This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
Paper proposes dp-VAE for preserving spatial context in gene expression data.
problem Inaccessibility of spatial context in single-cell gene expression data.
method Generic representation learning and transfer learning framework with a distance-preserving regularizer.
result dp-VAE effectively reconstructs and imputes spatial context from gene expression data.
In this work we study the properties of deep neural networks (DNN) with random weights. We formally prove that these networks perform a distance-preserving embedding of the data. Based on this we then draw conclusions on the size of the training data and the networks' structure. A longer version of this paper with more…
These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (\url{http://www.math.psu.edu/mass/}). The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphas…
ResNets can approximate input distances under certain conditions, but existing theory is flawed.
problem Theoretical justification for regularizing ResNets to preserve input distances is flawed.
method Frequency analysis perspective to explain effectiveness of regularization schemes.
result Regularization schemes enforce a lower Lipschitz bound on low-frequency projections of images.
We show that the group of isometries (i.e., distance-preserving homeomorphisms) of an equiregular subRiemannian manifold is a finite-dimensional Lie group of smooth transformations. The proof is based on a new PDE argument, in the spirit of harmonic coordinates, establishing that in an arbitrary subRiemannian manifold …
Two new algorithms select matrix rows and columns to preserve distances.
problem Preserving distances in large matrix visualizations.
method Selects rows and columns to preserve distances.
result Preserves distances as closely as possible.
Smooth maps preserve distances on specific revolution surfaces.
problem Existence of smooth maps on revolution surfaces.
method Proving existence of maps preserving distances on meridians and parallels.
result Smooth maps exist from revolution surfaces to Euclidean plane.
New method preserves distances in time series data.
problem Preserving distances in time series data under interpolation.
method Developed lines-preserving terminal embeddings.
result First dimension-free coresets for Fréchet distance clustering.
Maps preserving Carathéodory distance between symmetric domains are rigid.
problem Rigidity of maps preserving Carathéodory distance between bounded symmetric domains.
method Large-scale geometry of Carathéodory distance, horocompactification, Gromov product.
result Maps preserving Carathéodory distance are rigid and either holomorphic or antiholomorphic.
We present Graph Random Neural Features (GRNF), a novel embedding method from graph-structured data to real vectors based on a family of graph neural networks. The embedding naturally deals with graph isomorphism and preserves the metric structure of the graph domain, in probability. In addition to being an explicit em…
As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study th…
Bayesian layer improves image segmentation and out-of-distribution detection.
problem Outlier detection in image segmentation.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates improve out-of-distribution detection.
Random projections help in representing sparse graphs efficiently.
problem Efficiently representing sparse graphs of varying sizes and vertex sets.
method Random projection of adjacency matrices to retain graph functionality and properties.
result Random projections can accurately represent graphs of different sizes and vertex sets in the same space.
TTRP method preserves distances in high-dimensional data with reduced storage and speed.
problem Preserving distances in high-dimensional datasets efficiently and accurately.
method Tensor train random projection (TTRP) using TT-ranks of one.
result TTRP is an expected isometric projection with bounded variance.
We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equ…
Landmark-based node embeddings approximate shortest path distances in random graphs.
problem Capturing global graph distances in node representations.
method Landmark-based node embeddings using shortest path distances from a subset of reference nodes (landmarks).
result Random graphs require lower dimensions in landmark-based embeddings compared to worst-case graphs.
Paper develops heavy-tailed embeddings for better text classification and augmentation.
problem Improving text classification, especially for extreme values.
method Develops heavy-tailed embeddings using multivariate extreme value theory and introduces a scale-invariant classifier.
result The classifier outperforms baselines and generates meaningful augmented text.
Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measur…
Proposes Isometric Graph Neural Networks to preserve graph distances.
problem Lack of faithful distance representation in graph neural networks.
method Introduces a new technique to modify GNNs' input space and loss function.
result Significant improvement in reflecting graph distances, as measured by KT.
Unified framework for multi-view learning with orthogonal projections.
problem Learning individual orthogonal projections for multiple views.
method Successive approximations via eigenvectors, iterative Krylov subspace method.
result Consistently competitive and often better than existing methods.
Three important properties of a classification machinery are: (i) the system preserves the core information of the input data; (ii) the training examples convey information about unseen data; and (iii) the system is able to treat differently points from different classes. In this work we show that these fundamental pro…
This work improves scalability of Wasserstein distances in high dimensions.
problem Scalability issues in computing Wasserstein distances in high dimensions.
method Empirical convergence rates, robustness to data contamination, and computational methods.
result Established fast rates and robust estimation risks for sliced Wasserstein distances.
This research converts visual information into audio for users to perceive.
problem Brevity in conveying visual information through spoken language.
method Pretrained image embedding network, GAN for metric space mapping, human subject testing.
result Users can accurately classify audio sonifications of faces.
Bayesian Gaussian Processes layer detects out-of-distribution data in medical imaging.
problem Detecting out-of-distribution data in medical imaging tasks.
method Parameter-efficient hierarchical convolutional Gaussian Processes in Wasserstein-2 space.
result Uncertainty estimates enable superior out-of-distribution detection compared to previous methods.
Optimization can learn Johnson-Lindenstrauss embeddings without randomization.
problem Achieving compact data representations with theoretical guarantees.
method A novel optimization-based approach over the space of random solution samplers.
result The method avoids bad stationary points and converges to a deterministic solution.
This study improves graph coarsening methods by preserving graph spectrum and distances.
problem Solving large-scale graph problems by working on a smaller graph.
method Developed a geometric approach using Gromov--Wasserstein distance to minimize the difference between graph distances and their coarsened versions.
result Minimizing the difference between graph distances and their coarsened versions can be achieved using the weighted kernel K-means method. Isometry regularizer improves autoencoder performance on manifold learning.
problem Bad generalization in autoencoders, especially extrinsic and intrinsic issues.
method Introduces an isometry regularizer that encourages the decoder to be an isometry and the encoder to be its pseudo-inverse.
result Isometry regularizer leads to better generalization and useful low-dimensional data representations.
This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset X⊂Rn, associated with the Euclidean metric, with points in the cube {±1}m and we associa…
A fast binary embedding method preserves Euclidean distances in high-dimensional data.
problem Preserving Euclidean distances in high-dimensional datasets.
method Stable noise-shaping quantization of Ax with A a sparse Gaussian random matrix, followed by a linear transformation. result Euclidean distances are approximated by the ℓ1 norm on binary sequences, leading to accurate binary codes. The objective in extreme multi-label learning is to train a classifier that can automatically tag a novel data point with the most relevant subset of labels from an extremely large label set. Embedding based approaches make training and prediction tractable by assuming that the training label matrix is low-rank and hen…
The paper characterizes global hyperbolicity in Lorentzian manifolds without relying on manifold topology.
problem Characterizing global hyperbolicity in smooth Lorentzian manifolds without assuming manifold topology.
method Two formulations of global hyperbolicity: one using chronological diamonds and the other using properties of the Lorentzian distance function.
result The second formulation is equivalent to the definition of `Lorentzian metric space' and introduces the concept of d-reflectivity. Graph-Laplacians and their spectral embeddings play an important role in multiple areas of machine learning. This paper is focused on graph-Laplacian dimension reduction for the spectral clustering of data as a primary application. Spectral embedding provides a low-dimensional parametrization of the data manifold which…
Unified analysis of multilabel Fisher discriminants with improved dimensionality and robustness.
problem Improving discriminant analysis for multilabel classification with enhanced dimensionality and robustness.
method Unified theoretical analysis of multilabel Fisher discriminants with algebraic and statistical guarantees.
result Unified characterization of multilabel Fisher objectives and their equivalence under orthogonality constraints.
Unified analysis of multilabel Fisher discriminants with improved dimensionality and robustness.
problem Improving discriminant analysis for multilabel classification with enhanced dimensionality and robustness.
method Unified algebraic and statistical analysis of multilabel Fisher discriminants with Stiefel orthogonality constraints.
result Equivalence of four Fisher objectives under the Stiefel constraint and improved discriminant dimensionality.