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 problem”
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.
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.
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…
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.
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 …
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.
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…
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.
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.
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.
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.
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…
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.
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.
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…
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…
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.
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…
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…
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 study distributed computing of the truncated singular value decomposition problem. We develop an algorithm that we call \texttt{LocalPower} for improving communication efficiency. Specifically, we uniformly partition the dataset among nodes and alternate between multiple (precisely ) local power iterations an…
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…
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.
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…
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.
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.
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.
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.
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.
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.
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.
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.
DFRot: Achieving Outlier-Free and Massive Activation-Free for Rotated LLMs with Refined Rotationcs.LG
DFRot improves LLMs by reducing outlier and massive activation effects.
problem Reducing outlier and massive activation effects in rotated LLMs.
method Weighted loss function and orthogonal Procrustes transforms for rotation matrix refinement.
result DFRot achieves dual free (Outlier-Free and Massive Activation-Free) with significant improvements in perplexity.
Novel tRSA combines geometry and topology for brain and model analysis.
problem Traditional RSA overlooks topological information in neural representations.
method Topological RSA (tRSA) using nonlinear monotonic transforms.
result Robust model comparisons and novel insights into neural computation.
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.
Study finds no significant alignment between whitepaper claims and market structure.
problem Correlation between cryptocurrency whitepaper narratives and market behavior.
method Developed a contamination-aware pipeline for measuring structural correspondence, combining NLP classification and market statistics.
result No significant claims-market alignment detected in the sample.
We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold gr…
Optimal transport reformulates multiple quantile hedging problem.
problem Multiple quantile hedging problem in incomplete markets.
method Reformulated as Monge optimal transport problem, introduced Kantorovitch version, proved no duality gap.
result Multiple quantile hedging problem can be seen as semi-discrete optimal transport problem.
Solves four problems related to circle families in the plane.
problem Four basic problems of circle families in the plane.
method Solves all four basic problems of circle families in the plane.
result All four basic problems are solved.
Solves four problems related to sphere families in 3D space.
problem Four basic problems of sphere families in Euclidean 3-space.
method Solves all four basic problems of sphere families in Euclidean 3-space.
result All four basic problems are solved.
Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometrymath.OC
The paper solves optimal control problems for various convex sets using convex trigonometry.
problem Optimal control problems with 2D convex compact sets.
method Using convex trigonometry to derive extremals for various problems.
result Geodesics in multiple sub-Finsler problems are derived.