The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be homologous (i.e. corresponding to each other), as far as possible, and the Procrust…
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48 results for “Procrustes distance”
A new method for computing shape barycenters from point clouds using Procrustes-Wasserstein distance.
problem Computing representative shapes from point clouds with precise alignment and shape preservation.
method Developed a new distance metric (Procrustes-Wasserstein) and algorithms for computing barycenters.
result Superior performance in precise alignment and shape preservation compared to existing OT approaches.
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
New measures link neural representation geometry to decoding ability.
problem Understanding how neural representations relate to decoding ability.
method Showed that popular similarity measures can be interpreted from a decoding perspective.
result Proved that measures like CKA and CCA quantify alignment between optimal linear readouts.
Modified Wasserstein metric for Gaussian distributions, invariant to isometries.
problem Distance measurement for latent Gaussian distributions invariant to isometries.
method Modified Benamou-Brenier approach leading to a Procrustes Wasserstein metric.
result For Gaussian distributions, the metric reduces to Euclidean distance between eigenvalues.
The paper introduces metrics for robust unsupervised learning of vehicle interactions.
problem Robust representation learning of temporal dynamic interactions in robotics.
method Geometric approach using Procrustes distance and optimal transport for comparing interaction distributions.
result Metrics for assessing stability and comparing interaction learning algorithms.
LDLE embeds manifolds in lower dimensions with low distortion.
problem Embedding manifolds in lower dimensions with low distortion.
method Constructs local views using global eigenvectors of the graph Laplacian, registers them using Procrustes analysis, and tears manifolds apart for intrinsic dimension embedding.
result LDLE preserves distances up to a constant scale with low distortion.
We solve a key problem in cross-lingual learning using a novel approach.
problem Aligning word embeddings across different languages.
method We devise a direct solution to the Wasserstein-Procrustes problem.
result Our method improves existing UCL approaches significantly.
Method preserves correlations in synthetic data.
problem Preserving dependence structure of original data.
method Orthogonal Procrustes problem for restoring Pearson correlation.
result Restores Pearson correlation structure while preserving feature distributions and downstream tasks performance.
Python package for SPD matrix distances, reproducible and extensible.
problem Computing distances between SPD matrices for various applications.
method Unified, extensible framework supporting multiple SPD metrics.
result Reproducible and accessible SPD matrix comparison tool.
Local Procrustes for Manifold Embedding: A Measure of Embedding Quality and Embedding Algorithmsstat.ML
We present the Procrustes measure, a novel measure based on Procrustes rotation that enables quantitative comparison of the output of manifold-based embedding algorithms (such as LLE (Roweis and Saul, 2000) and Isomap (Tenenbaum et al, 2000)). The measure also serves as a natural tool when choosing dimension-reduction …
In this letter, we consider two sets of observations defined as subspace signals embedded in noise and we wish to analyze the distance between these two subspaces. The latter entails evaluating the angles between the subspaces, an issue reminiscent of the well-known Procrustes problem. A Bayesian approach is investigat…
Multivariate Analysis (MVA) comprises a family of well-known methods for feature extraction that exploit correlations among input variables of the data representation. One important property that is enjoyed by most such methods is uncorrelation among the extracted features. Recently, regularized versions of MVA methods…
Paper proposes a KGE framework that reduces training time and carbon footprint.
problem Efficient KGE learning with reduced computational cost and environmental impact.
method Full batch learning, Orthogonal Procrustes Analysis, non-negative-sampling training.
result Significant reduction in training time and carbon footprint compared to state-of-the-art approaches.
Representational similarity analysis (RSA) has been shown to be an effective framework to characterize brain-activity profiles and deep neural network activations as representational geometry by computing the pairwise distances of the response patterns as a representational dissimilarity matrix (RDM). However, how to p…
One of the common tasks in unsupervised learning is dimensionality reduction, where the goal is to find meaningful low-dimensional structures hidden in high-dimensional data. Sometimes referred to as manifold learning, this problem is closely related to the problem of localization, which aims at embedding a weighted gr…
Study matches two noisy point clouds with geometric transformations and relabeling.
problem Matching two noisy point clouds with orthogonal transformations and relabeling.
method Information-theoretic results and Ping-Pong algorithm for computational alignment.
result The Ping-Pong algorithm retrieves the planted signal after one step.
Solves a fundamental problem in statistics and imaging with new methods.
problem Generalized Orthogonal Procrustes Problem (GOPP)
method Semidefinite relaxation (SDR) and generalized power method (GPM)
result GPM converges linearly to the global minimizer under large signal-to-noise ratio.
Algorithm recovers permutations of high-dimensional Gaussian vectors with constant correlation.
problem Recovering permutations of high-dimensional Gaussian vectors with constant correlation.
method Computing and comparing weighted counts of specially chosen wide trees.
result Polynomial-time algorithm for exact recovery at constant correlation.
A new algorithm reduces communication in distributed SVD by factors.
problem Efficiently compute SVD in distributed systems.
method LocalPower algorithm with weighted aggregation and periodic decay of iterations.
result Reduces communication cost by a factor of .
A new method for aligning datasets without known correspondences.
problem Aligning datasets from different domains without labeled correspondences.
method Integrates MDS and Wasserstein Procrustes for joint optimization of embeddings and correspondences.
result Maps datasets to a common low-dimensional space without labeled correspondences.
Mapping and translating professional but arcane clinical jargons to consumer language is essential to improve the patient-clinician communication. Researchers have used the existing biomedical ontologies and consumer health vocabulary dictionary to translate between the languages. However, such approaches are limited b…
Deep model predicts shapes of curves with multiple covariates.
problem Predicting shapes of planar curves with various covariates.
method Deep learning model using complex-valued functions, conditional covariance smoother with modality-specific encoders.
result Model accurately predicts shapes of curves with multimodal covariates.
The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty te…
We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth. Several results establish tightness of SDP based relaxations…
Gradient descent solves asymmetric low-rank matrix sensing without balancing.
problem Recovering asymmetric low-rank matrices from linear measurements.
method Gradient descent with spectral initialization, avoiding balancing term.
result Gradient descent converges linearly without balancing, factors stay balanced.
The Geometric Canary: Predicting Steerability and Detecting Drift via Representational Stabilitycs.LG
Geometric stability predicts steerability and detects drift in language models.
problem Predicting steerability and detecting drift in language models.
method Supervised and unsupervised geometric stability measures.
result Supervised geometric stability predicts steerability with high accuracy and detects drift earlier.
Proposes BONMI for integrating noisy matrices from multi-source data.
problem Integrating noisy matrices from multi-source data with block-wise missingness.
method Exploits orthogonal Procrustes problem to align eigenspaces and completes missing blocks.
result Statistical rate for eigenspace of underlying matrix comparable to independently missing assumption.
Change detection in dynamic networks is an important problem in many areas, such as fraud detection, cyber intrusion detection and health care monitoring. It is a challenging problem because it involves a time sequence of graphs, each of which is usually very large and sparse with heterogeneous vertex degrees, resultin…
Survey of Locally Linear Embedding and its variants.
problem Representing high-dimensional data in a lower-dimensional space while preserving local structure.
method Explains various LLE and variant methods, including kernel LLE, inverse LLE, feature fusion, out-of-sample embedding, incremental LLE, landmark LLE, supervised LLE, robust LLE, fusion with other methods, and weighted LLE.
result Comprehensive overview of LLE and its variants.
Robustly computes intrinsic coordinates on point clouds using resampling and averaging.
problem Computing intrinsic coordinates on noisy or outlier-prone point clouds.
method Subsample data, vary hyperparameters, cluster candidate embeddings, identify representative embeddings, and average them using Procrustes analysis.
result Robust to noise and outliers, validated on synthetic and real data.
Geometric stability measures neural network robustness, distinguishing from similarity metrics.
problem Lack of robustness in neural network representations.
method Introduces geometric stability, quantified by Shesha metric measuring self-consistency.
result Stability and similarity are uncorrelated, revealing distinct properties of neural network robustness.
We revisit the inductive matrix completion problem that aims to recover a rank- matrix with ambient dimension given features as the side prior information. The goal is to make use of the known features to reduce sample and computational complexities. We present and analyze a new gradient-based non-convex…
Proposes a method to align language and image data.
problem Aligning language and image data for better understanding.
method Uses triplet loss to learn consistent embeddings of language and images.
result Outperforms four baselines across multiple metrics.
Study finds whitepaper narratives do not predict market factor structure.
problem Predicting market behavior from cryptocurrency whitepaper claims.
method Zero-shot NLP classification combined with CP tensor decomposition of market data.
result Weak alignment between whitepaper claims and market statistics and latent factors.
We consider the task of aligning two sets of points in high dimension, which has many applications in natural language processing and computer vision. As an example, it was recently shown that it is possible to infer a bilingual lexicon, without supervised data, by aligning word embeddings trained on monolingual data. …
We propose a data aggregation-based algorithm with monotonic convergence to a global optimum for a generalized version of the L1-norm error fitting model with an assumption of the fitting function. The proposed algorithm generalizes the recent algorithm in the literature, aggregate and iterative disaggregate (AID), whi…
FedPower improves eigenspace estimation privacy in federated learning.
problem Privacy breaches and communication challenges in federated eigenspace estimation.
method FedPower uses a power method with local power iterations and global aggregation, weighted by OPT, and adds Gaussian noise for privacy.
result FedPower provides convergence bounds and demonstrates effectiveness in experiments.
New metrics defined on SPD matrices link to divergences and curvature.
problem Defining and characterizing metrics on SPD matrices.
method Developed a principle of deformed metrics and introduced balanced bilinear forms.
result Introduce Mixed-Euclidean metrics with negative sectional curvature.
New algorithm for robust circular coordinates in recurrent time series data.
problem Inefficient and sensitive methods for finding circular coordinates on recurrent data.
method Subsampling, aligning, and averaging to correct uneven sampling density.
result More robust and efficient circular coordinates for neuronal recordings.
New method preserves privacy while detecting communities in distributed networks.
problem Privacy-preserving community detection in locally distributed multi-layer networks.
method Privacy-preserving Distributed Spectral Clustering (ppDSC) using randomized response mechanism.
result Developed a novel algorithm that maintains community structure while protecting privacy.
The paper tightens bounds on distances between Reeb graphs.
problem Certifying quasi-universality of distances between Reeb graphs.
method Establishes tight bi-Lipschitz bounds for various distances.
result Proves strict universality of the functional contortion distance for contour trees and coincides with interleaving distance for merge trees.
Extends Teichmüller distance concept to non-distance maps.
problem Defining distance metrics for non-distance functions.
method Generalizes horofunction compactification to non-distance maps.
result Defines horofunction counterpart to Teichmüller distance.
The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in vario…
We define a novel class of distances between statistical multivariate distributions by modeling an optimal transport problem on their marginals with respect to a ground distance defined on their conditionals. These new distances are metrics whenever the ground distance between the marginals is a metric, generalize both…
In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called -distance spaces). As a corollary, a complete solution to generalized Borsuk p…
New toolkit for directed distances improves flexibility of OT problems.
problem Optimal transport problems with constraints.
method Directed distances between quantile functions.
result Flexibility in solving OT problems enhanced.
Developments in the theory of randomized shortest paths with a comparison of graph node distancesstat.ML
There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…