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.
Neural networks' feature geometry evolves like discrete Ricci flow.
problem Understanding neural feature representations and their geometric transformations.
method Approximating input manifold with geometric graphs and analyzing their evolution during training.
result Neural feature geometry evolves like discrete Ricci flow, with nonlinear activations playing a crucial role.
The paper matches features in images using centro-affine invariants and heat flow.
problem Feature matching in images with invariant algorithms.
method Developed an invariant algorithm using centro-affine invariants and heat flow.
result The algorithm compares favorably with existing feature matching methods.
Adapts IG for better feature attributions and robustness.
problem Reliability concerns in feature attributions for deep learning models.
method Adaptation of path-based feature attribution to Riemannian geometry of data manifolds.
result IG along geodesics generates more intuitive and robust explanations.
Training shapes the geometry of neural network feature maps, revealing local area magnification.
problem Understanding how training affects the geometric structure of neural network feature maps.
method Analyzing the Riemannian geometry induced by neural network feature maps at infinite width and after training.
result Training breaks the symmetry of the geometry induced by random neural network feature maps, magnifying local areas along decision boundaries.
SQFA learns features maximizing Fisher-Rao distance for better classification.
problem Improving classification accuracy through feature learning.
method SQFA learns linear features maximizing Fisher-Rao distance between class-conditional distributions.
result SQFA-H features achieve the best classification accuracy.
Study reveals LLM personas have two distinct components: frame-robust aggregated traits and frame-dependent geometric features.
problem Evaluation of LLM personas via psychometric questionnaires discards within-instance correlation structure.
method Constructed within-instance correlation matrices from IPIP-50 responses and analyzed geometry on SPD manifolds under manipulated question orderings.
result Persona expression comprises two dissociable components: aggregated features (Big Five scores) and geometric features (SPD manifold).
As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is th…
Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.
Sparse GEMINI selects relevant features for clustering without assumptions.
problem Feature selection in clustering with relevant clusters and variables.
method Discriminative clustering model maximizing GEMINI with l1 penalty.
result Sparse GEMINI selects relevant subsets of variables without prior hypotheses.
The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions.…
Support vector machines (SVMs) rely on the inherent geometry of a data set to classify training data. Because of this, we believe SVMs are an excellent candidate to guide the development of an analytic feature selection algorithm, as opposed to the more commonly used heuristic methods. We propose a filter-based feature…
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
Introduces neural point-forms for learning geometric features from noisy point clouds.
problem Learning geometric features from noisy point clouds with missing tangency information.
method Uses Laplacian-based techniques to build comparison matrices for point clouds, proving consistency under various assumptions.
result Neural point-forms provide a competitive and interpretable representation, especially beneficial for dense or manifold-like structures.
Quantum field theory connects Riemannian geometry to quantum fluctuations.
problem Generating Riemannian structures from quantum fluctuations.
method QFT approach to Riemannian Geometry, focusing on Ricci curvature.
result Ricci curvature is crucial in generating Riemannian structures.
HKConv learns hyperbolic features by aggregating kernel points.
problem Challenges in learning good hyperbolic representations using Euclidean operations.
method Proposes HKConv, a trainable hyperbolic convolution that correlates local features with kernel points and aggregates them.
result HKConv learns expressive local features according to hyperbolic geometry and enjoys equivariance to permutation and invariance to parallel transport.
A new method integrates autoencoders with geometry regularization for manifold learning.
problem Extracting simplified low-dimensional representations that capture intrinsic geometry in data.
method Integrates autoencoders with a geometric regularization term based on diffusion potential distances.
result The method preserves intrinsic structure, enables out-of-sample extension, and faithful reconstruction.
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.
3D Convolutional Neural Networks (3D-CNN) have been used for object recognition based on the voxelized shape of an object. However, interpreting the decision making process of these 3D-CNNs is still an infeasible task. In this paper, we present a unique 3D-CNN based Gradient-weighted Class Activation Mapping method (3D…
RNNs compute by warping neural representations over time.
problem Understanding how RNNs perform task computations.
method Developed a Riemannian geometric framework to derive the manifold topology and geometry of RNNs.
result Dynamic warping is a fundamental feature of RNN computations.
We propose a novel framework for combining datasets via alignment of their intrinsic geometry. This alignment can be used to fuse data originating from disparate modalities, or to correct batch effects while preserving intrinsic data structure. Importantly, we do not assume any pointwise correspondence between datasets…
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
Develops a smooth operator framework for analyzing neural network representations.
problem Analyzing the geometry of feedforward neural network representations.
method Introduces a smooth operator-theoretic approach based on diffusion Markov operators derived from feature clouds.
result Establishes a stable operator-geometric framework for tracking training, width, and perturbation stability.
Theory explains how deep nets learn features from data.
problem Understanding how deep neural networks learn features from data.
method Developed a noise-nonlinearity phase diagram and a mechanical theory.
result Links feature learning across layers to generalization.
AEN-SAEs address feature starvation in sparse autoencoders by stabilizing the geometric alignment of sparse coding.
problem Feature starvation in sparse autoencoders, leading to unstable and misaligned representations.
method Adaptive Elastic Net SAEs (AEN-SAEs) combine ℓ2 and ℓ1 terms to stabilize the sparse coding map and control feature interactions. result AEN-SAEs mitigate feature starvation without heuristic resampling, maintaining competitive reconstruction abilities.
MSA compares neural representations' intrinsic geometry for better understanding.
problem Existing similarity measures fail to capture subtle distinctions between neural network solutions.
method Metric similarity analysis (MSA) using Riemannian geometry.
result MSA can disentangle features of neural computations and compare nonlinear dynamics.
A new tree-Wasserstein distance for high-dimensional data with latent feature hierarchy.
problem Finding meaningful distances between high-dimensional data samples with latent feature hierarchy.
method Proposes a new tree-Wasserstein distance (TWD) for high-dimensional data with a latent feature hierarchy, using diffusion geometry and tree decoding.
result The proposed TWD effectively recovers the latent feature hierarchy and is efficient and scalable.
Persistent homology reveals geometric features of metric spaces, especially geodesic circles.
problem Detecting geometric features in metric spaces using persistent homology.
method Analyzing algebraic elements (footprints) in persistent homology of metric spaces and subspace.
result Higher-dimensional persistent homology captures lower-dimensional geometric features.
Survey of global geometry for double field theory.
problem Global description of double field theory geometry.
method Review of Courant algebroids, metric algebroids, AKSZ construction, para-Hermitian geometry.
result Global description of doubled geometry and topological models.
Triangular, overlapping Mel-scaled filters ("f-banks") are the current standard input for acoustic models that exploit their input's time-frequency geometry, because they provide a psycho-acoustically motivated time-frequency geometry for a speech signal. F-bank coefficients are provably robust to small deformations in…
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. …
A new learning method using hyperbolic geometry for class labels.
problem Class label representation and learning in machine learning.
method Hyperbolic Prototype Learning with a new loss function based on hyperbolic geometry.
result Hyperbolic Prototype Learning is equivalent to logistic regression in the one-dimensional case.
New method learns shape correspondences robustly from raw geometry.
problem Inaccurate and poor generalization of shape correspondences.
method Learning-based approach with feature-extraction network and functional map representation.
result Robust and accurate shape correspondence learning with less training data.
The physics of classical particles in a Lorentz-breaking spacetime has numerous features resembling the properties of Finsler geometry. In particular, the Lagrange function plays a role similar to that of a Finsler structure function. A summary is presented of recent results, including new calculable Finsler structures…
New geometric interpretation explains over-parameterized models and adversarial perturbations.
problem Geometric understanding of over-parameterized regression and adversarial perturbations.
method Alternative geometric interpretation of regression in feature space.
result Adversarial perturbations are a natural feature of biased models due to underlying geometry.
Transformer architectures show significant promise for natural language processing. Given that a single pretrained model can be fine-tuned to perform well on many different tasks, these networks appear to extract generally useful linguistic features. A natural question is how such networks represent this information in…
A new Lagrangian formulation of the Raychaudhuri equation in non-Riemannian geometry.
problem Formulating the Raychaudhuri equation in non-Riemannian geometries.
method Established a formal connection between the expansion scalar and the cross-sectional volume of the congruence. Derived a Lagrangian and Hamiltonian formulation.
result The expansion scalar equals the fractional rate of change of volume, weighted by a scalar factor.
In this paper, we evaluate the accuracy of deep learning approaches on geospatial vector geometry classification tasks. The purpose of this evaluation is to investigate the ability of deep learning models to learn from geometry coordinates directly. Previous machine learning research applied to geospatial polygon data …
QCML uses quantum geometry to represent data.
problem Data representation and the curse of dimensionality.
method QCML encodes data as Hermitian matrices in Hilbert space.
result Data geometry reveals intrinsic dimension and topological properties.
A new framework learns system design using neural features in function space.
problem Learning system design with neural feature extractors.
method Introduces feature geometry in function space, nesting technique for optimal feature approximation.
result Optimal features found from data samples using off-the-shelf architectures and optimizers.
Solves multi-class imbalanced data problem with geometry-based sampling and synthetic data.
problem Handling imbalanced multi-class data in classification problems.
method Two novel methods: undersampling and oversampling.
result Efficacy demonstrated through comparison with state-of-the-art methods.
Study characterizes cryospheric spectral feature space using joint PC+t-SNE approach.
problem Characterize cryospheric spectral feature space for remote sensing applications.
method Compare and contrast two approaches for identifying feature space basis vectors via dimensionality reduction (PCA and t-SNE).
result Joint characterization reveals distinct continua and clusters of ice reflectance properties.
Adversarial examples have attracted significant attention in machine learning, but the reasons for their existence and pervasiveness remain unclear. We demonstrate that adversarial examples can be directly attributed to the presence of non-robust features: features derived from patterns in the data distribution that ar…
We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidea…
This work studies adversarial training in high dimensions, revealing key feature trade-offs.
problem Understanding adversarial robustness in high-dimensional settings.
method Introduces a tractable model to study the geometry of data and adversarial attacks.
result Characterizes directions in data associated with robustness vs. usefulness trade-offs.
Study shows neural collapse is invariant to class imbalances under certain conditions.
problem Neural collapse properties are only valid for balanced data.
method Adopted UFM and introduced SELI for invariant characterization.
result Embeddings and classifiers always interpolate a simplex-encoded label matrix regardless of class imbalances.
In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are co…
New insights into how neural networks learn features, especially when they are very wide.
problem Understanding how gradient flow in wide neural networks selects solutions, especially in the feature-learning regime.
method Axiomatizing the canonical regularizer as a function-space energy and lift, and deriving geodesic ridge for the feature-learning regime.
result Gradient flow in feature-learning networks biases towards ridge regularization, distorting the inductive bias and damaging pretrained networks.