS-SVD for multimodal datasets.
method Replica-symmetric analysis of spiked rectangular random matrices with missing entries.
result PLS-SVD performance transitions from uninformative to informative singular vectors at a critical signal-to-noise threshold.
Dense Associative Memories outperform classical networks in robustness and signal processing.
problem Improving neural network performance in adversarial attacks and weak signal processing.
method Relaxing replica symmetry in statistical mechanics of spin glasses to analyze unsupervised and supervised learning.
result Explicit analytical investigation of phase diagrams and storage capacities for Dense Associative Memories.
We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show tha…
We investigate the signal reconstruction performance of sparse linear regression in the presence of noise when piecewise continuous nonconvex penalties are used. Among such penalties, we focus on the SCAD penalty. The contributions of this study are three-fold: We first present a theoretical analysis of a typical recon…
We study Generalised Restricted Boltzmann Machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as Generalised Hopfield models. We underline the role of the r…
Statistical physics helps solve complex machine learning problems.
problem Large dimensional inference problems in machine learning.
method Replica symmetric level analysis and cavity methods.
result General framework for solving various problems with weak long-range interactions.
Binary perceptron's instability linked to replica symmetry breaking.
problem Understanding the relationship between algorithmic instability and replica symmetry breaking in binary perceptron learning.
method Established the connection between algorithmic instability and replica symmetry breaking by comparing the instability condition around the fixed point to the instability for breaking the replica symmetric solution of the free energy function.
result The instability condition around the algorithmic fixed point is identical to the instability for breaking the replica symmetric saddle point solution of the free energy function.
Minimizing a convex, quadratic objective of the form fA,b(x):=21x⊤Ax−⟨b,x⟩ for A≻0 is a fundamental problem in machine learning and optimization. In this work, we prove gradient-query complexity lower bounds for minimizing conv…
A fast, approximate method for variable selection in GLMs tackles correlated data.
problem Variable selection in generalized linear models with correlated data.
method Replica method of statistical mechanics and vector approximate message passing.
result The proposed algorithm provides fast convergence and high approximation accuracy.
Analyzes unsupervised neural networks using statistical mechanics and Monte Carlo simulations.
problem Understanding computational capabilities of unsupervised neural networks.
method Statistical mechanics approach and Monte Carlo simulations.
result Obtained a phase diagram summarizing network performance.
The inverse Ising problem seeks to reconstruct the parameters of an Ising Hamiltonian on the basis of spin configurations sampled from the Boltzmann measure. Over the last decade, many applications of the inverse Ising problem have arisen, driven by the advent of large-scale data across different scientific disciplines…
We study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a high-dimensional regime where the spike has fixed strength and the aspect ratio of the matrix converges to a finite limit. This setup includes Johnstone's spiked covariance model. We analyze the likelihood ratio …
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.
Neural networks can detect weak patterns hidden in noise.
problem Detecting weak patterns in noisy data.
method Developed a three-layer Sejnowski machine with redundant representation, showing patterns can be stored and retrieved efficiently.
result Neural networks can retrieve information with intensity O(1) even in the presence of noise O(\sqrt{N}) in the large N limit.
Injectivity of ReLU networks studied using statistical physics.
problem When can the input of a ReLU neural network be inferred from its output?
method Connection to spherical integral geometry and statistical physics.
result Replica symmetry-breaking theory and Gordon's min--max theorem provide insights into the injectivity threshold.
Dense neural networks learn efficiently with large datasets and noise.
problem Training neural networks with large, noisy datasets.
method Statistical mechanics and Monte Carlo simulations.
result Dense neural networks can handle large amounts of patterns and recognize patterns at high signal-to-noise ratios.
Study neural networks learning from noisy examples via reverberation.
problem Learning from noisy examples in neural networks.
method Adapted Guerra's interpolation technique to provide statistical mechanics of supervised and unsupervised learning.
result Full phase diagrams and thresholds for learning are analytically obtained.
New method improves sparse signal reconstruction using 1RSB-AMP.
problem Sparse signal reconstruction with improved accuracy.
method Developed 1RSB-AMP and 1RSB-SE for SCAD penalty minimization.
result 1RSB-AMP achieves improved reconstruction compared to RS-AMP.
Recently a daily routine for associative neural networks has been proposed: the network Hebbian-learns during the awake state (thus behaving as a standard Hopfield model), then, during its sleep state, optimizing information storage, it consolidates pure patterns and removes spurious ones: this forces the synaptic matr…
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…
We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, …
Note on new cancellation formulas for manifolds.
problem Generalizing anomaly cancellation formulas to manifolds.
method Proving new (a, b) type cancellation formulas and using transgression.
result Obtained characteristic forms with modularity properties.
The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …
Formula calculates volume of two-bridge knots.
problem Calculating the volume of two-bridge knots.
method Derived from Hopf formula and Fox derivatives.
result Closed formula for the volume of two-bridge knots.
Introduces a universal Bochner formula for scalar curvature.
problem None explicitly stated; focuses on a new formula.
method Introduces a universal Bochner formula.
result Contains special cases like stability inequality and Schrödinger-Lichnerowicz-type formula.
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
It has been shown that the Alvarez-Gaumeˊ-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…
Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.
Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
New Crofton formulae derived from existing ones.
problem Generalizing Crofton formulae for products.
method Calculations in the ring of normal densities.
result Generalizations of Crofton formulae in terms of mixed Riemannian volume.
Formulae for non-symmetric connections derived from covariant derivatives.
problem Deriving commutation formulae for non-symmetric affine connections.
method Covariant derivatives of tensors with respect to symmetric and non-symmetric affine connections.
result Formulae for non-symmetric connections derived from covariant derivatives.
Formula connects curvature to volume in special geometric spaces.
problem Deriving formulas for curvature in specific geometric spaces.
method Used strong locality of Laplacian and eigenfunction approximation.
result Proved integral type Gauss-Green formula linking curvature to volume.
Kenmotsu's formula describes surfaces in Euclidean 3-space by their mean curvature functions and Gauss maps. In Lorentzian 3-space, Akutagawa-Nishikawa's formula and Magid's formula are Kenmotsu-type formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rota…
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
New formulas for measuring geometric properties of definable sets.
problem Measuring geometric properties of definable sets in a non-standard setting.
method Proved two kinematic formulas integrating on SO(n)imesSn−1. result Generalized Cauchy-Crofton and infinitesimal linear kinematic formulas.
Guillemin trace formula adapted for group actions.
problem Distributional trace for proper, cocompact group actions.
method Developing an equivariant version of the distributional trace.
result Equivariant Guillemin trace formula for group actions.
Unified entropy formula for real, complex, and quaternionic DLNs.
problem Deriving a formula for DLNs over different fields.
method Extending Menon and Yu's formula to complex and quaternionic DLNs.
result Unified entropy formula for DLNs over R, C, and H. Proves an Euler-type formula for Möbius strip partitions.
problem No specific problem stated; focuses on a mathematical formula.
method Analyzes partitions of the Möbius strip.
result Proves an Euler-type formula for Möbius strip partitions.
We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.
Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on non-orientable surfaces.
New formulas for coassociative submanifolds' volume variation.
problem Understanding volume changes in coassociative submanifolds.
method Proved new variation formulae using G2 data. result Highlight the role of ambient torsion and Ricci curvature in volume changes.
New methods derive a generalized Frenkel trace formula for Lie groups.
problem Deriving a generalized Frenkel trace formula for Lie groups.
method Applying supersymmetric localization to quantum mechanical and gauged sigma models.
result Presented two complementary approaches for the derivation of the trace formula.