New distances for comparing multivariate normal distributions.
problem Comparing multivariate normal distributions efficiently and accurately.
method Approximated Fisher-Rao distance and pullback SPD cone distances.
result Efficient computation of distances between normal distributions.
Distance, normals, and double normals for real plane curves with singularities
problem Relation between normals and double normals and critical points of the squared distance function for real algebraic curves with singularities
method Investigate the topological discriminant of the distance function
result The topological discriminant consists of the evolute and distinguished normal lines at algebraic singular points
Constructs portfolios based on Hellinger distance to normal, finding market invariance.
problem Finding a market invariant for portfolio construction.
method Uses Hellinger distance to normal distribution for portfolio construction and analysis.
result Minimum Hellinger distance varies drastically between markets, suggesting market invariance.
New algorithms for efficient matrix profile computation using various Euclidean distances.
problem Efficiently computing matrix profile for all-pairs-similarity search on time series.
method Proposed AAMP, ACAMP, and extended algorithms for p-norm distance.
result AAMP and ACAMP algorithms outperform existing methods for specific Euclidean distances.
We improve density-based distances using normalizing flows and score matching.
problem Inaccurate density estimates and poor convergence in graph-based methods for high-dimensional spaces.
method Learn densities with normalizing flows and refine geodesics with a score model.
result Improved density-based distances that scale to high dimensions and improve numerical stability.
Paper introduces Normalized Wasserstein measure for better handling of imbalanced mixture distributions.
problem Wasserstein distance fails for mixture distributions with imbalanced proportions.
method Introduce mixture proportions as optimization variables to normalize Wasserstein formulation.
result Normalized Wasserstein measure leads to significant performance gains for mixture distributions.
Paper connects surface shape analysis and unbalanced optimal transport.
problem Computing the SRNF shape distance on piecewise linear surfaces.
method Characterizes SRNF shape distance as WFR distance pullback, proposes new algorithm for WFR distance computation.
result Direct computation of SRNF shape distance on piecewise linear surfaces.
Transforms distance-based outlier scores into interpretable probabilistic estimates.
problem Difficult interpretation of distance-based outlier scores.
method Generic transformation of scores into probabilistic estimates using distance probability distributions.
result Probabilistic transformation improves interpretability without impacting detection performance.
Revisits fuzzy neural networks with generalized Hamming distance, simplifying BN and ReLU.
problem Improving neural network techniques using fuzzy logic and generalized Hamming distance.
method Introducing generalized Hamming distance to reinterpret BN and ReLU, proposing GHN.
result Batch normalization and ReLU can be simplified or removed without loss of performance.
New model improves data augmentation for causal tasks.
problem Optimizing causal models robustly under Wasserstein distances.
method Proposes a new G-Causal Normalizing Flow architecture.
result Empirically outperforms standard generative models.
In this paper we study the main geometric properties of the Carnot-Carathéodory (abbreviated CC) distance $\dc$ in the setting of k-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formul…
For time series comparisons, it has often been observed that z-score normalized Euclidean distances far outperform the unnormalized variant. In this paper we show that a z-score normalized, squared Euclidean Distance is, in fact, equal to a distance based on Pearson Correlation. This has profound impact on many distanc…
Proves minimum number of normals to curves in 3D space.
problem Finding the minimum number of normals to closed curves in 3D.
method Morse theory for squared distance function and self intersections of the focal surface.
result For generic curves, points have at least 6, 8, or 10 normals depending on knotting.
Bayesian neural flows improve Gaia distance estimates and dust modeling.
problem Improving precision of distance estimates from Gaia DR2 data.
method Normalizing flow for learning flexible color-magnitude diagrams.
result Distance posteriors improved by more than 48% over raw Gaia data.
We present a new similarity measure based on information theoretic measures which is superior than Normalized Compression Distance for clustering problems and inherits the useful properties of conditional Kolmogorov complexity. We show that Normalized Compression Dictionary Size and Normalized Compression Dictionary En…
The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.
problem The expressivity of bi-Lipschitz Normalizing Flows in approximating specific target distributions.
method Characterization of expressivity through lower bounds on Total Variation distance and discussion of potential remedies.
result Several target distributions are difficult to approximate using bi-Lipschitz Normalizing Flows, and lower bounds on their approximation are provided.
Characterizes convexity of distance functions on Riemannian manifolds.
problem Understanding convexity of distance functions on Riemannian manifolds.
method Characterization of proximal normal cones, separation theorems, and analysis of convex subsets' boundaries.
result Convexity of distance functions for various boundary conditions on Riemannian manifolds.
Paper discusses sliced generative models for improved sample discrimination.
problem Improving sample discrimination in generative models.
method One-dimensional slicing of AutoEncoder-based generative models, focusing on normality tests and classical distances.
result The second group of methods based on classical distances gives a faster decrease rate of Fréchet Inception Distance (FID).
Paper derives normal approximations for singular subspaces with i.i.d. noise.
problem Normal approximation of singular subspaces under i.i.d. noise.
method Explicit representation formula, expected projection distance calculation, non-asymptotic normal approximation, bias corrections.
result Non-asymptotic normal approximation with optimal SNR condition and comprehensive simulation results.
The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.
problem Recovering the homology of submanifolds with narrow cycles.
method Embedding submanifolds into scaled oriented Grassmannian bundles to reduce normal curvature and stabilize Čech persistent homology.
result The Čech persistent homology is stable with respect to the interleaving distance and provides lower bounds on scales for homology recovery.
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respe…
The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.
problem Non-asymptotic bounds for accuracy of normal approximation in linear two-timescale stochastic approximation algorithms.
method Established bounds for normal approximation in terms of convex distance, focusing on last iterate and Polyak-Ruppert averaging.
result Normal approximation rate for the last iterate improves with increased timescale separation, while it decreases in the averaged setting.
In case of a standard form vN-algebra, the Bures distance is the natural distance between the fibres of implementing vectors at normal positive linear forms. Thereby, it is well-known that to each two normal positive linear forms implementing vectors exist such that the Bures distance is attained by the metric distance…
This work compares meta-embeddings for natural language processing tasks.
problem Improving distributed word representations in natural language processing.
method Meta-embeddings trained with angular and normalized distance loss functions compared to standard loss functions.
result Normalization methods outperform standard loss functions on word similarity datasets.
The study confirms conjectures about normals to convex polytopes in 3D space.
problem Concurrent normals problem for convex polytopes in 3D.
method Analyzes the PL concurrent normals problem for convex polytopes, proving conjectures for specific cases.
result Polytopes in 3D have points with 10 normals from interior points, confirmed for all tetrahedra and triangular prisms.
SNGP improves DNNs' uncertainty estimation with minimal changes.
problem Uncertainty estimation in deep learning models for real-time applications.
method Formalizing uncertainty as a minimax problem, SNGP adds weight normalization and replaces the output layer with a Gaussian process.
result SNGP outperforms other single-model approaches in uncertainty estimation across vision and language tasks.
We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any N≥2 we provide examples of N-dimensional normal currents whose associated vector fields are simple, and whose supports are purely 2-unrectifiable and have Nagata dimension N. We show that in l∞ norm…
Geometric analysis of normal distributions using Fisher and Killing metrics.
problem Quantifying the difference between Fisher and Killing metrics on the space of normal distributions.
method Riemannian geometry, Fisher information metric, Killing metric, asymptotic geodesics.
result Approximation of Fisher metric by Killing metric for long distances is justified.
Generative models create paintings that match training data.
problem Creating realistic paintings using machine learning.
method Used Spectral Normalization GAN (SN-GAN) and SN-GAN with Gradient Penalty to generate paintings.
result SN-GAN produced paintings most comparable to the training dataset.
This paper improves normalizing flows by combining MLE and sliced-Wasserstein distance for better data fidelity.
problem Normalizing flows struggle with generating realistic data and detecting out-of-distribution data.
method Proposes a hybrid objective function combining MLE and sliced-Wasserstein distance.
result Shows better generative abilities and lower likelihood of out-of-distribution data.
In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to the contractivity of normalized L-Wasserstein distance under backward Ricci flow.
This paper proposes a new method for feature scaling in K-Nearest Neighbors.
problem Feature scaling issues in K-Nearest Neighbors algorithm.
method Assign weights to individual features using out-of-bag errors from decision tree models.
result Improves prediction accuracy by assigning weights to features based on out-of-bag errors.
New statistical Minkowski distances for Gaussian mixtures with closed-form formulas.
problem Computing distances for Gaussian mixture models efficiently.
method Proposed novel statistical distances based on Minkowski's inequality for Gaussian mixtures.
result Closed-form formula for Gaussian mixture models with integer exponents.
For a pinched Hadamard manifold X and a discrete group of isometries Γ of X, the critical exponent δΓ is the exponential growth rate of the orbit of a point in X under the action of Γ. We show that the critical exponent for any family N of normal subgroups of Γ0 has the same coarse behaviour…
SNGP improves single-model deep uncertainty by enhancing distance-awareness.
problem Improving uncertainty estimation in deep learning models, especially for real-time applications.
method SNGP improves distance-awareness of DNNs through spectral normalization and Gaussian process layers.
result SNGP outperforms other single-model approaches in prediction, calibration, and out-of-domain detection.
New method calculates Ricci curvature from distances between weighted volumes.
problem Calculating Ricci curvature for weighted Riemannian manifolds.
method Asymptotic retrieval of generalized Ricci tensor from scaled metric derivatives of Wasserstein 1-distances.
result Limiting coarse curvature of random graphs converges to generalized Ricci tensor.
MN-PCA models structured noise in data and feature spaces.
problem Complex and structured noise in real-world data.
method Matrix normal distribution for structured noise modeling, generalized Mahalanobis distance approximation.
result MN-PCA obtains a low-rank data representation and structured noise simultaneously.
We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff the Hausdorff distance between A and gA is finite. We show that a quasi-normal su…
GANs struggle with density estimation on simple datasets, while normalizing flows perform well.
problem Comparing GANs and normalizing flows for density estimation on non-image data.
method An extensive grid search over GAN architectures, hyperparameters, and training procedures; comparison on low-dimensional synthetic datasets.
result Normalizing flows outperform GANs in density estimation, especially in terms of Wasserstein distance.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
A formula that relates triple points, branch points, and their distances from infinity is presented. We recover trivial normal Euler classes for oriented surfaces, and formulas on signed triple points.
For a Riemannian manifold Mn+1 and a compact domain Ω⊂Mn+1 bounded by a hypersurface ∂Ω with normal curvature bounded below, estimates are obtained in terms of the distance from O to ∂Ω for the angle between the geodesic line joining a fixed interior point O in Ω to a point on…
Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…
Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
A new method uses normalizing flows for gradual domain adaptation.
problem Difficulty in domain adaptation when source and target domains have a large gap.
method Proposes using normalizing flows to learn a transformation from target to Gaussian mixture distribution.
result Improves classification performance and mitigates the problem of gradual self-training failure.
Optimal transport theory applied to quantum states on Grassmannians.
problem Developing optimal transport for quantum states.
method Metric geometry of Grassmannians and spectral theorem for density matrices.
result Wasserstein distance for normal states of von Neumann algebras.