Bayesian deep learning avoids underfitting by projecting onto null space of generalized Gauss-Newton matrix.
problem Bayesian deep learning often underfits, leading to less accurate predictions than point estimates.
method Proposes a matrix-free algorithm to project onto the null space of the generalized Gauss-Newton matrix, ensuring Bayesian predictions do not underfit.
result The method scales to large models, including vision transformers with 28 million parameters, and avoids underfitting.
Meta-learner reduces few-shot learning errors by nulling out error signals.
problem Few-shot learning accuracy issues in neural networks.
method Linear transformer for null-space projection of neural network outputs.
result Meta-learner achieves best or near-best image classification accuracies.
Paper analyzes null space for class-specific discriminant learning.
problem Improving discriminant learning for class-specific problems.
method Null space analysis for Class-Specific Discriminant Analysis (CSDA).
result Proposed solutions outperform standard CSDA methods.
Integrates outlier detection into neural networks for improved performance.
problem Lack of competency awareness in machine learning systems, especially in detecting outliers.
method Null Space Analysis (NuSA) of neural networks, computing and controlling null space projection.
result NuSA-trained networks maintain classification performance and detect outliers effectively.
The paper addresses fairness in machine learning models through structural econometrics, projecting indexes into null spaces to find fair solutions.
problem Fairness concerns in machine learning, especially regarding disadvantaged groups.
method Model fairness as a linear operator, projecting indexes into null spaces to find fair solutions, balancing status quo and full fairness.
result Achieving approximate fairness by introducing a fairness penalty and balancing influences.
Paper refines null space conditions for nuclear norm minimization in low-rank matrix recovery.
problem Establishing conditions for successful nuclear norm minimization recovery of low-rank matrices.
method Developed new null space conditions for nuclear norm minimization, proving their necessity and sufficiency.
result Weak null space condition is sufficient but not necessary for nuclear norm minimization recovery, providing a new necessary and sufficient condition.
New findings on curvature and null spaces of Laplacians.
problem Relationship between sectional curvature and Laplacian null spaces.
method Analysis of curvature operators and Laplacians on Riemannian manifolds.
result Curvature operator's positivity implies sectional curvature positivity.
Study reveals hidden null components in overparametrized neural networks.
problem Hidden null components in overparametrized neural networks.
method Structure theorem of null space for neural networks using ridgelet transforms.
result Null components can be uniquely written as linear combinations of ridgelet transforms.
There is a class of Laplacian like conformally invariant differential operators on differential forms Lkℓ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explic…
Study improves distributed linear estimation under adversarial conditions.
problem Mean estimation of a random vector with adversarial measurements and asynchrony.
method Two-timescale ℓ1-minimization algorithm with tight convergence rates.
result Unified finite-time characterization of robustness, identifiability, and statistical efficiency.
In the Compressed Sensing community, it is well known that given a matrix X∈Rn×p with ℓ2 normalized columns, the Restricted Isometry Property (RIP) implies the Null Space Property (NSP). It is also well known that a small Coherence μ implies a weak RIP, i.e. the singular values of XT l…
Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in \cite{arxiv,Recht_Xu_Hassibi}. However simulations show that the thresholds are far fro…
The paper proposes a novel MKL approach for OCC using ℓp-norm constraints.
problem Addressing the MKL problem for one-class classification.
method A min-max saddle point Lagrangian optimisation problem is formulated and solved efficiently.
result The proposed method outperforms baselines and other algorithms on various data sets.
Enhances robustness of one-class classification framework.
problem Susceptibility to training data corruptions and inability to rank observations.
method Regularization of null-space kernel Fisher methodology in OC-KSR.
result Enhanced robustness against contamination in training set.
Machine learning identifies chimera states in complex dynamical systems.
problem Chimera states are hard to identify due to their varied appearance and peculiar nature.
method Machine learning techniques, specifically random forest and oblique random forest with null space regularization.
result High accuracy in identifying chimera states across different dynamical models.
Deep learning improves ROI reconstruction in low-dose CT.
problem Severe cupping artifacts in standard analytic reconstruction.
method Proposes a deep learning architecture to remove null space signals from FBP reconstruction.
result Near-perfect reconstruction with 7-10 dB improvement in PSNR.
Extends OC-KSR for multi-task one-class classification.
problem Improving one-class classification performance with shared information.
method Linear and non-linear structure learning mechanisms for multi-task one-class classification.
result Improved performance on multiple one-class problems.
ZDP detects drift in large language models without labels, proving key theorems and metrics.
problem Detecting drift in large language models without task labels or output evaluations.
method Zero-Direction Probing (ZDP) framework based on null directions of transformer activations, proving theoretical guarantees.
result Proves the Variance--Leak Theorem, Fisher Null-Conservation, Rank--Leak bound, and logarithmic-regret guarantee.
Performance of nuclear threat detection systems based on gamma-ray spectrometry often strongly depends on the ability to identify the part of measured signal that can be attributed to background radiation. We have successfully applied a method based on Principal Component Analysis (PCA) to obtain a compact null-space m…
The paper explores properties of the Radon transform in relation to neural networks and ridges.
problem Understanding the Radon transform and its application to neural networks and ridges.
method Investigates properties of the Radon transform, introduces new subspaces, and characterizes ridges for any distributional profile.
result Clarifies and simplifies results on the optimality of ReLU networks using the Radon transform.
For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are elliptic in the case of Riemannian signature); the cohomology spaces of these; conformally stable form spaces that we may…
In this paper, we propose new efficient algorithms to verify the null space condition in compressed sensing (CS). Given an (n−m)×n (m>0) CS matrix A and a positive k, we are interested in computing αk={z:Az=0,z=0}max{K:∣K∣≤k}max ∥zK∥1∥z∥1, where …
Physics-informed neural networks improve by measuring effective dimensionality of constraints.
problem Task interference in physics-informed neural networks due to shared parameter space.
method Introduce effective dimensionality (deff) as an operator invariant to quantify constraints. result Effective dimensionality measures unconstrained parameter directions, independent of network architecture.
The study classifies quasi-minimal surfaces in 4D pseudo-Riemannian space-forms with positive nullity.
problem Characterizing surfaces in pseudo-Riemannian space forms with positive nullity.
method Analyzing the relative null space and classifying quasi-minimal surfaces.
result Classifications of quasi-minimal surfaces with positive relative nullity.
In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical me…
The study of higher-order homology embeddings for manifold topology.
problem Understanding the structure of higher-order homology embeddings to disclose geometric or topological information.
method Analysis of the null space of the k-th order Laplacian and proposing an algorithm to factorize the homology embedding. result The proposed spectral loop detection algorithm is more efficient and effective on various data types.
On an even conformal manifold (M,c), such that the critical GJMS operator has non-trivial kernel, we identify and discuss the role of a finite dimensional vector space N(Q) of functions determined by the conformal structure. Using these we describe an infinite dimensional class of functions that cannot be the Q-cur…
Study recasts learning non-linear functions from noisy data as robust regression, proving reconstruction guarantees.
problem Learning non-linear functions from corrupted and dependent data.
method Sparse robust linear regression with ℓ1-optimization, incorporating unknown coefficients and corruptions. result Reconstruction guarantees for ℓ1-optimization problem with dependent data, proving null and stable null space properties. Quantum method detects financial stress regimes from market data.
problem Detecting financial stress regimes from market data.
method Adapted Pauli Correlation Encoding to quantum topological data analysis.
result Quantum method can recover Betti numbers exactly at every scale.
We propose a definition for analytic torsion of the Rumin complex on contact manifolds. This is given by the derivative at zero of a well-chosen combination of zeta functions of a fourth-order modified Rumin Laplacian. The regular value at zero (before differentiation) of this well-chosen combination of zeta functions …
New curvature concept preserves graph distances under operations.
problem Preserving graph distances under graph operations.
method Characterization of distance matrix and its null space.
result Linear system Dx=1 may not have a solution. The paper analyzes how gradient descent implicitly regularizes solutions in overparameterized neural networks, revealing depth-dependent regularization effects.
problem Understanding implicit regularization in overparameterized linear neural networks for regression problems.
method Analyzing the approximation error between gradient flow limit points and ℓ1-minimization solutions, deriving tight upper and lower bounds. result The approximation error decreases linearly for D≥3 and at a slower rate for D=2, linked to null space property constants. We address some theoretical guarantees for Schatten-p quasi-norm minimization (p∈(0,1]) in recovering low-rank matrices from compressed linear measurements. Firstly, using null space properties of the measurement operator, we provide a sufficient condition for exact recovery of low-rank matrices. This condition…
Max-convolution is an important problem closely resembling standard convolution; as such, max-convolution occurs frequently across many fields. Here we extend the method with fastest known worst-case runtime, which can be applied to nonnegative vectors by numerically approximating the Chebyshev norm $\| \cdot \|_\infty…
A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of…
New binary matrices improve compressed sensing with faster and less storage requirements.
problem Achieving robust sparse recovery with binary measurement matrices.
method Derived bounds and conditions for binary matrices to satisfy the robust null space property (RNSP).
result Binary matrices with girth six are nearly optimal for compressed sensing.
A financial contract's value is determined by a quantum measurement outcome, and a pricing state exists to value it.
problem Valuing financial contracts contingent on quantum measurement outcomes.
method Proving the existence of a pricing state equivalent to the physical state on null spaces, and solving optimization problems for optimal contract payouts.
result There exists a pricing state equivalent to the physical state on null spaces, leading to a pricing function for financial contracts.
On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense, and for any such operator we construct a partial connection on a (finite ra…
Einstein's non-symmetric geometry uses Bochner's technique to prove decomposition and vanishing results.
problem Analyzing Einstein's non-symmetric geometry with Bochner's technique.
method Defining concepts, proving decomposition formula, and showing vanishing results.
result Vanishing results about the null space of Bochner and Hodge type Laplacians.
We introduce a simple and very fast algorithm that computes Weil-Petersson metrics on moduli spaces of polarized Calabi-Yau manifolds. Also, by using Donaldson's quantization link between the infinite and finite dimensional G.I.T quotients that describe moduli spaces of varieties, we define a natural sequence of Kaehle…
Study on Hermitian manifolds with curvature, finding geometric properties.
problem Understanding the structure of Hermitian manifolds with semipositive Griffiths curvature.
method Combining HCF, torsion-twisted connection properties, and geometric observations.
result Null spaces of the Chern-Ricci form generate a holomorphic, integrable distribution.
The study examines the geometry of Lichnerowicz Laplacian's kernel on various spaces.
problem Understanding the kernel of the Lichnerowicz Laplacian on different types of spaces.
method Analytical method of Bochner to prove vanishing theorems for null space of Laplace operator.
result Applications to theories of infinitesimal Einstein deformations and stability of Einstein manifolds.
Proposes a new sparse recovery method using generalized error function.
problem Sparse recovery in signal processing and imaging.
method Introduces a penalty function with shape and scale parameters for sparse recovery.
result The method improves MRI reconstruction and is theoretically sound.
The paper extends Bochner's technique to singular distributions on manifolds.
problem Analyzing the curvature and null space of Hodge Laplacian on singular distributions.
method Defining modified statistical connection, exterior derivative, and Weitzenbock type curvature operator.
result Derivation of Bochner-Weitzenbock type formula leading to vanishing theorems.
The alignment of a set of objects by means of transformations plays an important role in computer vision. Whilst the case for only two objects can be solved globally, when multiple objects are considered usually iterative methods are used. In practice the iterative methods perform well if the relative transformations b…
We analyze the resolvent R(k)=(P+k2)−1 of Schrödinger operators P=Δ+V with short range potential V on asymptotically conic manifolds (M,g) (this setting includes asymptotically Euclidean manifolds) near k=0. We make the assumption that the dimension is greater or equal to 3 and that P has no L2 null …
A new algorithm solves constrained optimization problems with stochastic gradients.
problem Nonlinear equality constrained optimization with rank-deficient Jacobians.
method Step decomposition strategy combining normal and tangential steps.
result Convergence guarantees in rank-deficient Jacobian cases.
New method discovers symmetries in differential equations from data.
problem Directly identifying Lie symmetries from scattered data without explicit equations.
method Numerical scheme using manifold learning and linear system construction.
result Accuracy and robustness demonstrated in various differential equations.